Bézier curve - Knowledge

7136: 6349: 8439: 7131:{\displaystyle {\begin{aligned}(1-t)^{2}\mathbf {P} _{0}+2(1-t)t\mathbf {P} _{1}+t^{2}\mathbf {P} _{2}&=(1-t)^{3}\mathbf {P} _{0}+2(1-t)^{2}t\mathbf {P} _{1}+(1-t)t^{2}\mathbf {P} _{2}+(1-t)^{2}t\mathbf {P} _{0}+2(1-t)t^{2}\mathbf {P} _{1}+t^{3}\mathbf {P} _{2}\\&=(1-t)^{3}\mathbf {P} _{0}+(1-t)^{2}t\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)+(1-t)t^{2}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)+t^{3}\mathbf {P} _{2}\\&=(1-t)^{3}\mathbf {P} _{0}+3(1-t)^{2}t{\tfrac {1}{3}}\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)+3(1-t)t^{2}{\tfrac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)+t^{3}\mathbf {P} _{2}\end{aligned}}} 7798: 9623: 9634:) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ 5445: 9006: 10869:

TrueType fonts. CFF stands for the Type1 font format. Strictly speaking, it refers to the Compact Font Format, which is used in the compression processes for the Type2 fonts. (...) the cubic Bézier format of the Type1 fonts is more space-saving compared to the quadratic format of the TrueType fonts. Some kilobytes can be saved in large, elaborate fonts which may represent an advantage on the Web. On the other hand, the more detailed hinting information of the TrueType fonts is useful for very extensive optimization for screen use.

553: 4656: 8434:{\displaystyle {\begin{aligned}\mathbf {B} (t)&=(1-t)\sum _{i=0}^{n}\mathbf {b} _{i,n}(t)\mathbf {P} _{i}+t\sum _{i=0}^{n}\mathbf {b} _{i,n}(t)\mathbf {P} _{i}\\&=\sum _{i=0}^{n}{\frac {n+1-i}{n+1}}\mathbf {b} _{i,n+1}(t)\mathbf {P} _{i}+\sum _{i=0}^{n}{\frac {i+1}{n+1}}\mathbf {b} _{i+1,n+1}(t)\mathbf {P} _{i}\\&=\sum _{i=0}^{n+1}\left({\frac {i}{n+1}}\mathbf {P} _{i-1}+{\frac {n+1-i}{n+1}}\mathbf {P} _{i}\right)\mathbf {b} _{i,n+1}(t)\\&=\sum _{i=0}^{n+1}\mathbf {b} _{i,n+1}(t)\mathbf {P'} _{i}\end{aligned}}} 7787: 5874: 6055: 5953: 3322: 6094: 9814:. Typically font engines and vector graphics engines render Bézier curves by splitting them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. The "smooth curve" feature of charts in 7332: 28: 20: 3692: 2919: 2942: 7782:{\displaystyle {\begin{cases}{n+1 \choose i}(1-t)\mathbf {b} _{i,n}={n \choose i}\mathbf {b} _{i,n+1}\\{n+1 \choose i+1}t\mathbf {b} _{i,n}={n \choose i}\mathbf {b} _{i+1,n+1}\end{cases}}\quad \implies \quad {\begin{cases}(1-t)\mathbf {b} _{i,n}={\frac {n+1-i}{n+1}}\mathbf {b} _{i,n+1}\\t\mathbf {b} _{i,n}={\frac {i+1}{n+1}}\mathbf {b} _{i+1,n+1}\end{cases}}} 5883: 6064: 5735: 5962: 4570: 9717:. Within each segment, either horizontal or vertical movement dominates, and the total number of steps in either direction can be read off from the endpoint coordinates; in for example the 0–45° sector horizontal movement to the right dominates, so it only remains to decide between which steps to the right the curve should make a step up. 3385: 1846: 2689: 2302: 3317:{\displaystyle {\begin{aligned}\mathbf {B} (t)&=\sum _{i=0}^{n}{n \choose i}(1-t)^{n-i}t^{i}\mathbf {P} _{i}\\&=(1-t)^{n}\mathbf {P} _{0}+{n \choose 1}(1-t)^{n-1}t\mathbf {P} _{1}+\cdots +{n \choose n-1}(1-t)t^{n-1}\mathbf {P} _{n-1}+t^{n}\mathbf {P} _{n},&&0\leqslant t\leqslant 1\end{aligned}}} 2484: 10868:

The OpenType format was formulated in 1996. By 2003, it began to replace two competing formats: the Type1 fonts, developed by Adobe and based on ostcript, and the TrueType fonts, specified by Microsoft and Apple. (...) TTF stands for TrueTypeFont and indicates that the font data is the same as in the

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and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is

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by Zingl that performs this rasterization by subdividing the curve into rational pieces and calculating the error at each pixel location such that it either travels at a 45° angle or straight depending on compounding error as it iterates through the curve. This reduces the next step calculation to a

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cannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with

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design and smoothing cursor trajectory in eye gaze controlled interfaces. For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or

8666: 3687:{\displaystyle {\begin{aligned}\mathbf {B} (t)&=(1-t)^{5}\mathbf {P} _{0}+5t(1-t)^{4}\mathbf {P} _{1}+10t^{2}(1-t)^{3}\mathbf {P} _{2}+10t^{3}(1-t)^{2}\mathbf {P} _{3}+5t^{4}(1-t)\mathbf {P} _{4}+t^{5}\mathbf {P} _{5},&&0\leqslant t\leqslant 1.\end{aligned}}} 8921: 4283: 5247: 9214: 976: 1655: 9842:. Furthermore, joint space trajectories can be accurately differentiated using Bézier curves. Consequently, the derivatives of joint space trajectories are used in the calculation of the dynamics and control effort (torque profiles) of the robotic manipulator. 2914:{\displaystyle \mathbf {B} (t)=\mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{n}}(t)=(1-t)\mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{n-1}}(t)+t\mathbf {B} _{\mathbf {P} _{1}\mathbf {P} _{2}\ldots \mathbf {P} _{n}}(t)} 1309: 302:= 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve. The sums in the following sections are to be understood as 9515:} generate a full circle with radius one. For curves with points and weights on a circle, the weights can be scaled without changing the curve's shape. Scaling the central weight of the above curve by 1.35508 gives a more uniform parameterization. 5634: 9830:, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also preserves the curve definition under all linear or perspective 2D and 3D transforms and projections. 7317: 7228: 1475: 5427:

Bézier curves—meaning that any change to a control point requires recalculation of and thus affects the aspect of the entire curve, "although the further that one is from the control point that was changed, the smaller is the change in the

3822: 2100: 190:, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limits of 6190:. This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation. 9225: 2588: 2078:

Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order. Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to

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E is either point on the curve with a tangent at 45° to CD (dashed green). If G is the intersection of this tangent and the axis, the line passing through G and perpendicular to CD is the directrix (solid

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The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of

4565:{\displaystyle \mathbf {C} _{j}={\frac {n!}{(n-j)!}}\sum _{i=0}^{j}{\frac {(-1)^{i+j}\mathbf {P} _{i}}{i!(j-i)!}}=\prod _{m=0}^{j-1}(n-m)\sum _{i=0}^{j}{\frac {(-1)^{i+j}\mathbf {P} _{i}}{i!(j-i)!}}.} 8520: 787: 8764: 6333: 4118: 5360: 7803: 6354: 5407: 3390: 2947: 9745:, Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path. 5310: 4876: 9748:

In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation. Bézier curves are now very frequently used to control the animation easing in

1841:{\displaystyle \mathbf {B} (t)=(1-t)\mathbf {B} _{\mathbf {P} _{0},\mathbf {P} _{1},\mathbf {P} _{2}}(t)+t\mathbf {B} _{\mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}}(t),\ 0\leq t\leq 1.} 538: 9050: 8997:

Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree. A number of approximation algorithms have been proposed and used in practice.

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The focus (F) is at the intersection of the axis and a line passing through E and perpendicular to CD (dotted yellow). The latus rectum is the line segment within the curve (solid yellow).

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moves" that are purely vertical or purely horizontal, along the pixel boundaries. To that end, the plane is first split into eight 45° sectors (by the coordinate axes and the two lines

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designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.

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arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into

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Abstract composition of cubic Bézier curves ray-traced in 3D. Ray intersection with swept volumes along curves is calculated with Phantom Ray-Hair Intersector algorithm.

6158:(the Bézier curve or in general a path of several Bézier segments). The conversion from offset curves to filled Bézier contours is of practical importance in converting 5460:, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows: 5416:. What this means in intuitive terms is that a Bézier curve does not "undulate" more than the polygon of its control points, and may actually "undulate" less than that. 9687: 7233: 7144: 2297:{\displaystyle \mathbf {B} '(t)=3(1-t)^{2}(\mathbf {P} _{1}-\mathbf {P} _{0})+6(1-t)t(\mathbf {P} _{2}-\mathbf {P} _{1})+3t^{2}(\mathbf {P} _{3}-\mathbf {P} _{2})\,.} 4980: 4913: 4664:

2. placing its 2 middle control points (yellow circles) 2/3 along line segments from the end points to the quadratic curve's middle control point (black rectangle).

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Alexander Reshetov and David Luebke, Phantom Ray-Hair Intersector. In Proceedings of the ACM on Computer Graphics and Interactive Techniques (August 1, 2018).

9454:{\displaystyle \mathbf {B} (t)={\frac {\sum _{i=0}^{n}{n \choose i}t^{i}(1-t)^{n-i}\mathbf {P} _{i}w_{i}}{\sum _{i=0}^{n}{n \choose i}t^{i}(1-t)^{n-i}w_{i}}}.} 1384: 2528: 11089: 6170:, which only require (for efficiency purposes) the mathematically simpler operation of filling a contour defined by (non-self-intersecting) Bézier curves. 245:. De Casteljau's method was patented in France but not published until the 1980s while the Bézier polynomials were widely publicised in the 1960s by the 9838:

Because the control polygon allows to tell whether or not the path collides with any obstacles, Bézier curves are used in producing trajectories of the

2479:{\displaystyle \mathbf {B} ''(t)=6(1-t)(\mathbf {P} _{2}-2\mathbf {P} _{1}+\mathbf {P} _{0})+6t(\mathbf {P} _{3}-2\mathbf {P} _{2}+\mathbf {P} _{1})\,.} 2047:{\displaystyle \mathbf {B} (t)=(1-t)^{3}\mathbf {P} _{0}+3(1-t)^{2}t\mathbf {P} _{1}+3(1-t)t^{2}\mathbf {P} _{2}+t^{3}\mathbf {P} _{3},\ 0\leq t\leq 1.} 6131:), cannot be exactly formed by a Bézier curve (except in some trivial cases). In general, the two-sided offset curve of a cubic Bézier is a 10th-order 2619: 1155:{\displaystyle \mathbf {B} (t)=\mathbf {P} _{1}+(1-t)^{2}(\mathbf {P} _{0}-\mathbf {P} _{1})+t^{2}(\mathbf {P} _{2}-\mathbf {P} _{1}),\ 0\leq t\leq 1.} 9886: 9619:

suffices to force the control point at which two constituent Bézier curves meet to lie on the line defined by the two control points on either side.

11257: 10847: 10401: 3833: 5420: 10213: 11033: 11061: 482:{\displaystyle \mathbf {B} (t)=\mathbf {P} _{0}+t(\mathbf {P} _{1}-\mathbf {P} _{0})=(1-t)\mathbf {P} _{0}+t\mathbf {P} _{1},\ 0\leq t\leq 1} 6196: 4198: 5471:

to CD (dashed cyan line) defines its vertex (V). Its axis of symmetry (dash-dot cyan) passes through V and is perpendicular to the tangent.

10302: 9614: 4771: 8936: 10830: 5064: 4768:), with a maximum radial error of less than one part in a thousand, when each inner control point (or offline point) is the distance 11118: – an open source online book explaining Bézier curves and associated graphics algorithms, with interactive graphics 5910:

For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points

11232: 6093: 5008: 8661:{\displaystyle \mathbf {P'} _{i}={\frac {i}{n+1}}\mathbf {P} _{i-1}+{\frac {n+1-i}{n+1}}\mathbf {P} _{i},\quad i=0,\ldots ,n+1.} 8916:{\displaystyle \mathbf {P} _{i,r}=\sum _{j=0}^{n}\mathbf {P} _{j}{\tbinom {n}{j}}{\frac {\tbinom {r}{i-j}}{\tbinom {n+r}{i}}}.} 4753:

A curve can be split at any point into two subcurves, or into arbitrarily many subcurves, each of which is also a Bézier curve.

11202: 11164: 10934: 10545: 10335: 10329: 10272: 10245: 10144: 9988: 614: 10510: 9784: 6282: 4040: 10881: 6343:, thus increasing a degree by one, without changing the value. Here is the example of increasing degree from 2 to 3. 5315: 213:

This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.

9209:{\displaystyle \mathbf {B} (t)={\frac {\sum _{i=0}^{n}b_{i,n}(t)\mathbf {P} _{i}w_{i}}{\sum _{i=0}^{n}b_{i,n}(t)w_{i}}},} 5882: 5365: 5252: 4829: 971:{\displaystyle \mathbf {B} (t)=(1-t)^{2}\mathbf {P} _{0}+2(1-t)t\mathbf {P} _{1}+t^{2}\mathbf {P} _{2},\ 0\leq t\leq 1.} 9539:

Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the

6063: 499: 10423: 5734: 5242:{\displaystyle \mathbf {P} '_{k}={\tfrac {k}{n+1}}\mathbf {P} _{k-1}+\left(1-{\tfrac {k}{n+1}}\right)\mathbf {P} _{k}} 2936:

The formula can be expressed explicitly as follows (where t and (1-t) are extended continuously to be 1 throughout ):

10975: 10723: 10196: 10169: 10047: 10015: 9883:– Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails. 5961: 5873: 138: 10354: 6054: 9649:

of cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.

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method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker

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of a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the

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TinySpline: Open source C-library for NURBS, B-splines and Bézier curves with bindings for various languages

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to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a

10770: 10448:"US20110285719A1 Approximation of stroked higher-order curved segments by quadratic bèzier curve segments" 11142: 9544: 6112: 4640: 275: 234: 10855: 8479: 9935: 9803:

fonts can use either kind of curve, depending on which font technology underlies the OpenType wrapper.

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can be applied on the curve by applying the respective transform on the control points of the curve.

5629:{\displaystyle \mathbf {B} '(t)=n\sum _{i=0}^{n-1}b_{i,n-1}(t)(\mathbf {P} _{i+1}-\mathbf {P} _{i}).} 4607: 10915:"Trajectory Generation for a Multibody Robotic System using the Product of Exponentials Formulation" 9574: 7593: 7341: 5464:

Tangents to the parabola at the endpoints of the curve (A and B) intersect at its control point (C).

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Excellent discussion of implementation details; available for free as part of the TeX distribution.

11039: 10437: 9792: 9586: 207: 11137: 11070: 10634: 10235: 4826:-piece cubic Bézier curve can approximate a circle, when each inner control point is the distance 11355: 10696: 9613:). In order to join Bézier curves into a composite Bézier curve without kinks, a property called 1649:, the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves: 10090:

Biswas, Pradipta; Langdon, Pat (2015-04-03). "Multimodal Intelligent Eye-Gaze Tracking System".

7312:{\displaystyle \mathbf {P'} _{2}={\tfrac {1}{3}}\left(2\mathbf {P} _{1}+\mathbf {P} _{2}\right)} 7223:{\displaystyle \mathbf {P'} _{1}={\tfrac {1}{3}}\left(\mathbf {P} _{0}+2\mathbf {P} _{1}\right)} 11360: 10432: 9552: 577: 10306: 10291: 10262: 10037: 10005: 10819: 10186: 10066: 9663: 9622: 9548: 9528: 3817:{\displaystyle \mathbf {B} (t)=\sum _{i=0}^{n}b_{i,n}(t)\mathbf {P} _{i},\ \ \ 0\leq t\leq 1} 1513:

in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at

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Hovey, Chad (2022). Formulation and Python Implementation of Bézier and B-Spline Geometry.

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In other words, the original start and end points are unchanged. The new control points are

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Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree

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horizontally or vertically from an outer control point on a unit circle. More generally, an

11266: 11124: – video showing how computers render a cubic Bézier curve, by Peter Nowell 10606:

Rababah, Abedallah; Ibrahim, Salisu (2018). "Geometric Degree Reduction of Bézier Curves".

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in mathematics (lying "parallel" to the original curve, like the offset between rails in a

4881: 4185: 3990: 3972: 3369: 2515: 794: 493: 222: 11127: 8: 10231: 9931: 9594: 1470:{\displaystyle \mathbf {B} ''(t)=2(\mathbf {P} _{2}-2\mathbf {P} _{1}+\mathbf {P} _{0}).} 541: 238: 147: 11270: 10958:

Gross, Renan (2014). "Bridges, String Art, and Bézier Curves". In Pitici, Mircea (ed.).

9547:, the points can be graphically displayed and used to manipulate the curve intuitively. 9495: 5444: 11239: 10981: 10940: 10379: 10115: 9939: 9811: 9761: 9714: 9696: 9646: 9562: 9475: 9005: 5432: 4141: 2511: 2072: 1555:; these points are only there to provide directional information. The distance between 9810:, draw the font's curves (and lines) on a pixellated surface using a process known as 2583:{\displaystyle \mathbf {B} _{\mathbf {P} _{0}\mathbf {P} _{1}\ldots \mathbf {P} _{k}}} 180: 11213: 11198: 11160: 11005: 10971: 10944: 10930: 10752: 10740: 10719: 10592: 10541: 10325: 10268: 10241: 10192: 10165: 10140: 10107: 10043: 10011: 9984: 9645:

Analytical methods where a Bézier is intersected with each scan line involve finding

9598: 9590: 9533: 4636: 2087: = 2/3, the control points for the original Bézier curve can be recovered. 270: 230: 151: 10119: 9851: 176: 11298: 11274: 11178: 11020: 10963: 10922: 10914: 10663: 10615: 10588: 10476: 10099: 9947: 6167: 4764:

Bézier curve; though a four-piece cubic Bézier curve can approximate a circle (see

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A cubic Bézier curve (yellow) can be made identical to a quadratic one (black) by

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is based on discretising the curve, so that it is approximated by a sequence of "

9578: 9017:. Rational Bézier curves can, among other uses, be used to represent segments of 6151: 6132: 2676:{\displaystyle \mathbf {B} _{\mathbf {P} _{0}}(t)=\mathbf {P} _{0}{\text{, and}}} 191: 187: 10619: 10461: 11289: 11024: 10889: 10667: 9566: 6128: 6124: 4736: 202: 163:(1910–1999), who used it in the 1960s for designing curves for the bodywork of 32: 11279: 11252: 10985: 10967: 10230: 10184: 11349: 10111: 9976: 9943: 9870: 9631: 9469: 9018: 5468: 5457: 4644: 329: 10237:

Computing Handbook, Third Edition: Computer Science and Software Engineering

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methods that usually give an adequate approximation for practical purposes.

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in in red. The Bézier curves for each of the lower orders are also shown.

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Computer Graphics and Geometric Modelling: Implementation & Algorithms

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This can be written in a way that highlights the symmetry with respect to

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were not applied to graphics until some 50 years later when mathematician

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values where it is parallel to one of these lines can be done by solving

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to the definition of the curve followed by some rearrangement will yield

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Segments of conic sections represented exactly by rational Bézier curves

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For fifth-order curves, one can construct similar intermediate points.

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A Bézier curve of order higher than two may intersect itself or have a

4181: 1166: 557: 254: 226: 11335: 10480: 6272:{\displaystyle \mathbf {B} (t)=(1-t)\mathbf {B} (t)+t\mathbf {B} (t).} 4267:{\displaystyle \mathbf {B} (t)=\sum _{j=0}^{n}{t^{j}\mathbf {C} _{j}}} 242: 11221: 11115: 9602: 8676:

The concept of degree elevation can be repeated on a control polygon

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Some terminology is associated with these parametric curves. We have

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Malik, Aryslan; Henderson, Troy; Prazenica, Richard (January 2021).

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J. Sánchez-Reyes (November 2009). "Complex rational Bézier curves".

9689:), then the curve is decomposed into smaller segments such that the 10537: 10042:. Vol. 1. Springer Science & Business Media. p. 119. 9906: 9856: 9807: 9800: 9788: 9772: 9653: 9606: 9556: 6166:, which require stroking of Bézier curves, to the more widely used 6163: 5453: 4815:{\displaystyle \textstyle {\frac {4\left({\sqrt {2}}-1\right)}{3}}} 2518:) of a pair of corresponding points in two Bézier curves of degree 10579:

Eck, Matthias (August 1993). "Degree reduction of Bézier curves".

10303:"FreeType Glyph Conventions – Version 2.1 / VI. FreeType outlines" 9577:. A composite Bézier curve is commonly referred to as a "path" in 8982:{\displaystyle \mathbf {\lim _{r\to \infty }R_{r}} =\mathbf {B} .} 5757:

For quadratic Bézier curves one can construct intermediate points

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to the first and last section of the Bézier polygon, respectively.

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Simple Bézier curve implementation via recursive method in Python

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The curve at a fixed offset from a given Bézier curve, called an

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Curves and Surfaces in Geometric Modeling: Theory and Algorithms

5118:{\displaystyle \mathbf {P} '_{0},\,\dots ,\,\mathbf {P} '_{n+1}} 197:

Bézier curves are also used in the time domain, particularly in

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This book is out of print and freely available from the author.

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The expression can be extended by using number systems besides

6154:, painting two symmetrically distanced offset curves is called 5448:

Equivalence of a quadratic Bézier curve and a parabolic segment

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For fourth-order curves one can construct intermediate points

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If D is the midpoint of AB, the tangent to the curve which is

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denote the Bézier curve determined by any selection of points

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Prautzsch, Hartmut; Boehm, Wolfgang; Paluszny, Marco (2002).

10699:. Zhengzhou, China: School of Software, Zhengzhou University. 10446:

Kilgard, Mark J.; Moreton, Henry Packard (24 November 2011).

10355:"Finding a Point on a Bézier Curve: De Casteljau's Algorithm" 9875: 9865: 9861: 9569:

Bézier curves are most common. Higher degree curves are more

11006:"The Bernstein polynomial basis: A centennial retrospective" 10635:"Some Mathematical Elements of Graphics: Rational B-splines" 10324:. Springer Undergraduate Mathematics Series (2nd ed.). 6139:

the two-sided offset curve is an algebraic curve of degree 4

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is equivalent (including the parametrization) to the degree

5050:{\displaystyle \mathbf {P} _{0},\,\dots ,\,\mathbf {P} _{n}} 1569:

determines "how far" and "how fast" the curve moves towards

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Mario A. Gutiérrez; Frédéric Vexo; Daniel Thalmann (2023).

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Sometimes it is desirable to express the Bézier curve as a

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Bézier curves. Other languages and imaging tools (such as

8926:

It can also be shown that for the underlying Bézier curve

2307:

The second derivative of the Bézier curve with respect to

1374:

The second derivative of the Bézier curve with respect to

9749: 6118: 2090:

The derivative of the cubic Bézier curve with respect to

825:

respectively. Rearranging the preceding equation yields:

797:

of corresponding points on the linear Bézier curves from

11336:

C++ library to generate Bézier functions at compile time

10224: 5653:

denote the fraction of progress (from 0 to 1) the point

10912: 10534:

Curves and surfaces for computer-aided geometric design

782:{\displaystyle \mathbf {B} (t)=(1-t)+t,\ 0\leq t\leq 1} 171:

and animation. Bézier curves can be combined to form a

11154: 10409:, particularly p. 16 "taxonomy of offset curves". 10267:. Springer Science & Business Media. p. 404. 10133:

Gerald E. Farin; Josef Hoschek; Myung-Soo Kim (2002).

8878: 8848: 8823: 7258: 7169: 7048: 6965: 5203: 5151: 4884: 4833: 4775: 4000: 3334: 2506:

A recursive definition for the Bézier curve of degree

11211: 9699: 9666: 9498: 9478: 9228: 9053: 8939: 8767: 8523: 8482: 8450: 7801: 7335: 7236: 7147: 6352: 6328:{\displaystyle \mathbf {b} _{i,n}(t)\mathbf {P} _{i}} 6285: 6199: 5500: 5368: 5318: 5255: 5131: 5067: 5011: 4962: 4921: 4878:

from an outer control point on a unit circle, where

4832: 4774: 4705: 4676: 4610: 4581: 4286: 4201: 4113:{\displaystyle {n \choose i}={\frac {n!}{i!(n-i)!}}.} 4043: 3999: 3836: 3711: 3388: 3333: 2945: 2692: 2622: 2531: 2320: 2103: 1860: 1658: 1387: 1182: 997: 834: 617: 502: 341: 139: 118: 10292:"FreeType Glyph Conventions / VI. FreeType outlines" 5355:{\displaystyle \mathbf {P} _{n+1}:=\mathbf {P} _{0}} 112: 10092:

International Journal of Human-Computer Interaction

5402:{\displaystyle \mathbf {P} _{-1}:=\mathbf {P} _{n}} 1622:) for the quadratic Bézier curve defined by points 576:A quadratic Bézier curve is the path traced by the 109: 10694: 10425:Hermite Approximation for Offset Curve Computation 10162:Mathématiques et CAO. Tome 2 : Formes à pôles 9705: 9681: 9507: 9484: 9453: 9208: 8981: 8915: 8660: 8503: 8468: 8433: 7781: 7311: 7222: 7130: 6327: 6271: 6111:These representations rest on the process used in 6033:that describe quadratic Bézier curves, and points 5628: 5401: 5354: 5305:{\displaystyle \forall k=0,\,1,\,\dots ,\,n,\,n+1} 5304: 5241: 5117: 5049: 4974: 4948: 4907: 4871:{\displaystyle \textstyle {\frac {4}{3}}\tan(t/4)} 4870: 4814: 4720: 4691: 4627: 4596: 4564: 4266: 4112: 4026: 3960: 3816: 3686: 3360: 3316: 2913: 2675: 2582: 2478: 2296: 2046: 1840: 1469: 1303: 1154: 970: 781: 532: 481: 16:Curve used in computer graphics and related fields 10496:"MetaFog: Converting Metafont shapes to contours" 4172:of the Bézier polygon contains the Bézier curve. 4060: 4047: 3881: 3868: 3209: 3188: 3127: 3114: 3008: 2995: 533:{\displaystyle \mathbf {P} _{1}-\mathbf {P} _{0}} 11347: 11258:Journal of Computational and Applied Mathematics 10653: 10493: 9044:, the rational Bézier curve can be described by 8942: 5452:A quadratic Bézier curve is also a segment of a 332:between those two points. The curve is given by 167:cars. Other uses include the design of computer 11194:Curves and Surfaces for CAGD: A Practical Guide 11179:"Chapter 5. Polynomial Curves as Bézier Curves" 11091:Computer Graphics: Principles and Practice in C 11003: 10829:. University of Cambridge Computer Laboratory. 10697:"Complex Quadratic Bézier Curve on Unit Circle" 10126: 9887:Variation diminishing property of Bézier curves 6182:can be converted into a Bézier curve of degree 5931:that describe linear Bézier curves, and points 2925: 10962:. Princeton University Press. pp. 77–89. 10605: 10462:"Comparing offset curve approximation methods" 10445: 10395:"Introduction to Pythagorean-hodograph curves" 10322:Applied Geometry for Computer Graphics and CAD 10260: 10153: 10007:Mathematics for Computer Graphics Applications 9775:fonts use composite Bézier curves composed of 4643:should be used if this occurs). Note that the 1343:increases from 0 to 1, the curve departs from 175:, or generalized to higher dimensions to form 11098: 10380:"CS 354 Vector Graphics & Path Rendering" 10159: 10089: 9391: 9378: 9286: 9273: 8902: 8881: 8872: 8851: 8839: 8826: 7538: 7525: 7492: 7463: 7426: 7413: 7368: 7347: 5639: 4760:, cannot be described exactly by a Bézier or 4604:can be computed prior to many evaluations of 4017: 4004: 3351: 3338: 2071:the curve may intersect itself, or contain a 221:The mathematical basis for Bézier curves—the 11122:Cubic Bezier Curves – Under the Hood (video) 11059: 11046: 10908: 10906: 10632: 10527: 10525: 10523: 10509:(3–Proceedings of the 1995 Annual Meeting). 10377: 10319: 10067:"Cubic class - animation library - Dart API" 10031: 10029: 10027: 9472:the points {1}, {-1}, and {1} with weights { 8671: 6193:To do degree elevation, we use the equality 4140:formed by connecting the Bézier points with 2494:Bézier curves can be defined for any degree 11087: 11031: 10690: 10688: 10392: 10286: 10284: 10254: 10234:; Jorge Diaz-Herrera; Allen Tucker (2014). 10136:Handbook of Computer Aided Geometric Design 6012:that describe linear Bézier curves, points 127: 121: 23:Cubic Bézier curve with four control points 10741:A Rasterizing Algorithm for Drawing Curves 10373: 10371: 10035: 7586: 7582: 6135:and more generally for a Bézier of degree 5435:for certain choices of the control points. 4756:Some curves that seem simple, such as the 4027:{\displaystyle \scriptstyle {n \choose i}} 3361:{\displaystyle \scriptstyle {n \choose i}} 547: 492:This is the simplest and is equivalent to 11278: 11159:. Springer Science & Business Media. 10903: 10520: 10436: 10211: 10024: 10003: 9799:Bézier curves for drawing curved shapes. 9630:The simplest method for scan converting ( 9000: 6101:Animation of a fifth-order Bézier curve, 5865:) and describes a quadratic Bézier curve. 5440:Second-order curve is a parabolic segment 5292: 5285: 5278: 5271: 5093: 5086: 5034: 5027: 4184:instead of a sum of less straightforward 2472: 2290: 10685: 10554: 10421: 10281: 10039:Encyclopaedia of Mathematics: Supplement 9997: 9621: 9527: 9004: 5890:Construction of a quadratic Bézier curve 5443: 4654: 1314:from which it can be concluded that the 551: 26: 18: 11173: 10817: 10469:IEEE Computer Graphics and Applications 10368: 9720:There is also a modified curve form of 5945:that describe quadratic Bézier curves: 5895:Animation of a quadratic Bézier curve, 5687:) is one quarter of the way from point 2501: 544:from the start point to the end point. 309: 11348: 10882:"smooth_curve_bezier_example_file.xls" 8680:to get a sequence of control polygons 6335:is multiplied by (1 − 6119:Offsets (or stroking) of Bézier curves 6071:Construction of a quartic Bézier curve 5905: 2931: 1480: 306:– that is, the coefficients sum to 1. 11296: 11212: 11190: 11101:"Implementing Bezier Curves in games" 10957: 10750: 10738: 10732: 10710: 10531: 10459: 10010:. Industrial Press Inc. p. 264. 9975: 9589:) and vector graphics programs (like 6076:Animation of a quartic Bézier curve, 2489: 568:) define the quadratic Bézier curve ( 11230: 10960:The Best Writing on Mathematics 2013 10848:"The difference between CFF and TTF" 10061: 10059: 9905:Image manipulation programs such as 9795:) use composite Béziers composed of 9652:The rasterisation algorithm used in 9523: 5969:Construction of a cubic Bézier curve 5832:and describes a linear Bézier curve. 5804:and describes a linear Bézier curve. 5742:Animation of a linear Bézier curve, 5661:) has made along its traversal from 5487:The derivative for a curve of order 2924:This recursion is elucidated in the 1541:. Usually, it will not pass through 1169:of the Bézier curve with respect to 328:, a linear Bézier curve is simply a 194:images and are intuitive to modify. 11250: 11054:. Addison-Wesley. pp. 123–131. 10578: 9737:In animation applications, such as 9585:), vector graphics standards (like 8992: 6173: 6143: − 2. However, there are 6047:that describe cubic Bézier curves: 5974:Animation of a cubic Bézier curve, 5843:) is interpolated linearly between 5752: 1851:The explicit form of the curve is: 13: 11287: 11109: 10771:"Using motion paths in animations" 9926:In animation applications such as 9722:Bresenham's line drawing algorithm 9382: 9277: 9021:exactly, including circular arcs. 8952: 8885: 8855: 8830: 8514:Therefore, new control points are 8504:{\displaystyle \mathbf {P} _{n+1}} 7529: 7467: 7417: 7351: 5256: 4746:The start and end of the curve is 4175: 4051: 4008: 3872: 3342: 3192: 3118: 2999: 298:is called the order of the curve ( 225:—was established in 1912, but the 14: 11382: 11319: 11197:(5th ed.). Morgan Kaufmann. 11088:J. D. Foley; et al. (1992). 10820:"Advanced Graphics Lecture Notes" 10352: 10056: 9983:(3rd ed.). Pearson Longman. 8469:{\displaystyle \mathbf {P} _{-1}} 2510:expresses it as a point-to-point 264: 183:is a special case of the latter. 10886:Rotating Machinery Analysis, Inc 10836:from the original on 2022-10-09. 10566:FontForge 20230101 documentation 10540:Science & Technology Books. 10516:from the original on 2022-10-09. 9981:Longman Pronunciation Dictionary 9332: 9230: 9122: 9055: 8972: 8962: 8958: 8946: 8812: 8770: 8617: 8564: 8527: 8485: 8453: 8413: 8379: 8309: 8292: 8239: 8167: 8128: 8066: 8033: 7958: 7931: 7892: 7865: 7807: 7744: 7697: 7666: 7613: 7546: 7503: 7434: 7391: 7294: 7279: 7240: 7205: 7187: 7151: 7114: 7084: 7069: 7001: 6983: 6923: 6879: 6849: 6834: 6781: 6763: 6718: 6674: 6649: 6606: 6566: 6526: 6483: 6442: 6417: 6381: 6315: 6288: 6253: 6233: 6201: 6092: 6062: 6053: 5960: 5951: 5881: 5872: 5733: 5644: 5610: 5589: 5503: 5389: 5371: 5342: 5321: 5229: 5172: 5134: 5096: 5070: 5037: 5014: 4949:{\displaystyle t=360^{\circ }/n} 4721:{\displaystyle \mathbf {P} _{n}} 4708: 4692:{\displaystyle \mathbf {P} _{0}} 4679: 4612: 4597:{\displaystyle \mathbf {C} _{j}} 4584: 4520: 4385: 4289: 4253: 4203: 3777: 3713: 3649: 3624: 3581: 3531: 3481: 3438: 3394: 3279: 3248: 3166: 3098: 3054: 2951: 2890: 2875: 2863: 2856: 2821: 2806: 2794: 2787: 2746: 2731: 2719: 2712: 2694: 2658: 2632: 2625: 2568: 2553: 2541: 2534: 2459: 2444: 2426: 2399: 2384: 2366: 2323: 2277: 2262: 2228: 2213: 2171: 2156: 2106: 2013: 1988: 1945: 1902: 1862: 1796: 1781: 1766: 1759: 1730: 1715: 1700: 1693: 1660: 1451: 1436: 1418: 1390: 1285: 1270: 1243: 1228: 1185: 1118: 1103: 1072: 1057: 1017: 999: 937: 912: 876: 836: 793:which can be interpreted as the 745: 727: 688: 670: 619: 520: 505: 448: 430: 397: 382: 361: 343: 105: 11094:(2nd ed.). Addison Wesley. 11013:Computer Aided Geometric Design 11004:Rida T. Farouki (August 2012). 10951: 10874: 10840: 10811: 10787: 10763: 10703: 10695:Xuexiang Li & Junxiao Xue. 10674: 10656:Computer Aided Geometric Design 10647: 10626: 10599: 10581:Computer Aided Geometric Design 10572: 10487: 10412: 10386: 10378:Mark Kilgard (April 10, 2012). 10346: 10313: 10205: 10191:. Springer Nature. p. 33. 10178: 9518: 8702:degree elevations, the polygon 8630: 7587: 7581: 4628:{\displaystyle \mathbf {B} (t)} 3985: = 1, (1 − 269:A Bézier curve is defined by a 11157:Bézier and B-Spline Techniques 10305:. 6 March 2011. Archived from 10214:"Forcing Bezier Interpolation" 10083: 10004:Mortenson, Michael E. (1999). 9969: 9920: 9899: 9420: 9407: 9315: 9302: 9240: 9234: 9187: 9181: 9117: 9111: 9065: 9059: 8949: 8407: 8401: 8337: 8331: 8162: 8156: 8061: 8055: 7953: 7947: 7887: 7881: 7839: 7827: 7817: 7811: 7608: 7596: 7583: 7386: 7374: 7034: 7022: 6952: 6939: 6912: 6899: 6811: 6799: 6744: 6731: 6707: 6694: 6634: 6622: 6592: 6579: 6551: 6539: 6512: 6499: 6472: 6459: 6409: 6397: 6370: 6357: 6310: 6304: 6263: 6257: 6243: 6237: 6229: 6217: 6211: 6205: 5620: 5584: 5581: 5575: 5517: 5511: 5414:variation diminishing property 5061:+ 1 curve with control points 4989:Bézier curve is also a degree 4864: 4850: 4661:1. copying the end points, and 4622: 4616: 4550: 4538: 4503: 4493: 4466: 4454: 4415: 4403: 4368: 4358: 4325: 4313: 4213: 4207: 4098: 4086: 3910: 3897: 3859: 3853: 3772: 3766: 3723: 3717: 3697: 3619: 3607: 3570: 3557: 3520: 3507: 3470: 3457: 3427: 3414: 3404: 3398: 3227: 3215: 3146: 3133: 3087: 3074: 3027: 3014: 2961: 2955: 2908: 2902: 2845: 2839: 2782: 2770: 2764: 2758: 2704: 2698: 2650: 2644: 2469: 2421: 2409: 2361: 2358: 2346: 2337: 2331: 2287: 2257: 2238: 2208: 2202: 2190: 2181: 2151: 2142: 2129: 2120: 2114: 1973: 1961: 1931: 1918: 1891: 1878: 1872: 1866: 1814: 1808: 1748: 1742: 1688: 1676: 1670: 1664: 1461: 1413: 1404: 1398: 1295: 1265: 1253: 1223: 1220: 1208: 1199: 1193: 1128: 1098: 1082: 1052: 1043: 1030: 1009: 1003: 904: 892: 865: 852: 846: 840: 755: 722: 710: 707: 698: 665: 653: 650: 647: 635: 629: 623: 425: 413: 407: 377: 353: 347: 1: 11132:American Mathematical Society 10188:Stepping into Virtual Reality 10104:10.1080/10447318.2014.1001301 9957: 5482: 4650: 1534:coming from the direction of 11099:Rajiv Chandel (2014-03-20). 10846: 10593:10.1016/0167-8396(93)90039-6 10036:Hazewinkel, Michiel (1997). 9962: 9821:Because arcs of circles and 9732: 9729:additions and subtractions. 6115:to calculate Bézier curves. 1165:Which immediately gives the 216: 7: 11297:Hovey, Chad (20 May 2022). 11143:Encyclopedia of Mathematics 10633:Neil Dodgson (2000-09-25). 10620:10.1007/978-981-13-2095-8_8 9845: 9833: 4739:all the control points are 4575:This could be practical if 3973:Bernstein basis polynomials 10: 11387: 11032:Paul Bourke (2009-07-19). 11025:10.1016/j.cagd.2012.03.001 10996: 10854:. Linotype. Archived from 10668:10.1016/j.cagd.2009.06.003 10139:. Elsevier. pp. 4–6. 9936:Microsoft Expression Blend 9818:also uses this algorithm. 5640:Constructing Bézier curves 5005:curve with control points 4136:for the Bézier curve. The 1357:, then bends to arrive at 11280:10.1016/j.cam.2003.10.008 11191:Farin, Gerald E. (2002). 11116:A Primer on Bézier Curves 10968:10.1515/9781400847990-011 10818:Dodgson, Neil A. (1999). 10608:Mathematics and Computing 10494:Richard J. Kinch (1995). 10240:. CRC Press. page 32-14. 9571:computationally expensive 9028: + 1 control points 8672:Repeated degree elevation 6178:A Bézier curve of degree 3989:) = 1, and the 42:for cubic Bézier curves: 11290:"Animated Bézier curves" 11251:Ahn, Young Joon (2004). 11128:From Bézier to Bernstein 10749:HTML abstract and demo: 9892: 9767: 9468:for the weights. In the 6125:offset or parallel curve 6113:De Casteljau's algorithm 4728:; this is the so-called 4641:de Casteljau's algorithm 235:de Casteljau's algorithm 11035:Bézier Surfaces (in 3D) 10919:AIAA Scitech 2021 Forum 10261:Max K. Agoston (2005). 9682:{\displaystyle y=\pm x} 6168:PostScript type 1 fonts 5412:Bézier curves have the 1576:before turning towards 548:Quadratic Bézier curves 10532:Farin, Gerald (1997). 10460:Elber, G. (May 1997). 10422:Ostromoukhov, Victor. 9707: 9683: 9627: 9575:composite Bézier curve 9549:Affine transformations 9536: 9509: 9486: 9455: 9374: 9269: 9210: 9164: 9094: 9010: 9001:Rational Bézier curves 8983: 8917: 8809: 8662: 8505: 8470: 8444:introducing arbitrary 8435: 8376: 8213: 8099: 7998: 7928: 7862: 7783: 7313: 7224: 7132: 6329: 6273: 5630: 5552: 5449: 5403: 5356: 5306: 5243: 5119: 5051: 5001:. In detail, a degree 4976: 4975:{\displaystyle n>2} 4950: 4909: 4908:{\textstyle t=2\pi /n} 4872: 4816: 4766:composite Bézier curve 4730:endpoint interpolation 4722: 4693: 4666: 4629: 4598: 4566: 4489: 4453: 4354: 4268: 4239: 4114: 4028: 3962: 3827:where the polynomials 3818: 3749: 3688: 3362: 3318: 2991: 2915: 2677: 2584: 2480: 2298: 2048: 1842: 1471: 1364:from the direction of 1305: 1156: 972: 783: 573: 534: 483: 314:Given distinct points 96: 24: 11052:Metafont: the Program 10751:Zingl, Alois (2016). 10739:Zingl, Alois (2012). 10716:Metafont: The Program 10320:Duncan Marsh (2005). 10296:The Free Type Project 9708: 9684: 9625: 9531: 9510: 9487: 9456: 9354: 9249: 9211: 9144: 9074: 9015:Bernstein polynomials 9008: 8984: 8918: 8789: 8663: 8506: 8471: 8436: 8350: 8187: 8079: 7978: 7908: 7842: 7784: 7314: 7225: 7133: 6330: 6274: 5631: 5526: 5456:. As a parabola is a 5447: 5404: 5357: 5307: 5244: 5120: 5052: 4977: 4951: 4910: 4873: 4817: 4723: 4694: 4658: 4630: 4599: 4567: 4469: 4427: 4334: 4269: 4219: 4188:. Application of the 4186:Bernstein polynomials 4115: 4029: 3963: 3819: 3729: 3689: 3370:binomial coefficients 3363: 3319: 2971: 2916: 2678: 2585: 2481: 2299: 2049: 1843: 1472: 1306: 1157: 973: 784: 564:) and control point ( 556:Quadratic Béziers in 555: 535: 484: 223:Bernstein polynomials 30: 22: 11130:Feature Column from 10795:"Following a Spline" 9697: 9664: 9496: 9476: 9226: 9051: 8937: 8765: 8521: 8480: 8448: 7799: 7333: 7234: 7145: 6350: 6283: 6197: 5775:varies from 0 to 1: 5713:) draws a line from 5705:varies from 0 to 1, 5675:. For example, when 5498: 5366: 5316: 5253: 5129: 5065: 5009: 4960: 4919: 4882: 4830: 4772: 4735:The curve is a line 4703: 4674: 4670:The curve begins at 4608: 4579: 4284: 4199: 4041: 3997: 3991:binomial coefficient 3834: 3709: 3386: 3331: 2943: 2690: 2620: 2529: 2516:linear interpolation 2502:Recursive definition 2318: 2101: 2057:For some choices of 1858: 1656: 1385: 1350:in the direction of 1180: 995: 832: 615: 500: 494:linear interpolation 339: 310:Linear Bézier curves 11271:2004JCoAM.167..405A 10927:10.2514/6.2021-2016 10298:. 13 February 2018. 10160:Paul de Casteljau. 9932:Adobe After Effects 9806:Font engines, like 9715:quadratic equations 9595:Timeworks Publisher 8698:, and so on. After 6188:with the same shape 5906:Higher-order curves 5146: 5114: 5082: 4151:and finishing with 2932:Explicit definition 1481:Cubic Bézier curves 542:displacement vector 304:affine combinations 11231:Hoffmann, Gernot. 11214:Weisstein, Eric W. 11185:. Morgan Kaufmann. 11060:Thomas Sederberg. 10718:. Addison-Wesley. 9812:font rasterization 9703: 9679: 9628: 9537: 9508:{\displaystyle -i} 9505: 9482: 9451: 9206: 9011: 8979: 8956: 8913: 8907: 8877: 8844: 8658: 8501: 8466: 8431: 8429: 7779: 7774: 7576: 7326:we use equalities 7309: 7267: 7220: 7178: 7128: 7126: 7057: 6974: 6325: 6269: 5626: 5450: 5399: 5352: 5302: 5239: 5220: 5168: 5132: 5115: 5094: 5068: 5047: 4972: 4946: 4905: 4868: 4867: 4812: 4811: 4718: 4689: 4667: 4625: 4594: 4562: 4264: 4110: 4024: 4023: 3958: 3814: 3684: 3682: 3375:For example, when 3358: 3357: 3314: 3312: 2911: 2673: 2580: 2512:linear combination 2490:General definition 2476: 2294: 2044: 1838: 1467: 1301: 1152: 968: 795:linear interpolant 779: 574: 560:: The end points ( 530: 479: 239:numerically stable 233:in 1959 developed 97: 25: 11299:"Bézier Geometry" 11204:978-1-55860-737-8 11166:978-3-540-43761-1 10936:978-1-62410-609-5 10757:members.chello.at 10547:978-0-12-249054-5 10481:10.1109/38.586019 10458:For a survey see 10393:Rida T. Farouki. 10331:978-1-85233-801-5 10274:978-1-84628-108-2 10247:978-1-4398-9852-9 10146:978-0-444-51104-1 9990:978-1-4058-8118-0 9706:{\displaystyle t} 9599:Adobe Illustrator 9534:Adobe Illustrator 9524:Computer graphics 9485:{\displaystyle i} 9446: 9389: 9284: 9201: 8941: 8908: 8900: 8870: 8837: 8711:has the vertices 8613: 8560: 8288: 8235: 8124: 8029: 7740: 7662: 7536: 7490: 7424: 7366: 7266: 7177: 7056: 6973: 6109: 6108: 6084: 6083: 5982: 5981: 5903: 5902: 5750: 5749: 5219: 5167: 4842: 4809: 4792: 4637:numeric stability 4557: 4422: 4332: 4105: 4058: 4015: 3936: 3933: 3930: 3879: 3798: 3795: 3792: 3349: 3207: 3125: 3006: 2671: 2613:. Then to start, 2604:, ..., 2028: 1822: 1136: 952: 763: 463: 231:Paul de Casteljau 152:computer graphics 11378: 11308: 11293: 11284: 11282: 11246: 11244: 11238:. Archived from 11237: 11227: 11226: 11208: 11186: 11170: 11151: 11104: 11095: 11084: 11082: 11081: 11075: 11069:. Archived from 11068: 11055: 11043: 11038:. Archived from 11028: 11010: 10990: 10989: 10955: 10949: 10948: 10910: 10901: 10900: 10898: 10897: 10888:. Archived from 10878: 10872: 10871: 10865: 10863: 10844: 10838: 10837: 10835: 10824: 10815: 10809: 10808: 10806: 10805: 10791: 10785: 10784: 10782: 10781: 10767: 10761: 10760: 10747: 10745: 10736: 10730: 10729: 10712:Knuth, Donald E. 10707: 10701: 10700: 10692: 10683: 10678: 10672: 10671: 10651: 10645: 10644: 10642: 10641: 10630: 10624: 10623: 10603: 10597: 10596: 10587:(3–4): 237–251. 10576: 10570: 10569: 10562:"Bézier Splines" 10558: 10552: 10551: 10529: 10518: 10517: 10515: 10500: 10491: 10485: 10484: 10466: 10455: 10442: 10440: 10430: 10416: 10410: 10408: 10407:on June 5, 2015. 10406: 10400:. Archived from 10399: 10390: 10384: 10383: 10375: 10366: 10365: 10363: 10361: 10350: 10344: 10343: 10317: 10311: 10310: 10299: 10288: 10279: 10278: 10258: 10252: 10251: 10232:Teofilo Gonzalez 10228: 10222: 10221: 10216:. Archived from 10209: 10203: 10202: 10182: 10176: 10175: 10157: 10151: 10150: 10130: 10124: 10123: 10087: 10081: 10080: 10078: 10077: 10063: 10054: 10053: 10033: 10022: 10021: 10001: 9995: 9994: 9979:(3 April 2008). 9973: 9951: 9948:Autodesk 3ds Max 9924: 9918: 9903: 9712: 9710: 9709: 9704: 9688: 9686: 9685: 9680: 9638:≤ 0.5 and 0.5 ≤ 9581:languages (like 9514: 9512: 9511: 9506: 9491: 9489: 9488: 9483: 9460: 9458: 9457: 9452: 9447: 9445: 9444: 9443: 9434: 9433: 9406: 9405: 9396: 9395: 9394: 9381: 9373: 9368: 9352: 9351: 9350: 9341: 9340: 9335: 9329: 9328: 9301: 9300: 9291: 9290: 9289: 9276: 9268: 9263: 9247: 9233: 9215: 9213: 9212: 9207: 9202: 9200: 9199: 9198: 9180: 9179: 9163: 9158: 9142: 9141: 9140: 9131: 9130: 9125: 9110: 9109: 9093: 9088: 9072: 9058: 8993:Degree reduction 8988: 8986: 8985: 8980: 8975: 8967: 8966: 8965: 8955: 8922: 8920: 8919: 8914: 8909: 8906: 8905: 8896: 8884: 8876: 8875: 8869: 8854: 8847: 8845: 8843: 8842: 8829: 8821: 8820: 8815: 8808: 8803: 8785: 8784: 8773: 8667: 8665: 8664: 8659: 8626: 8625: 8620: 8614: 8612: 8601: 8584: 8579: 8578: 8567: 8561: 8559: 8545: 8540: 8539: 8534: 8533: 8510: 8508: 8507: 8502: 8500: 8499: 8488: 8475: 8473: 8472: 8467: 8465: 8464: 8456: 8440: 8438: 8437: 8432: 8430: 8426: 8425: 8420: 8419: 8400: 8399: 8382: 8375: 8364: 8343: 8330: 8329: 8312: 8306: 8302: 8301: 8300: 8295: 8289: 8287: 8276: 8259: 8254: 8253: 8242: 8236: 8234: 8220: 8212: 8201: 8180: 8176: 8175: 8170: 8155: 8154: 8131: 8125: 8123: 8112: 8101: 8098: 8093: 8075: 8074: 8069: 8054: 8053: 8036: 8030: 8028: 8017: 8000: 7997: 7992: 7971: 7967: 7966: 7961: 7946: 7945: 7934: 7927: 7922: 7901: 7900: 7895: 7880: 7879: 7868: 7861: 7856: 7810: 7788: 7786: 7785: 7780: 7778: 7777: 7771: 7770: 7747: 7741: 7739: 7728: 7717: 7712: 7711: 7700: 7687: 7686: 7669: 7663: 7661: 7650: 7633: 7628: 7627: 7616: 7580: 7579: 7573: 7572: 7549: 7543: 7542: 7541: 7528: 7518: 7517: 7506: 7497: 7496: 7495: 7489: 7478: 7466: 7455: 7454: 7437: 7431: 7430: 7429: 7416: 7406: 7405: 7394: 7373: 7372: 7371: 7362: 7350: 7318: 7316: 7315: 7310: 7308: 7304: 7303: 7302: 7297: 7288: 7287: 7282: 7268: 7259: 7253: 7252: 7247: 7246: 7229: 7227: 7226: 7221: 7219: 7215: 7214: 7213: 7208: 7196: 7195: 7190: 7179: 7170: 7164: 7163: 7158: 7157: 7137: 7135: 7134: 7129: 7127: 7123: 7122: 7117: 7111: 7110: 7098: 7094: 7093: 7092: 7087: 7078: 7077: 7072: 7058: 7049: 7046: 7045: 7015: 7011: 7010: 7009: 7004: 6992: 6991: 6986: 6975: 6966: 6960: 6959: 6932: 6931: 6926: 6920: 6919: 6892: 6888: 6887: 6882: 6876: 6875: 6863: 6859: 6858: 6857: 6852: 6843: 6842: 6837: 6823: 6822: 6795: 6791: 6790: 6789: 6784: 6772: 6771: 6766: 6752: 6751: 6727: 6726: 6721: 6715: 6714: 6687: 6683: 6682: 6677: 6671: 6670: 6658: 6657: 6652: 6646: 6645: 6615: 6614: 6609: 6600: 6599: 6575: 6574: 6569: 6563: 6562: 6535: 6534: 6529: 6520: 6519: 6492: 6491: 6486: 6480: 6479: 6451: 6450: 6445: 6439: 6438: 6426: 6425: 6420: 6390: 6389: 6384: 6378: 6377: 6334: 6332: 6331: 6326: 6324: 6323: 6318: 6303: 6302: 6291: 6278: 6276: 6275: 6270: 6256: 6236: 6204: 6174:Degree elevation 6150:In the field of 6096: 6089: 6088: 6066: 6057: 6050: 6049: 5964: 5955: 5948: 5947: 5885: 5876: 5869: 5868: 5753:Quadratic curves 5737: 5730: 5729: 5635: 5633: 5632: 5627: 5619: 5618: 5613: 5604: 5603: 5592: 5574: 5573: 5551: 5540: 5510: 5506: 5408: 5406: 5405: 5400: 5398: 5397: 5392: 5383: 5382: 5374: 5361: 5359: 5358: 5353: 5351: 5350: 5345: 5336: 5335: 5324: 5311: 5309: 5308: 5303: 5248: 5246: 5245: 5240: 5238: 5237: 5232: 5226: 5222: 5221: 5218: 5204: 5187: 5186: 5175: 5169: 5166: 5152: 5142: 5137: 5124: 5122: 5121: 5116: 5110: 5099: 5078: 5073: 5056: 5054: 5053: 5048: 5046: 5045: 5040: 5023: 5022: 5017: 4981: 4979: 4978: 4973: 4955: 4953: 4952: 4947: 4942: 4937: 4936: 4914: 4912: 4911: 4906: 4901: 4877: 4875: 4874: 4869: 4860: 4843: 4835: 4821: 4819: 4818: 4813: 4810: 4805: 4804: 4800: 4793: 4788: 4777: 4727: 4725: 4724: 4719: 4717: 4716: 4711: 4698: 4696: 4695: 4690: 4688: 4687: 4682: 4634: 4632: 4631: 4626: 4615: 4603: 4601: 4600: 4595: 4593: 4592: 4587: 4571: 4569: 4568: 4563: 4558: 4556: 4530: 4529: 4528: 4523: 4517: 4516: 4491: 4488: 4483: 4452: 4441: 4423: 4421: 4395: 4394: 4393: 4388: 4382: 4381: 4356: 4353: 4348: 4333: 4331: 4311: 4303: 4298: 4297: 4292: 4273: 4271: 4270: 4265: 4263: 4262: 4261: 4256: 4250: 4249: 4238: 4233: 4206: 4190:binomial theorem 4160:, is called the 4144:, starting with 4119: 4117: 4116: 4111: 4106: 4104: 4078: 4070: 4065: 4064: 4063: 4050: 4033: 4031: 4030: 4025: 4022: 4021: 4020: 4007: 3967: 3965: 3964: 3959: 3934: 3931: 3928: 3924: 3923: 3896: 3895: 3886: 3885: 3884: 3871: 3852: 3851: 3823: 3821: 3820: 3815: 3796: 3793: 3790: 3786: 3785: 3780: 3765: 3764: 3748: 3743: 3716: 3693: 3691: 3690: 3685: 3683: 3663: 3658: 3657: 3652: 3646: 3645: 3633: 3632: 3627: 3606: 3605: 3590: 3589: 3584: 3578: 3577: 3556: 3555: 3540: 3539: 3534: 3528: 3527: 3506: 3505: 3490: 3489: 3484: 3478: 3477: 3447: 3446: 3441: 3435: 3434: 3397: 3379: = 5: 3367: 3365: 3364: 3359: 3356: 3355: 3354: 3341: 3323: 3321: 3320: 3315: 3313: 3293: 3288: 3287: 3282: 3276: 3275: 3263: 3262: 3251: 3245: 3244: 3214: 3213: 3212: 3206: 3191: 3175: 3174: 3169: 3160: 3159: 3132: 3131: 3130: 3117: 3107: 3106: 3101: 3095: 3094: 3067: 3063: 3062: 3057: 3051: 3050: 3041: 3040: 3013: 3012: 3011: 2998: 2990: 2985: 2954: 2926:animations below 2920: 2918: 2917: 2912: 2901: 2900: 2899: 2898: 2893: 2884: 2883: 2878: 2872: 2871: 2866: 2859: 2838: 2837: 2836: 2835: 2824: 2815: 2814: 2809: 2803: 2802: 2797: 2790: 2757: 2756: 2755: 2754: 2749: 2740: 2739: 2734: 2728: 2727: 2722: 2715: 2697: 2682: 2680: 2679: 2674: 2672: 2669: 2667: 2666: 2661: 2643: 2642: 2641: 2640: 2635: 2628: 2589: 2587: 2586: 2581: 2579: 2578: 2577: 2576: 2571: 2562: 2561: 2556: 2550: 2549: 2544: 2537: 2522: − 1. 2485: 2483: 2482: 2477: 2468: 2467: 2462: 2453: 2452: 2447: 2435: 2434: 2429: 2408: 2407: 2402: 2393: 2392: 2387: 2375: 2374: 2369: 2330: 2326: 2303: 2301: 2300: 2295: 2286: 2285: 2280: 2271: 2270: 2265: 2256: 2255: 2237: 2236: 2231: 2222: 2221: 2216: 2180: 2179: 2174: 2165: 2164: 2159: 2150: 2149: 2113: 2109: 2083: = 1/3 and 2053: 2051: 2050: 2045: 2026: 2022: 2021: 2016: 2010: 2009: 1997: 1996: 1991: 1985: 1984: 1954: 1953: 1948: 1939: 1938: 1911: 1910: 1905: 1899: 1898: 1865: 1847: 1845: 1844: 1839: 1820: 1807: 1806: 1805: 1804: 1799: 1790: 1789: 1784: 1775: 1774: 1769: 1762: 1741: 1740: 1739: 1738: 1733: 1724: 1723: 1718: 1709: 1708: 1703: 1696: 1663: 1476: 1474: 1473: 1468: 1460: 1459: 1454: 1445: 1444: 1439: 1427: 1426: 1421: 1397: 1393: 1318:to the curve at 1310: 1308: 1307: 1302: 1294: 1293: 1288: 1279: 1278: 1273: 1252: 1251: 1246: 1237: 1236: 1231: 1192: 1188: 1161: 1159: 1158: 1153: 1134: 1127: 1126: 1121: 1112: 1111: 1106: 1097: 1096: 1081: 1080: 1075: 1066: 1065: 1060: 1051: 1050: 1026: 1025: 1020: 1002: 977: 975: 974: 969: 950: 946: 945: 940: 934: 933: 921: 920: 915: 885: 884: 879: 873: 872: 839: 788: 786: 785: 780: 761: 754: 753: 748: 736: 735: 730: 697: 696: 691: 679: 678: 673: 622: 587:), given points 539: 537: 536: 531: 529: 528: 523: 514: 513: 508: 488: 486: 485: 480: 461: 457: 456: 451: 439: 438: 433: 406: 405: 400: 391: 390: 385: 370: 369: 364: 346: 148:parametric curve 142: 137: 136: 133: 132: 129: 126: 123: 120: 117: 114: 111: 94: 83: 68: 53: 41: 11386: 11385: 11381: 11380: 11379: 11377: 11376: 11375: 11346: 11345: 11322: 11288:Davies, Jason. 11242: 11235: 11233:"Bézier Curves" 11205: 11167: 11136: 11112: 11110:Further reading 11107: 11079: 11077: 11073: 11066: 11008: 10999: 10994: 10993: 10978: 10956: 10952: 10937: 10911: 10904: 10895: 10893: 10880: 10879: 10875: 10861: 10859: 10845: 10841: 10833: 10822: 10816: 10812: 10803: 10801: 10793: 10792: 10788: 10779: 10777: 10769: 10768: 10764: 10748: 10743: 10737: 10733: 10726: 10709:Parts 19–22 of 10708: 10704: 10693: 10686: 10679: 10675: 10652: 10648: 10639: 10637: 10631: 10627: 10604: 10600: 10577: 10573: 10560: 10559: 10555: 10548: 10530: 10521: 10513: 10498: 10492: 10488: 10464: 10428: 10417: 10413: 10404: 10397: 10391: 10387: 10376: 10369: 10359: 10357: 10351: 10347: 10332: 10318: 10314: 10301: 10300: 10290: 10289: 10282: 10275: 10259: 10255: 10248: 10229: 10225: 10212:John Burkardt. 10210: 10206: 10199: 10183: 10179: 10172: 10158: 10154: 10147: 10131: 10127: 10088: 10084: 10075: 10073: 10071:api.flutter.dev 10065: 10064: 10057: 10050: 10034: 10025: 10018: 10002: 9998: 9991: 9974: 9970: 9965: 9960: 9955: 9954: 9925: 9921: 9911:Adobe Photoshop 9904: 9900: 9895: 9848: 9836: 9816:Microsoft Excel 9770: 9735: 9698: 9695: 9694: 9665: 9662: 9661: 9579:vector graphics 9532:Bézier path in 9526: 9521: 9497: 9494: 9493: 9477: 9474: 9473: 9439: 9435: 9423: 9419: 9401: 9397: 9390: 9377: 9376: 9375: 9369: 9358: 9353: 9346: 9342: 9336: 9331: 9330: 9318: 9314: 9296: 9292: 9285: 9272: 9271: 9270: 9264: 9253: 9248: 9246: 9229: 9227: 9224: 9223: 9194: 9190: 9169: 9165: 9159: 9148: 9143: 9136: 9132: 9126: 9121: 9120: 9099: 9095: 9089: 9078: 9073: 9071: 9054: 9052: 9049: 9048: 9043: 9034: 9003: 8995: 8971: 8961: 8957: 8945: 8940: 8938: 8935: 8934: 8901: 8886: 8880: 8879: 8871: 8859: 8850: 8849: 8846: 8838: 8825: 8824: 8822: 8816: 8811: 8810: 8804: 8793: 8774: 8769: 8768: 8766: 8763: 8762: 8757: 8740: 8730: 8720: 8710: 8697: 8690: 8674: 8621: 8616: 8615: 8602: 8585: 8583: 8568: 8563: 8562: 8549: 8544: 8535: 8526: 8525: 8524: 8522: 8519: 8518: 8489: 8484: 8483: 8481: 8478: 8477: 8457: 8452: 8451: 8449: 8446: 8445: 8428: 8427: 8421: 8412: 8411: 8410: 8383: 8378: 8377: 8365: 8354: 8341: 8340: 8313: 8308: 8307: 8296: 8291: 8290: 8277: 8260: 8258: 8243: 8238: 8237: 8224: 8219: 8218: 8214: 8202: 8191: 8178: 8177: 8171: 8166: 8165: 8132: 8127: 8126: 8113: 8102: 8100: 8094: 8083: 8070: 8065: 8064: 8037: 8032: 8031: 8018: 8001: 7999: 7993: 7982: 7969: 7968: 7962: 7957: 7956: 7935: 7930: 7929: 7923: 7912: 7896: 7891: 7890: 7869: 7864: 7863: 7857: 7846: 7820: 7806: 7802: 7800: 7797: 7796: 7773: 7772: 7748: 7743: 7742: 7729: 7718: 7716: 7701: 7696: 7695: 7689: 7688: 7670: 7665: 7664: 7651: 7634: 7632: 7617: 7612: 7611: 7589: 7588: 7575: 7574: 7550: 7545: 7544: 7537: 7524: 7523: 7522: 7507: 7502: 7501: 7491: 7479: 7468: 7462: 7461: 7460: 7457: 7456: 7438: 7433: 7432: 7425: 7412: 7411: 7410: 7395: 7390: 7389: 7367: 7352: 7346: 7345: 7344: 7337: 7336: 7334: 7331: 7330: 7298: 7293: 7292: 7283: 7278: 7277: 7273: 7269: 7257: 7248: 7239: 7238: 7237: 7235: 7232: 7231: 7209: 7204: 7203: 7191: 7186: 7185: 7184: 7180: 7168: 7159: 7150: 7149: 7148: 7146: 7143: 7142: 7125: 7124: 7118: 7113: 7112: 7106: 7102: 7088: 7083: 7082: 7073: 7068: 7067: 7063: 7059: 7047: 7041: 7037: 7005: 7000: 6999: 6987: 6982: 6981: 6980: 6976: 6964: 6955: 6951: 6927: 6922: 6921: 6915: 6911: 6890: 6889: 6883: 6878: 6877: 6871: 6867: 6853: 6848: 6847: 6838: 6833: 6832: 6828: 6824: 6818: 6814: 6785: 6780: 6779: 6767: 6762: 6761: 6760: 6756: 6747: 6743: 6722: 6717: 6716: 6710: 6706: 6685: 6684: 6678: 6673: 6672: 6666: 6662: 6653: 6648: 6647: 6641: 6637: 6610: 6605: 6604: 6595: 6591: 6570: 6565: 6564: 6558: 6554: 6530: 6525: 6524: 6515: 6511: 6487: 6482: 6481: 6475: 6471: 6452: 6446: 6441: 6440: 6434: 6430: 6421: 6416: 6415: 6385: 6380: 6379: 6373: 6369: 6353: 6351: 6348: 6347: 6319: 6314: 6313: 6292: 6287: 6286: 6284: 6281: 6280: 6279:Each component 6252: 6232: 6200: 6198: 6195: 6194: 6176: 6152:vector graphics 6133:algebraic curve 6121: 6046: 6039: 6032: 6025: 6018: 6011: 6004: 5997: 5990: 5944: 5937: 5930: 5923: 5916: 5908: 5860: 5849: 5831: 5824: 5813: 5803: 5796: 5785: 5770: 5763: 5755: 5726: 5719: 5700: 5693: 5674: 5667: 5647: 5642: 5614: 5609: 5608: 5593: 5588: 5587: 5557: 5553: 5541: 5530: 5502: 5501: 5499: 5496: 5495: 5485: 5442: 5393: 5388: 5387: 5375: 5370: 5369: 5367: 5364: 5363: 5346: 5341: 5340: 5325: 5320: 5319: 5317: 5314: 5313: 5254: 5251: 5250: 5233: 5228: 5227: 5208: 5202: 5195: 5191: 5176: 5171: 5170: 5156: 5150: 5138: 5133: 5130: 5127: 5126: 5100: 5095: 5074: 5069: 5066: 5063: 5062: 5041: 5036: 5035: 5018: 5013: 5012: 5010: 5007: 5006: 4961: 4958: 4957: 4938: 4932: 4928: 4920: 4917: 4916: 4897: 4883: 4880: 4879: 4856: 4834: 4831: 4828: 4827: 4787: 4786: 4782: 4778: 4776: 4773: 4770: 4769: 4712: 4707: 4706: 4704: 4701: 4700: 4683: 4678: 4677: 4675: 4672: 4671: 4665: 4662: 4653: 4611: 4609: 4606: 4605: 4588: 4583: 4582: 4580: 4577: 4576: 4531: 4524: 4519: 4518: 4506: 4502: 4492: 4490: 4484: 4473: 4442: 4431: 4396: 4389: 4384: 4383: 4371: 4367: 4357: 4355: 4349: 4338: 4312: 4304: 4302: 4293: 4288: 4287: 4285: 4282: 4281: 4257: 4252: 4251: 4245: 4241: 4240: 4234: 4223: 4202: 4200: 4197: 4196: 4178: 4176:Polynomial form 4166:control polygon 4159: 4150: 4131: 4079: 4071: 4069: 4059: 4046: 4045: 4044: 4042: 4039: 4038: 4016: 4003: 4002: 4001: 3998: 3995: 3994: 3913: 3909: 3891: 3887: 3880: 3867: 3866: 3865: 3841: 3837: 3835: 3832: 3831: 3781: 3776: 3775: 3754: 3750: 3744: 3733: 3712: 3710: 3707: 3706: 3700: 3681: 3680: 3662: 3653: 3648: 3647: 3641: 3637: 3628: 3623: 3622: 3601: 3597: 3585: 3580: 3579: 3573: 3569: 3551: 3547: 3535: 3530: 3529: 3523: 3519: 3501: 3497: 3485: 3480: 3479: 3473: 3469: 3442: 3437: 3436: 3430: 3426: 3407: 3393: 3389: 3387: 3384: 3383: 3350: 3337: 3336: 3335: 3332: 3329: 3328: 3311: 3310: 3292: 3283: 3278: 3277: 3271: 3267: 3252: 3247: 3246: 3234: 3230: 3208: 3196: 3187: 3186: 3185: 3170: 3165: 3164: 3149: 3145: 3126: 3113: 3112: 3111: 3102: 3097: 3096: 3090: 3086: 3065: 3064: 3058: 3053: 3052: 3046: 3042: 3030: 3026: 3007: 2994: 2993: 2992: 2986: 2975: 2964: 2950: 2946: 2944: 2941: 2940: 2934: 2894: 2889: 2888: 2879: 2874: 2873: 2867: 2862: 2861: 2860: 2855: 2854: 2825: 2820: 2819: 2810: 2805: 2804: 2798: 2793: 2792: 2791: 2786: 2785: 2750: 2745: 2744: 2735: 2730: 2729: 2723: 2718: 2717: 2716: 2711: 2710: 2693: 2691: 2688: 2687: 2668: 2662: 2657: 2656: 2636: 2631: 2630: 2629: 2624: 2623: 2621: 2618: 2617: 2612: 2603: 2596: 2572: 2567: 2566: 2557: 2552: 2551: 2545: 2540: 2539: 2538: 2533: 2532: 2530: 2527: 2526: 2504: 2492: 2463: 2458: 2457: 2448: 2443: 2442: 2430: 2425: 2424: 2403: 2398: 2397: 2388: 2383: 2382: 2370: 2365: 2364: 2322: 2321: 2319: 2316: 2315: 2281: 2276: 2275: 2266: 2261: 2260: 2251: 2247: 2232: 2227: 2226: 2217: 2212: 2211: 2175: 2170: 2169: 2160: 2155: 2154: 2145: 2141: 2105: 2104: 2102: 2099: 2098: 2070: 2063: 2017: 2012: 2011: 2005: 2001: 1992: 1987: 1986: 1980: 1976: 1949: 1944: 1943: 1934: 1930: 1906: 1901: 1900: 1894: 1890: 1861: 1859: 1856: 1855: 1800: 1795: 1794: 1785: 1780: 1779: 1770: 1765: 1764: 1763: 1758: 1757: 1734: 1729: 1728: 1719: 1714: 1713: 1704: 1699: 1698: 1697: 1692: 1691: 1659: 1657: 1654: 1653: 1648: 1639: 1630: 1617: 1616: 1607: 1598: 1582: 1575: 1568: 1561: 1554: 1547: 1540: 1533: 1527:and arrives at 1526: 1519: 1512: 1505: 1498: 1491: 1483: 1455: 1450: 1449: 1440: 1435: 1434: 1422: 1417: 1416: 1389: 1388: 1386: 1383: 1382: 1370: 1363: 1356: 1349: 1338: 1331: 1324: 1289: 1284: 1283: 1274: 1269: 1268: 1247: 1242: 1241: 1232: 1227: 1226: 1184: 1183: 1181: 1178: 1177: 1122: 1117: 1116: 1107: 1102: 1101: 1092: 1088: 1076: 1071: 1070: 1061: 1056: 1055: 1046: 1042: 1021: 1016: 1015: 998: 996: 993: 992: 987: 941: 936: 935: 929: 925: 916: 911: 910: 880: 875: 874: 868: 864: 835: 833: 830: 829: 824: 817: 810: 803: 749: 744: 743: 731: 726: 725: 692: 687: 686: 674: 669: 668: 618: 616: 613: 612: 607: 600: 593: 550: 540:represents the 524: 519: 518: 509: 504: 503: 501: 498: 497: 496:. The quantity 452: 447: 446: 434: 429: 428: 401: 396: 395: 386: 381: 380: 365: 360: 359: 342: 340: 337: 336: 327: 320: 312: 293: 284: 267: 219: 188:vector graphics 181:Bézier triangle 177:Bézier surfaces 140: 108: 104: 85: 70: 55: 43: 36: 33:basis functions 17: 12: 11: 5: 11384: 11374: 11373: 11368: 11363: 11358: 11356:Graphic design 11344: 11343: 11338: 11333: 11327: 11326: 11321: 11320:External links 11318: 11317: 11316: 11313:SAND2022-7702C 11309: 11294: 11285: 11265:(2): 405–416. 11248: 11245:on 2006-12-02. 11228: 11217:"Bézier Curve" 11209: 11203: 11188: 11171: 11165: 11152: 11138:"Bézier curve" 11134: 11125: 11119: 11111: 11108: 11106: 11105: 11096: 11085: 11057: 11044: 11042:on 2009-07-19. 11029: 11019:(6): 379–419. 11000: 10998: 10995: 10992: 10991: 10986:j.ctt4cgb74.13 10976: 10950: 10935: 10902: 10873: 10839: 10810: 10786: 10762: 10731: 10724: 10702: 10684: 10673: 10662:(8): 865–876. 10646: 10625: 10598: 10571: 10553: 10546: 10536:(4 ed.). 10519: 10486: 10457: 10456: 10452:Google Patents 10443: 10438:10.1.1.43.1724 10411: 10385: 10367: 10345: 10330: 10312: 10309:on 2011-09-29. 10280: 10273: 10253: 10246: 10223: 10220:on 2013-12-25. 10204: 10197: 10177: 10170: 10152: 10145: 10125: 10098:(4): 277–294. 10082: 10055: 10048: 10023: 10016: 9996: 9989: 9967: 9966: 9964: 9961: 9959: 9956: 9953: 9952: 9919: 9897: 9896: 9894: 9891: 9890: 9889: 9884: 9878: 9873: 9868: 9859: 9854: 9852:Bézier surface 9847: 9844: 9835: 9832: 9769: 9766: 9734: 9731: 9702: 9678: 9675: 9672: 9669: 9545:control points 9525: 9522: 9520: 9517: 9504: 9501: 9481: 9462: 9461: 9450: 9442: 9438: 9432: 9429: 9426: 9422: 9418: 9415: 9412: 9409: 9404: 9400: 9393: 9388: 9385: 9380: 9372: 9367: 9364: 9361: 9357: 9349: 9345: 9339: 9334: 9327: 9324: 9321: 9317: 9313: 9310: 9307: 9304: 9299: 9295: 9288: 9283: 9280: 9275: 9267: 9262: 9259: 9256: 9252: 9245: 9242: 9239: 9236: 9232: 9217: 9216: 9205: 9197: 9193: 9189: 9186: 9183: 9178: 9175: 9172: 9168: 9162: 9157: 9154: 9151: 9147: 9139: 9135: 9129: 9124: 9119: 9116: 9113: 9108: 9105: 9102: 9098: 9092: 9087: 9084: 9081: 9077: 9070: 9067: 9064: 9061: 9057: 9039: 9032: 9019:conic sections 9002: 8999: 8994: 8991: 8990: 8989: 8978: 8974: 8970: 8964: 8960: 8954: 8951: 8948: 8944: 8924: 8923: 8912: 8904: 8899: 8895: 8892: 8889: 8883: 8874: 8868: 8865: 8862: 8858: 8853: 8841: 8836: 8833: 8828: 8819: 8814: 8807: 8802: 8799: 8796: 8792: 8788: 8783: 8780: 8777: 8772: 8745: 8735: 8725: 8715: 8706: 8695: 8688: 8673: 8670: 8669: 8668: 8657: 8654: 8651: 8648: 8645: 8642: 8639: 8636: 8633: 8629: 8624: 8619: 8611: 8608: 8605: 8600: 8597: 8594: 8591: 8588: 8582: 8577: 8574: 8571: 8566: 8558: 8555: 8552: 8548: 8543: 8538: 8532: 8529: 8498: 8495: 8492: 8487: 8463: 8460: 8455: 8442: 8441: 8424: 8418: 8415: 8409: 8406: 8403: 8398: 8395: 8392: 8389: 8386: 8381: 8374: 8371: 8368: 8363: 8360: 8357: 8353: 8349: 8346: 8344: 8342: 8339: 8336: 8333: 8328: 8325: 8322: 8319: 8316: 8311: 8305: 8299: 8294: 8286: 8283: 8280: 8275: 8272: 8269: 8266: 8263: 8257: 8252: 8249: 8246: 8241: 8233: 8230: 8227: 8223: 8217: 8211: 8208: 8205: 8200: 8197: 8194: 8190: 8186: 8183: 8181: 8179: 8174: 8169: 8164: 8161: 8158: 8153: 8150: 8147: 8144: 8141: 8138: 8135: 8130: 8122: 8119: 8116: 8111: 8108: 8105: 8097: 8092: 8089: 8086: 8082: 8078: 8073: 8068: 8063: 8060: 8057: 8052: 8049: 8046: 8043: 8040: 8035: 8027: 8024: 8021: 8016: 8013: 8010: 8007: 8004: 7996: 7991: 7988: 7985: 7981: 7977: 7974: 7972: 7970: 7965: 7960: 7955: 7952: 7949: 7944: 7941: 7938: 7933: 7926: 7921: 7918: 7915: 7911: 7907: 7904: 7899: 7894: 7889: 7886: 7883: 7878: 7875: 7872: 7867: 7860: 7855: 7852: 7849: 7845: 7841: 7838: 7835: 7832: 7829: 7826: 7823: 7821: 7819: 7816: 7813: 7809: 7805: 7804: 7790: 7789: 7776: 7769: 7766: 7763: 7760: 7757: 7754: 7751: 7746: 7738: 7735: 7732: 7727: 7724: 7721: 7715: 7710: 7707: 7704: 7699: 7694: 7691: 7690: 7685: 7682: 7679: 7676: 7673: 7668: 7660: 7657: 7654: 7649: 7646: 7643: 7640: 7637: 7631: 7626: 7623: 7620: 7615: 7610: 7607: 7604: 7601: 7598: 7595: 7594: 7592: 7585: 7578: 7571: 7568: 7565: 7562: 7559: 7556: 7553: 7548: 7540: 7535: 7532: 7527: 7521: 7516: 7513: 7510: 7505: 7500: 7494: 7488: 7485: 7482: 7477: 7474: 7471: 7465: 7459: 7458: 7453: 7450: 7447: 7444: 7441: 7436: 7428: 7423: 7420: 7415: 7409: 7404: 7401: 7398: 7393: 7388: 7385: 7382: 7379: 7376: 7370: 7365: 7361: 7358: 7355: 7349: 7343: 7342: 7340: 7322:For arbitrary 7307: 7301: 7296: 7291: 7286: 7281: 7276: 7272: 7265: 7262: 7256: 7251: 7245: 7242: 7218: 7212: 7207: 7202: 7199: 7194: 7189: 7183: 7176: 7173: 7167: 7162: 7156: 7153: 7139: 7138: 7121: 7116: 7109: 7105: 7101: 7097: 7091: 7086: 7081: 7076: 7071: 7066: 7062: 7055: 7052: 7044: 7040: 7036: 7033: 7030: 7027: 7024: 7021: 7018: 7014: 7008: 7003: 6998: 6995: 6990: 6985: 6979: 6972: 6969: 6963: 6958: 6954: 6950: 6947: 6944: 6941: 6938: 6935: 6930: 6925: 6918: 6914: 6910: 6907: 6904: 6901: 6898: 6895: 6893: 6891: 6886: 6881: 6874: 6870: 6866: 6862: 6856: 6851: 6846: 6841: 6836: 6831: 6827: 6821: 6817: 6813: 6810: 6807: 6804: 6801: 6798: 6794: 6788: 6783: 6778: 6775: 6770: 6765: 6759: 6755: 6750: 6746: 6742: 6739: 6736: 6733: 6730: 6725: 6720: 6713: 6709: 6705: 6702: 6699: 6696: 6693: 6690: 6688: 6686: 6681: 6676: 6669: 6665: 6661: 6656: 6651: 6644: 6640: 6636: 6633: 6630: 6627: 6624: 6621: 6618: 6613: 6608: 6603: 6598: 6594: 6590: 6587: 6584: 6581: 6578: 6573: 6568: 6561: 6557: 6553: 6550: 6547: 6544: 6541: 6538: 6533: 6528: 6523: 6518: 6514: 6510: 6507: 6504: 6501: 6498: 6495: 6490: 6485: 6478: 6474: 6470: 6467: 6464: 6461: 6458: 6455: 6453: 6449: 6444: 6437: 6433: 6429: 6424: 6419: 6414: 6411: 6408: 6405: 6402: 6399: 6396: 6393: 6388: 6383: 6376: 6372: 6368: 6365: 6362: 6359: 6356: 6355: 6322: 6317: 6312: 6309: 6306: 6301: 6298: 6295: 6290: 6268: 6265: 6262: 6259: 6255: 6251: 6248: 6245: 6242: 6239: 6235: 6231: 6228: 6225: 6222: 6219: 6216: 6213: 6210: 6207: 6203: 6175: 6172: 6129:railroad track 6120: 6117: 6107: 6106: 6098: 6097: 6082: 6081: 6074: 6072: 6068: 6067: 6060: 6058: 6044: 6037: 6030: 6023: 6016: 6009: 6002: 5995: 5988: 5980: 5979: 5972: 5970: 5966: 5965: 5958: 5956: 5942: 5935: 5928: 5921: 5914: 5907: 5904: 5901: 5900: 5893: 5891: 5887: 5886: 5879: 5877: 5867: 5866: 5858: 5847: 5833: 5829: 5822: 5818:) varies from 5811: 5805: 5801: 5794: 5790:) varies from 5783: 5768: 5761: 5754: 5751: 5748: 5747: 5739: 5738: 5724: 5717: 5698: 5691: 5672: 5665: 5646: 5643: 5641: 5638: 5637: 5636: 5625: 5622: 5617: 5612: 5607: 5602: 5599: 5596: 5591: 5586: 5583: 5580: 5577: 5572: 5569: 5566: 5563: 5560: 5556: 5550: 5547: 5544: 5539: 5536: 5533: 5529: 5525: 5522: 5519: 5516: 5513: 5509: 5505: 5484: 5481: 5480: 5479: 5476: 5472: 5465: 5441: 5438: 5437: 5436: 5429: 5417: 5410: 5396: 5391: 5386: 5381: 5378: 5373: 5349: 5344: 5339: 5334: 5331: 5328: 5323: 5301: 5298: 5295: 5291: 5288: 5284: 5281: 5277: 5274: 5270: 5267: 5264: 5261: 5258: 5236: 5231: 5225: 5217: 5214: 5211: 5207: 5201: 5198: 5194: 5190: 5185: 5182: 5179: 5174: 5165: 5162: 5159: 5155: 5149: 5145: 5141: 5136: 5113: 5109: 5106: 5103: 5098: 5092: 5089: 5085: 5081: 5077: 5072: 5044: 5039: 5033: 5030: 5026: 5021: 5016: 4993:curve for any 4983: 4971: 4968: 4965: 4945: 4941: 4935: 4931: 4927: 4924: 4904: 4900: 4896: 4893: 4890: 4887: 4866: 4863: 4859: 4855: 4852: 4849: 4846: 4841: 4838: 4808: 4803: 4799: 4796: 4791: 4785: 4781: 4754: 4751: 4744: 4737:if and only if 4733: 4715: 4710: 4686: 4681: 4663: 4660: 4652: 4649: 4624: 4621: 4618: 4614: 4591: 4586: 4573: 4572: 4561: 4555: 4552: 4549: 4546: 4543: 4540: 4537: 4534: 4527: 4522: 4515: 4512: 4509: 4505: 4501: 4498: 4495: 4487: 4482: 4479: 4476: 4472: 4468: 4465: 4462: 4459: 4456: 4451: 4448: 4445: 4440: 4437: 4434: 4430: 4426: 4420: 4417: 4414: 4411: 4408: 4405: 4402: 4399: 4392: 4387: 4380: 4377: 4374: 4370: 4366: 4363: 4360: 4352: 4347: 4344: 4341: 4337: 4330: 4327: 4324: 4321: 4318: 4315: 4310: 4307: 4301: 4296: 4291: 4275: 4274: 4260: 4255: 4248: 4244: 4237: 4232: 4229: 4226: 4222: 4218: 4215: 4212: 4209: 4205: 4177: 4174: 4162:Bézier polygon 4155: 4148: 4134:control points 4127: 4121: 4120: 4109: 4103: 4100: 4097: 4094: 4091: 4088: 4085: 4082: 4077: 4074: 4068: 4062: 4057: 4054: 4049: 4019: 4014: 4011: 4006: 3969: 3968: 3957: 3954: 3951: 3948: 3945: 3942: 3939: 3927: 3922: 3919: 3916: 3912: 3908: 3905: 3902: 3899: 3894: 3890: 3883: 3878: 3875: 3870: 3864: 3861: 3858: 3855: 3850: 3847: 3844: 3840: 3825: 3824: 3813: 3810: 3807: 3804: 3801: 3789: 3784: 3779: 3774: 3771: 3768: 3763: 3760: 3757: 3753: 3747: 3742: 3739: 3736: 3732: 3728: 3725: 3722: 3719: 3715: 3699: 3696: 3695: 3694: 3679: 3676: 3673: 3670: 3667: 3664: 3661: 3656: 3651: 3644: 3640: 3636: 3631: 3626: 3621: 3618: 3615: 3612: 3609: 3604: 3600: 3596: 3593: 3588: 3583: 3576: 3572: 3568: 3565: 3562: 3559: 3554: 3550: 3546: 3543: 3538: 3533: 3526: 3522: 3518: 3515: 3512: 3509: 3504: 3500: 3496: 3493: 3488: 3483: 3476: 3472: 3468: 3465: 3462: 3459: 3456: 3453: 3450: 3445: 3440: 3433: 3429: 3425: 3422: 3419: 3416: 3413: 3410: 3408: 3406: 3403: 3400: 3396: 3392: 3391: 3353: 3348: 3345: 3340: 3325: 3324: 3309: 3306: 3303: 3300: 3297: 3294: 3291: 3286: 3281: 3274: 3270: 3266: 3261: 3258: 3255: 3250: 3243: 3240: 3237: 3233: 3229: 3226: 3223: 3220: 3217: 3211: 3205: 3202: 3199: 3195: 3190: 3184: 3181: 3178: 3173: 3168: 3163: 3158: 3155: 3152: 3148: 3144: 3141: 3138: 3135: 3129: 3124: 3121: 3116: 3110: 3105: 3100: 3093: 3089: 3085: 3082: 3079: 3076: 3073: 3070: 3068: 3066: 3061: 3056: 3049: 3045: 3039: 3036: 3033: 3029: 3025: 3022: 3019: 3016: 3010: 3005: 3002: 2997: 2989: 2984: 2981: 2978: 2974: 2970: 2967: 2965: 2963: 2960: 2957: 2953: 2949: 2948: 2933: 2930: 2922: 2921: 2910: 2907: 2904: 2897: 2892: 2887: 2882: 2877: 2870: 2865: 2858: 2853: 2850: 2847: 2844: 2841: 2834: 2831: 2828: 2823: 2818: 2813: 2808: 2801: 2796: 2789: 2784: 2781: 2778: 2775: 2772: 2769: 2766: 2763: 2760: 2753: 2748: 2743: 2738: 2733: 2726: 2721: 2714: 2709: 2706: 2703: 2700: 2696: 2684: 2683: 2665: 2660: 2655: 2652: 2649: 2646: 2639: 2634: 2627: 2608: 2601: 2594: 2575: 2570: 2565: 2560: 2555: 2548: 2543: 2536: 2503: 2500: 2491: 2488: 2487: 2486: 2475: 2471: 2466: 2461: 2456: 2451: 2446: 2441: 2438: 2433: 2428: 2423: 2420: 2417: 2414: 2411: 2406: 2401: 2396: 2391: 2386: 2381: 2378: 2373: 2368: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2336: 2333: 2329: 2325: 2305: 2304: 2293: 2289: 2284: 2279: 2274: 2269: 2264: 2259: 2254: 2250: 2246: 2243: 2240: 2235: 2230: 2225: 2220: 2215: 2210: 2207: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2183: 2178: 2173: 2168: 2163: 2158: 2153: 2148: 2144: 2140: 2137: 2134: 2131: 2128: 2125: 2122: 2119: 2116: 2112: 2108: 2068: 2061: 2055: 2054: 2043: 2040: 2037: 2034: 2031: 2025: 2020: 2015: 2008: 2004: 2000: 1995: 1990: 1983: 1979: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1952: 1947: 1942: 1937: 1933: 1929: 1926: 1923: 1920: 1917: 1914: 1909: 1904: 1897: 1893: 1889: 1886: 1883: 1880: 1877: 1874: 1871: 1868: 1864: 1849: 1848: 1837: 1834: 1831: 1828: 1825: 1819: 1816: 1813: 1810: 1803: 1798: 1793: 1788: 1783: 1778: 1773: 1768: 1761: 1756: 1753: 1750: 1747: 1744: 1737: 1732: 1727: 1722: 1717: 1712: 1707: 1702: 1695: 1690: 1687: 1684: 1681: 1678: 1675: 1672: 1669: 1666: 1662: 1644: 1635: 1626: 1612: 1603: 1594: 1590: 1580: 1573: 1566: 1559: 1552: 1545: 1538: 1531: 1524: 1517: 1510: 1503: 1496: 1489: 1482: 1479: 1478: 1477: 1466: 1463: 1458: 1453: 1448: 1443: 1438: 1433: 1430: 1425: 1420: 1415: 1412: 1409: 1406: 1403: 1400: 1396: 1392: 1368: 1361: 1354: 1347: 1336: 1329: 1322: 1312: 1311: 1300: 1297: 1292: 1287: 1282: 1277: 1272: 1267: 1264: 1261: 1258: 1255: 1250: 1245: 1240: 1235: 1230: 1225: 1222: 1219: 1216: 1213: 1210: 1207: 1204: 1201: 1198: 1195: 1191: 1187: 1163: 1162: 1151: 1148: 1145: 1142: 1139: 1133: 1130: 1125: 1120: 1115: 1110: 1105: 1100: 1095: 1091: 1087: 1084: 1079: 1074: 1069: 1064: 1059: 1054: 1049: 1045: 1041: 1038: 1035: 1032: 1029: 1024: 1019: 1014: 1011: 1008: 1005: 1001: 985: 979: 978: 967: 964: 961: 958: 955: 949: 944: 939: 932: 928: 924: 919: 914: 909: 906: 903: 900: 897: 894: 891: 888: 883: 878: 871: 867: 863: 860: 857: 854: 851: 848: 845: 842: 838: 822: 815: 808: 801: 791: 790: 778: 775: 772: 769: 766: 760: 757: 752: 747: 742: 739: 734: 729: 724: 721: 718: 715: 712: 709: 706: 703: 700: 695: 690: 685: 682: 677: 672: 667: 664: 661: 658: 655: 652: 649: 646: 643: 640: 637: 634: 631: 628: 625: 621: 605: 598: 591: 549: 546: 527: 522: 517: 512: 507: 490: 489: 478: 475: 472: 469: 466: 460: 455: 450: 445: 442: 437: 432: 427: 424: 421: 418: 415: 412: 409: 404: 399: 394: 389: 384: 379: 376: 373: 368: 363: 358: 355: 352: 349: 345: 325: 318: 311: 308: 289: 282: 276:control points 266: 265:Specific cases 263: 218: 215: 203:user interface 15: 9: 6: 4: 3: 2: 11383: 11372: 11369: 11367: 11364: 11362: 11361:Interpolation 11359: 11357: 11354: 11353: 11351: 11342: 11339: 11337: 11334: 11332: 11329: 11328: 11325:Computer code 11324: 11323: 11315:. (153 pages) 11314: 11310: 11306: 11305: 11300: 11295: 11291: 11286: 11281: 11276: 11272: 11268: 11264: 11260: 11259: 11254: 11249: 11241: 11234: 11229: 11224: 11223: 11218: 11215: 11210: 11206: 11200: 11196: 11195: 11189: 11184: 11180: 11176: 11175:Gallier, Jean 11172: 11168: 11162: 11158: 11153: 11149: 11145: 11144: 11139: 11135: 11133: 11129: 11126: 11123: 11120: 11117: 11114: 11113: 11102: 11097: 11093: 11092: 11086: 11076:on 2006-02-21 11072: 11065: 11064: 11063:Bézier curves 11058: 11053: 11049: 11045: 11041: 11037: 11036: 11030: 11026: 11022: 11018: 11014: 11007: 11002: 11001: 10987: 10983: 10979: 10977:9780691160412 10973: 10969: 10965: 10961: 10954: 10946: 10942: 10938: 10932: 10928: 10924: 10920: 10916: 10909: 10907: 10892:on 2011-07-18 10891: 10887: 10883: 10877: 10870: 10858:on 2017-07-03 10857: 10853: 10849: 10843: 10832: 10828: 10821: 10814: 10800: 10796: 10790: 10776: 10772: 10766: 10758: 10754: 10742: 10735: 10727: 10725:0-201-13438-1 10721: 10717: 10713: 10706: 10698: 10691: 10689: 10682: 10677: 10669: 10665: 10661: 10657: 10650: 10636: 10629: 10621: 10617: 10613: 10609: 10602: 10594: 10590: 10586: 10582: 10575: 10567: 10563: 10557: 10549: 10543: 10539: 10535: 10528: 10526: 10524: 10512: 10508: 10504: 10497: 10490: 10482: 10478: 10474: 10470: 10463: 10453: 10449: 10444: 10439: 10434: 10427: 10426: 10420: 10419: 10418:For example: 10415: 10403: 10396: 10389: 10382:. p. 28. 10381: 10374: 10372: 10356: 10353:Shene, C. K. 10349: 10341: 10337: 10333: 10327: 10323: 10316: 10308: 10304: 10297: 10293: 10287: 10285: 10276: 10270: 10266: 10265: 10257: 10249: 10243: 10239: 10238: 10233: 10227: 10219: 10215: 10208: 10200: 10198:9783031364877 10194: 10190: 10189: 10181: 10173: 10171:9782866010423 10167: 10163: 10156: 10148: 10142: 10138: 10137: 10129: 10121: 10117: 10113: 10109: 10105: 10101: 10097: 10093: 10086: 10072: 10068: 10062: 10060: 10051: 10049:9780792347095 10045: 10041: 10040: 10032: 10030: 10028: 10019: 10017:9780831131111 10013: 10009: 10008: 10000: 9992: 9986: 9982: 9978: 9972: 9968: 9949: 9945: 9944:Autodesk Maya 9941: 9937: 9933: 9929: 9923: 9916: 9912: 9908: 9902: 9898: 9888: 9885: 9882: 9879: 9877: 9874: 9872: 9871:Hermite curve 9869: 9867: 9863: 9860: 9858: 9855: 9853: 9850: 9849: 9843: 9841: 9840:end effectors 9831: 9829: 9824: 9819: 9817: 9813: 9809: 9804: 9802: 9798: 9794: 9790: 9786: 9782: 9778: 9774: 9765: 9763: 9759: 9755: 9751: 9746: 9744: 9740: 9730: 9728: 9723: 9718: 9716: 9700: 9692: 9676: 9673: 9670: 9667: 9659: 9655: 9650: 9648: 9643: 9641: 9637: 9633: 9624: 9620: 9618: 9617: 9616:G1 continuity 9612: 9608: 9604: 9600: 9596: 9592: 9588: 9584: 9580: 9576: 9572: 9568: 9564: 9560: 9558: 9554: 9550: 9546: 9542: 9535: 9530: 9516: 9502: 9499: 9492:}, {1}, and { 9479: 9471: 9470:complex plane 9467: 9448: 9440: 9436: 9430: 9427: 9424: 9416: 9413: 9410: 9402: 9398: 9386: 9383: 9370: 9365: 9362: 9359: 9355: 9347: 9343: 9337: 9325: 9322: 9319: 9311: 9308: 9305: 9297: 9293: 9281: 9278: 9265: 9260: 9257: 9254: 9250: 9243: 9237: 9222: 9221: 9220: 9203: 9195: 9191: 9184: 9176: 9173: 9170: 9166: 9160: 9155: 9152: 9149: 9145: 9137: 9133: 9127: 9114: 9106: 9103: 9100: 9096: 9090: 9085: 9082: 9079: 9075: 9068: 9062: 9047: 9046: 9045: 9042: 9038: 9031: 9027: 9022: 9020: 9016: 9007: 8998: 8976: 8968: 8933: 8932: 8931: 8929: 8910: 8897: 8893: 8890: 8887: 8866: 8863: 8860: 8856: 8834: 8831: 8817: 8805: 8800: 8797: 8794: 8790: 8786: 8781: 8778: 8775: 8761: 8760: 8759: 8756: 8752: 8748: 8744: 8739: 8734: 8729: 8724: 8719: 8714: 8709: 8705: 8701: 8694: 8687: 8683: 8679: 8655: 8652: 8649: 8646: 8643: 8640: 8637: 8634: 8631: 8627: 8622: 8609: 8606: 8603: 8598: 8595: 8592: 8589: 8586: 8580: 8575: 8572: 8569: 8556: 8553: 8550: 8546: 8541: 8536: 8530: 8517: 8516: 8515: 8512: 8496: 8493: 8490: 8461: 8458: 8422: 8416: 8404: 8396: 8393: 8390: 8387: 8384: 8372: 8369: 8366: 8361: 8358: 8355: 8351: 8347: 8345: 8334: 8326: 8323: 8320: 8317: 8314: 8303: 8297: 8284: 8281: 8278: 8273: 8270: 8267: 8264: 8261: 8255: 8250: 8247: 8244: 8231: 8228: 8225: 8221: 8215: 8209: 8206: 8203: 8198: 8195: 8192: 8188: 8184: 8182: 8172: 8159: 8151: 8148: 8145: 8142: 8139: 8136: 8133: 8120: 8117: 8114: 8109: 8106: 8103: 8095: 8090: 8087: 8084: 8080: 8076: 8071: 8058: 8050: 8047: 8044: 8041: 8038: 8025: 8022: 8019: 8014: 8011: 8008: 8005: 8002: 7994: 7989: 7986: 7983: 7979: 7975: 7973: 7963: 7950: 7942: 7939: 7936: 7924: 7919: 7916: 7913: 7909: 7905: 7902: 7897: 7884: 7876: 7873: 7870: 7858: 7853: 7850: 7847: 7843: 7836: 7833: 7830: 7824: 7822: 7814: 7795: 7794: 7793: 7767: 7764: 7761: 7758: 7755: 7752: 7749: 7736: 7733: 7730: 7725: 7722: 7719: 7713: 7708: 7705: 7702: 7692: 7683: 7680: 7677: 7674: 7671: 7658: 7655: 7652: 7647: 7644: 7641: 7638: 7635: 7629: 7624: 7621: 7618: 7605: 7602: 7599: 7590: 7569: 7566: 7563: 7560: 7557: 7554: 7551: 7533: 7530: 7519: 7514: 7511: 7508: 7498: 7486: 7483: 7480: 7475: 7472: 7469: 7451: 7448: 7445: 7442: 7439: 7421: 7418: 7407: 7402: 7399: 7396: 7383: 7380: 7377: 7363: 7359: 7356: 7353: 7338: 7329: 7328: 7327: 7325: 7320: 7305: 7299: 7289: 7284: 7274: 7270: 7263: 7260: 7254: 7249: 7243: 7216: 7210: 7200: 7197: 7192: 7181: 7174: 7171: 7165: 7160: 7154: 7119: 7107: 7103: 7099: 7095: 7089: 7079: 7074: 7064: 7060: 7053: 7050: 7042: 7038: 7031: 7028: 7025: 7019: 7016: 7012: 7006: 6996: 6993: 6988: 6977: 6970: 6967: 6961: 6956: 6948: 6945: 6942: 6936: 6933: 6928: 6916: 6908: 6905: 6902: 6896: 6894: 6884: 6872: 6868: 6864: 6860: 6854: 6844: 6839: 6829: 6825: 6819: 6815: 6808: 6805: 6802: 6796: 6792: 6786: 6776: 6773: 6768: 6757: 6753: 6748: 6740: 6737: 6734: 6728: 6723: 6711: 6703: 6700: 6697: 6691: 6689: 6679: 6667: 6663: 6659: 6654: 6642: 6638: 6631: 6628: 6625: 6619: 6616: 6611: 6601: 6596: 6588: 6585: 6582: 6576: 6571: 6559: 6555: 6548: 6545: 6542: 6536: 6531: 6521: 6516: 6508: 6505: 6502: 6496: 6493: 6488: 6476: 6468: 6465: 6462: 6456: 6454: 6447: 6435: 6431: 6427: 6422: 6412: 6406: 6403: 6400: 6394: 6391: 6386: 6374: 6366: 6363: 6360: 6346: 6345: 6344: 6342: 6338: 6320: 6307: 6299: 6296: 6293: 6266: 6260: 6249: 6246: 6240: 6226: 6223: 6220: 6214: 6208: 6191: 6189: 6185: 6181: 6171: 6169: 6165: 6161: 6157: 6153: 6148: 6146: 6142: 6138: 6134: 6130: 6126: 6116: 6114: 6104: 6100: 6099: 6095: 6091: 6090: 6087: 6079: 6075: 6073: 6070: 6069: 6065: 6061: 6059: 6056: 6052: 6051: 6048: 6043: 6036: 6029: 6022: 6015: 6008: 6001: 5994: 5987: 5977: 5973: 5971: 5968: 5967: 5963: 5959: 5957: 5954: 5950: 5949: 5946: 5941: 5934: 5927: 5920: 5913: 5898: 5894: 5892: 5889: 5888: 5884: 5880: 5878: 5875: 5871: 5870: 5864: 5857: 5853: 5846: 5842: 5838: 5834: 5828: 5821: 5817: 5810: 5806: 5800: 5793: 5789: 5782: 5778: 5777: 5776: 5774: 5771:such that as 5767: 5760: 5745: 5741: 5740: 5736: 5732: 5731: 5728: 5723: 5716: 5712: 5708: 5704: 5697: 5690: 5686: 5682: 5678: 5671: 5664: 5660: 5656: 5652: 5645:Linear curves 5623: 5615: 5605: 5600: 5597: 5594: 5578: 5570: 5567: 5564: 5561: 5558: 5554: 5548: 5545: 5542: 5537: 5534: 5531: 5527: 5523: 5520: 5514: 5507: 5494: 5493: 5492: 5490: 5477: 5473: 5470: 5469:perpendicular 5466: 5463: 5462: 5461: 5459: 5458:conic section 5455: 5446: 5434: 5430: 5426: 5422: 5421:local control 5418: 5415: 5411: 5394: 5384: 5379: 5376: 5347: 5337: 5332: 5329: 5326: 5299: 5296: 5293: 5289: 5286: 5282: 5279: 5275: 5272: 5268: 5265: 5262: 5259: 5234: 5223: 5215: 5212: 5209: 5205: 5199: 5196: 5192: 5188: 5183: 5180: 5177: 5163: 5160: 5157: 5153: 5147: 5143: 5139: 5111: 5107: 5104: 5101: 5090: 5087: 5083: 5079: 5075: 5060: 5042: 5031: 5028: 5024: 5019: 5004: 5000: 4996: 4992: 4988: 4984: 4969: 4966: 4963: 4943: 4939: 4933: 4929: 4925: 4922: 4902: 4898: 4894: 4891: 4888: 4885: 4861: 4857: 4853: 4847: 4844: 4839: 4836: 4825: 4806: 4801: 4797: 4794: 4789: 4783: 4779: 4767: 4763: 4759: 4755: 4752: 4749: 4745: 4742: 4738: 4734: 4731: 4713: 4684: 4669: 4668: 4657: 4648: 4646: 4645:empty product 4642: 4638: 4619: 4589: 4559: 4553: 4547: 4544: 4541: 4535: 4532: 4525: 4513: 4510: 4507: 4499: 4496: 4485: 4480: 4477: 4474: 4470: 4463: 4460: 4457: 4449: 4446: 4443: 4438: 4435: 4432: 4428: 4424: 4418: 4412: 4409: 4406: 4400: 4397: 4390: 4378: 4375: 4372: 4364: 4361: 4350: 4345: 4342: 4339: 4335: 4328: 4322: 4319: 4316: 4308: 4305: 4299: 4294: 4280: 4279: 4278: 4258: 4246: 4242: 4235: 4230: 4227: 4224: 4220: 4216: 4210: 4195: 4194: 4193: 4191: 4187: 4183: 4173: 4171: 4167: 4163: 4158: 4154: 4147: 4143: 4139: 4135: 4130: 4126: 4107: 4101: 4095: 4092: 4089: 4083: 4080: 4075: 4072: 4066: 4055: 4052: 4037: 4036: 4035: 4012: 4009: 3992: 3988: 3984: 3980: 3978: 3974: 3971:are known as 3955: 3952: 3949: 3946: 3943: 3940: 3937: 3925: 3920: 3917: 3914: 3906: 3903: 3900: 3892: 3888: 3876: 3873: 3862: 3856: 3848: 3845: 3842: 3838: 3830: 3829: 3828: 3811: 3808: 3805: 3802: 3799: 3787: 3782: 3769: 3761: 3758: 3755: 3751: 3745: 3740: 3737: 3734: 3730: 3726: 3720: 3705: 3704: 3703: 3677: 3674: 3671: 3668: 3665: 3659: 3654: 3642: 3638: 3634: 3629: 3616: 3613: 3610: 3602: 3598: 3594: 3591: 3586: 3574: 3566: 3563: 3560: 3552: 3548: 3544: 3541: 3536: 3524: 3516: 3513: 3510: 3502: 3498: 3494: 3491: 3486: 3474: 3466: 3463: 3460: 3454: 3451: 3448: 3443: 3431: 3423: 3420: 3417: 3411: 3409: 3401: 3382: 3381: 3380: 3378: 3373: 3371: 3346: 3343: 3307: 3304: 3301: 3298: 3295: 3289: 3284: 3272: 3268: 3264: 3259: 3256: 3253: 3241: 3238: 3235: 3231: 3224: 3221: 3218: 3203: 3200: 3197: 3193: 3182: 3179: 3176: 3171: 3161: 3156: 3153: 3150: 3142: 3139: 3136: 3122: 3119: 3108: 3103: 3091: 3083: 3080: 3077: 3071: 3069: 3059: 3047: 3043: 3037: 3034: 3031: 3023: 3020: 3017: 3003: 3000: 2987: 2982: 2979: 2976: 2972: 2968: 2966: 2958: 2939: 2938: 2937: 2929: 2927: 2905: 2895: 2885: 2880: 2868: 2851: 2848: 2842: 2832: 2829: 2826: 2816: 2811: 2799: 2779: 2776: 2773: 2767: 2761: 2751: 2741: 2736: 2724: 2707: 2701: 2686: 2685: 2663: 2653: 2647: 2637: 2616: 2615: 2614: 2611: 2607: 2600: 2593: 2573: 2563: 2558: 2546: 2523: 2521: 2517: 2513: 2509: 2499: 2497: 2473: 2464: 2454: 2449: 2439: 2436: 2431: 2418: 2415: 2412: 2404: 2394: 2389: 2379: 2376: 2371: 2355: 2352: 2349: 2343: 2340: 2334: 2327: 2314: 2313: 2312: 2310: 2291: 2282: 2272: 2267: 2252: 2248: 2244: 2241: 2233: 2223: 2218: 2205: 2199: 2196: 2193: 2187: 2184: 2176: 2166: 2161: 2146: 2138: 2135: 2132: 2126: 2123: 2117: 2110: 2097: 2096: 2095: 2093: 2088: 2086: 2082: 2076: 2074: 2067: 2060: 2041: 2038: 2035: 2032: 2029: 2023: 2018: 2006: 2002: 1998: 1993: 1981: 1977: 1970: 1967: 1964: 1958: 1955: 1950: 1940: 1935: 1927: 1924: 1921: 1915: 1912: 1907: 1895: 1887: 1884: 1881: 1875: 1869: 1854: 1853: 1852: 1835: 1832: 1829: 1826: 1823: 1817: 1811: 1801: 1791: 1786: 1776: 1771: 1754: 1751: 1745: 1735: 1725: 1720: 1710: 1705: 1685: 1682: 1679: 1673: 1667: 1652: 1651: 1650: 1647: 1643: 1638: 1634: 1629: 1625: 1621: 1615: 1611: 1606: 1602: 1597: 1593: 1589: 1584: 1579: 1572: 1565: 1558: 1551: 1544: 1537: 1530: 1523: 1520:going toward 1516: 1509: 1502: 1495: 1488: 1464: 1456: 1446: 1441: 1431: 1428: 1423: 1410: 1407: 1401: 1394: 1381: 1380: 1379: 1377: 1372: 1367: 1360: 1353: 1346: 1342: 1335: 1332:intersect at 1328: 1321: 1317: 1298: 1290: 1280: 1275: 1262: 1259: 1256: 1248: 1238: 1233: 1217: 1214: 1211: 1205: 1202: 1196: 1189: 1176: 1175: 1174: 1172: 1168: 1149: 1146: 1143: 1140: 1137: 1131: 1123: 1113: 1108: 1093: 1089: 1085: 1077: 1067: 1062: 1047: 1039: 1036: 1033: 1027: 1022: 1012: 1006: 991: 990: 989: 984: 965: 962: 959: 956: 953: 947: 942: 930: 926: 922: 917: 907: 901: 898: 895: 889: 886: 881: 869: 861: 858: 855: 849: 843: 828: 827: 826: 821: 814: 807: 800: 796: 776: 773: 770: 767: 764: 758: 750: 740: 737: 732: 719: 716: 713: 704: 701: 693: 683: 680: 675: 662: 659: 656: 644: 641: 638: 632: 626: 611: 610: 609: 604: 597: 590: 586: 582: 579: 571: 567: 563: 559: 554: 545: 543: 525: 515: 510: 495: 476: 473: 470: 467: 464: 458: 453: 443: 440: 435: 422: 419: 416: 410: 402: 392: 387: 374: 371: 366: 356: 350: 335: 334: 333: 331: 324: 317: 307: 305: 301: 297: 292: 288: 281: 278: 277: 272: 262: 260: 256: 252: 251:Pierre Bézier 248: 244: 240: 236: 232: 228: 224: 214: 211: 209: 204: 200: 195: 193: 189: 184: 182: 178: 174: 173:Bézier spline 170: 166: 162: 161:Pierre Bézier 158: 153: 149: 145: 144: 135: 102: 93: 89: 82: 78: 74: 67: 63: 59: 51: 47: 39: 35:on the range 34: 29: 21: 11302: 11262: 11256: 11240:the original 11220: 11193: 11182: 11156: 11141: 11090: 11078:. Retrieved 11071:the original 11062: 11051: 11048:Donald Knuth 11040:the original 11034: 11016: 11012: 10959: 10953: 10918: 10894:. Retrieved 10890:the original 10885: 10876: 10867: 10860:. Retrieved 10856:the original 10851: 10842: 10827:cl.cam.ac.uk 10826: 10813: 10802:. Retrieved 10798: 10789: 10778:. Retrieved 10774: 10765: 10756: 10734: 10715: 10705: 10676: 10659: 10655: 10649: 10638:. Retrieved 10628: 10611: 10607: 10601: 10584: 10580: 10574: 10565: 10556: 10533: 10506: 10502: 10489: 10475:(3): 62–71. 10472: 10468: 10451: 10424: 10414: 10402:the original 10388: 10358:. Retrieved 10348: 10321: 10315: 10307:the original 10295: 10263: 10256: 10236: 10226: 10218:the original 10207: 10187: 10180: 10161: 10155: 10135: 10128: 10095: 10091: 10085: 10074:. 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5759:Q 5744:t 5725:1 5722:P 5718:0 5715:P 5711:t 5709:( 5707:B 5703:t 5699:1 5696:P 5692:0 5689:P 5685:t 5683:( 5681:B 5677:t 5673:1 5670:P 5666:0 5663:P 5659:t 5657:( 5655:B 5651:t 5624:. 5621:) 5616:i 5611:P 5601:1 5598:+ 5595:i 5590:P 5585:( 5582:) 5579:t 5576:( 5571:1 5565:n 5562:, 5559:i 5555:b 5549:1 5543:n 5538:0 5535:= 5532:i 5524:n 5521:= 5518:) 5515:t 5512:( 5504:B 5489:n 5425:n 5409:. 5395:n 5390:P 5380:1 5372:P 5348:0 5343:P 5333:1 5330:+ 5327:n 5322:P 5300:1 5297:+ 5294:n 5290:, 5287:n 5283:, 5276:, 5273:1 5269:, 5266:0 5263:= 5260:k 5235:k 5230:P 5224:) 5216:1 5213:+ 5210:n 5206:k 5197:1 5193:( 5189:+ 5184:1 5178:k 5173:P 5164:1 5161:+ 5158:n 5154:k 5148:= 5140:k 5135:P 5108:1 5105:+ 5102:n 5097:P 5091:, 5084:, 5076:0 5071:P 5059:n 5043:n 5038:P 5032:, 5025:, 5020:0 5015:P 5003:n 4999:n 4995:m 4991:m 4987:n 4982:. 4970:2 4964:n 4944:n 4940:/ 4926:= 4923:t 4903:n 4899:/ 4892:2 4889:= 4886:t 4865:) 4862:4 4858:/ 4854:t 4851:( 4840:3 4837:4 4824:n 4807:3 4802:) 4798:1 4790:2 4784:( 4780:4 4743:. 4714:n 4709:P 4685:0 4680:P 4639:( 4623:) 4620:t 4617:( 4613:B 4590:j 4585:C 4560:. 4554:! 4551:) 4548:i 4542:j 4539:( 4536:! 4533:i 4526:i 4521:P 4514:j 4511:+ 4508:i 4504:) 4500:1 4494:( 4486:j 4481:0 4478:= 4475:i 4467:) 4464:m 4458:n 4455:( 4450:1 4444:j 4439:0 4436:= 4433:m 4425:= 4419:! 4416:) 4413:i 4407:j 4404:( 4401:! 4398:i 4391:i 4386:P 4379:j 4376:+ 4373:i 4369:) 4365:1 4359:( 4351:j 4346:0 4343:= 4340:i 4329:! 4326:) 4323:j 4317:n 4314:( 4309:! 4306:n 4300:= 4295:j 4290:C 4259:j 4254:C 4247:j 4243:t 4236:n 4231:0 4228:= 4225:j 4217:= 4214:) 4211:t 4208:( 4204:B 4157:n 4153:P 4149:0 4146:P 4129:i 4125:P 4108:. 4102:! 4099:) 4096:i 4090:n 4087:( 4084:! 4081:i 4076:! 4073:n 4067:= 4061:) 4056:i 4053:n 4048:( 4018:) 4013:i 4010:n 4005:( 3987:t 3983:t 3977:n 3956:n 3953:, 3947:, 3944:0 3941:= 3938:i 3926:, 3921:i 3915:n 3911:) 3907:t 3901:1 3898:( 3893:i 3889:t 3882:) 3877:i 3874:n 3869:( 3863:= 3860:) 3857:t 3854:( 3849:n 3846:, 3843:i 3839:b 3812:1 3806:t 3800:0 3788:, 3783:i 3778:P 3773:) 3770:t 3767:( 3762:n 3759:, 3756:i 3752:b 3746:n 3741:0 3738:= 3735:i 3727:= 3724:) 3721:t 3718:( 3714:B 3672:t 3666:0 3660:, 3655:5 3650:P 3643:5 3639:t 3635:+ 3630:4 3625:P 3620:) 3617:t 3611:1 3608:( 3603:4 3599:t 3595:5 3592:+ 3587:3 3582:P 3575:2 3571:) 3567:t 3561:1 3558:( 3553:3 3549:t 3542:+ 3537:2 3532:P 3525:3 3521:) 3517:t 3511:1 3508:( 3503:2 3499:t 3492:+ 3487:1 3482:P 3475:4 3471:) 3467:t 3461:1 3458:( 3455:t 3452:5 3449:+ 3444:0 3439:P 3432:5 3428:) 3424:t 3418:1 3415:( 3412:= 3405:) 3402:t 3399:( 3395:B 3377:n 3352:) 3347:i 3344:n 3339:( 3308:1 3302:t 3296:0 3290:, 3285:n 3280:P 3273:n 3269:t 3265:+ 3260:1 3254:n 3249:P 3242:1 3236:n 3232:t 3228:) 3225:t 3219:1 3216:( 3210:) 3204:1 3198:n 3194:n 3189:( 3183:+ 3177:+ 3172:1 3167:P 3162:t 3157:1 3151:n 3147:) 3143:t 3137:1 3134:( 3128:) 3123:1 3120:n 3115:( 3109:+ 3104:0 3099:P 3092:n 3088:) 3084:t 3078:1 3075:( 3072:= 3060:i 3055:P 3048:i 3044:t 3038:i 3032:n 3028:) 3024:t 3018:1 3015:( 3009:) 3004:i 3001:n 2996:( 2988:n 2983:0 2980:= 2977:i 2969:= 2962:) 2959:t 2956:( 2952:B 2909:) 2906:t 2903:( 2896:n 2891:P 2881:2 2876:P 2869:1 2864:P 2857:B 2852:t 2849:+ 2846:) 2843:t 2840:( 2833:1 2827:n 2822:P 2812:1 2807:P 2800:0 2795:P 2788:B 2783:) 2780:t 2774:1 2771:( 2768:= 2765:) 2762:t 2759:( 2752:n 2747:P 2737:1 2732:P 2725:0 2720:P 2713:B 2708:= 2705:) 2702:t 2699:( 2695:B 2664:0 2659:P 2654:= 2651:) 2648:t 2645:( 2638:0 2633:P 2626:B 2610:k 2606:P 2602:1 2599:P 2595:0 2592:P 2574:k 2569:P 2559:1 2554:P 2547:0 2542:P 2535:B 2520:n 2514:( 2508:n 2496:n 2474:. 2470:) 2465:1 2460:P 2455:+ 2450:2 2445:P 2440:2 2432:3 2427:P 2422:( 2419:t 2416:6 2413:+ 2410:) 2405:0 2400:P 2395:+ 2390:1 2385:P 2380:2 2372:2 2367:P 2362:( 2359:) 2356:t 2350:1 2347:( 2344:6 2341:= 2338:) 2335:t 2332:( 2324:B 2309:t 2292:. 2288:) 2283:2 2278:P 2268:3 2263:P 2258:( 2253:2 2249:t 2245:3 2242:+ 2239:) 2234:1 2229:P 2219:2 2214:P 2209:( 2206:t 2203:) 2200:t 2194:1 2191:( 2188:6 2185:+ 2182:) 2177:0 2172:P 2162:1 2157:P 2152:( 2147:2 2143:) 2139:t 2133:1 2130:( 2127:3 2124:= 2121:) 2118:t 2115:( 2107:B 2092:t 2085:t 2081:t 2069:2 2066:P 2062:1 2059:P 2036:t 2030:0 2024:, 2019:3 2014:P 2007:3 2003:t 1999:+ 1994:2 1989:P 1982:2 1978:t 1974:) 1971:t 1965:1 1962:( 1959:3 1956:+ 1951:1 1946:P 1941:t 1936:2 1932:) 1928:t 1922:1 1919:( 1916:3 1913:+ 1908:0 1903:P 1896:3 1892:) 1888:t 1882:1 1879:( 1876:= 1873:) 1870:t 1867:( 1863:B 1830:t 1824:0 1818:, 1815:) 1812:t 1809:( 1802:3 1797:P 1792:, 1787:2 1782:P 1777:, 1772:1 1767:P 1760:B 1755:t 1752:+ 1749:) 1746:t 1743:( 1736:2 1731:P 1726:, 1721:1 1716:P 1711:, 1706:0 1701:P 1694:B 1689:) 1686:t 1680:1 1677:( 1674:= 1671:) 1668:t 1665:( 1661:B 1646:k 1642:P 1637:j 1633:P 1628:i 1624:P 1620:t 1618:( 1614:k 1610:P 1608:, 1605:j 1601:P 1599:, 1596:i 1592:P 1588:B 1581:2 1578:P 1574:1 1571:P 1567:2 1564:P 1560:1 1557:P 1553:2 1550:P 1546:1 1543:P 1539:2 1536:P 1532:3 1529:P 1525:1 1522:P 1518:0 1515:P 1511:3 1508:P 1504:2 1501:P 1497:1 1494:P 1490:0 1487:P 1465:. 1462:) 1457:0 1452:P 1447:+ 1442:1 1437:P 1432:2 1424:2 1419:P 1414:( 1411:2 1408:= 1405:) 1402:t 1399:( 1391:B 1376:t 1369:1 1366:P 1362:2 1359:P 1355:1 1352:P 1348:0 1345:P 1341:t 1337:1 1334:P 1330:2 1327:P 1323:0 1320:P 1299:, 1296:) 1291:1 1286:P 1276:2 1271:P 1266:( 1263:t 1260:2 1257:+ 1254:) 1249:0 1244:P 1234:1 1229:P 1224:( 1221:) 1218:t 1212:1 1209:( 1206:2 1203:= 1200:) 1197:t 1194:( 1186:B 1171:t 1144:t 1138:0 1132:, 1129:) 1124:1 1119:P 1109:2 1104:P 1099:( 1094:2 1090:t 1086:+ 1083:) 1078:1 1073:P 1063:0 1058:P 1053:( 1048:2 1044:) 1040:t 1034:1 1031:( 1028:+ 1023:1 1018:P 1013:= 1010:) 1007:t 1004:( 1000:B 986:1 983:P 960:t 954:0 948:, 943:2 938:P 931:2 927:t 923:+ 918:1 913:P 908:t 905:) 902:t 896:1 893:( 890:2 887:+ 882:0 877:P 870:2 866:) 862:t 856:1 853:( 850:= 847:) 844:t 841:( 837:B 823:2 820:P 816:1 813:P 809:1 806:P 802:0 799:P 789:, 777:1 771:t 765:0 759:, 756:] 751:2 746:P 741:t 738:+ 733:1 728:P 723:) 720:t 714:1 711:( 708:[ 705:t 702:+ 699:] 694:1 689:P 684:t 681:+ 676:0 671:P 666:) 663:t 657:1 654:( 651:[ 648:) 645:t 639:1 636:( 633:= 630:) 627:t 624:( 620:B 606:2 603:P 599:1 596:P 592:0 589:P 585:t 583:( 581:B 570:⋯ 566:× 562:• 526:0 521:P 511:1 506:P 477:1 471:t 465:0 459:, 454:1 449:P 444:t 441:+ 436:0 431:P 426:) 423:t 417:1 414:( 411:= 408:) 403:0 398:P 388:1 383:P 378:( 375:t 372:+ 367:0 362:P 357:= 354:) 351:t 348:( 344:B 326:1 323:P 319:0 316:P 300:n 296:n 291:n 287:P 283:0 280:P 134:/ 128:. 125:i 122:. 119:z 116:ɛ 113:b 110:ˈ 107:/ 103:( 95:. 92:t 88:y 81:t 79:) 77:t 73:y 66:t 64:) 62:t 58:y 52:) 50:t 46:y 38:t

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