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can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from
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lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are
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The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the affine plane and the line at infinity makes the
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In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are
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can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two
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The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a
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These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough
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the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
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Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the
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Solving the equation, therefore, we find that all circles 'pass through' the
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19:"Ideal line" redirects here. For the ideal line in racing, see
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and note that the line at infinity is specified by setting
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Making equations homogeneous by introducing powers of
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302:Elliptic geometry § Elliptic plane
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132:{\displaystyle \mathbb {R} P^{2}}
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84:y-intercept
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369:Categories
328:References
187:orientable
148:asymptotes
75:parallel.
60:ideal line
174:(so four
144:hyperbola
52:incidence
385:Infinity
296:See also
183:manifold
152:parabola
72:parallel
32:topology
28:geometry
268:= and
193:History
180:compact
48:closure
168:sphere
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