20:
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98:. In his conception, the angle used is the deviation from the direction of the curve at some fixed starting point, and this convention is sometimes used by other authors as well. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.
551:
78:
When the relation is a function, so that tangential angle is given as a function of arc length, certain properties become easy to manipulate. In particular, the derivative of the tangential angle with respect to arc length is equal to the
397:{\displaystyle {\frac {d{\vec {r}}}{ds}}={\begin{pmatrix}{\frac {dx}{ds}}\\{\frac {dy}{ds}}\end{pmatrix}}={\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad {\text{since}}\quad \left|{\frac {d{\vec {r}}}{ds}}\right|=1,}
765:
870:
Todhunter, Isaac. William
Whewell, D.D., An Account of His Writings, with Selections from His Literary and Scientific Correspondence. Vol. I. Macmillan and Co., 1876, London. Section 56: p. 317.
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59:-axis, and the arc length is the distance along the curve from a fixed point. These quantities do not depend on the coordinate system used except for the choice of the direction of the
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Whewell, W. Of the
Intrinsic Equation of a Curve, and its Application. Cambridge Philosophical Transactions, Vol. VIII, pp. 659-671, 1849.
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546:{\displaystyle {\begin{aligned}x&=\int \cos \varphi \,ds,\\y&=\int \sin \varphi \,ds.\end{aligned}}}
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760:{\displaystyle s={\frac {ae^{\varphi \tan \alpha }}{\sin \alpha }}}
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Parametric equations for the curve can be obtained by integrating:
71:
intrinsic equation. If one curve is obtained from another curve by
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83:. Thus, taking the derivative of the Whewell equation yields a
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is easily obtained by differentiating the
Whewell equation.
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598:{\displaystyle \kappa ={\frac {d\varphi }{ds}},}
451:{\displaystyle {\frac {dy}{dx}}=\tan \varphi .}
94:, who introduced it in 1849, in a paper in the
75:then their Whewell equations will be the same.
23:Important quantities in the Whewell equation
902:A Handbook on Curves and Their Properties
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188:{\displaystyle s\mapsto {\vec {r}},}
96:Cambridge Philosophical Transactions
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67:of the curve, or, less precisely,
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878:A catalog of special plane curves
846:{\displaystyle s=a\sin \varphi }
803:{\displaystyle s=a\tan \varphi }
146:{\displaystyle {\vec {r}}=(x,y)}
882:. Dover Publications. pp.
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157:in terms of the arc length,
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690:{\displaystyle s=a\varphi }
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653:{\displaystyle \varphi =c}
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16:Mathematical equation
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920:"Whewell Equation"
917:Weisstein, Eric W.
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85:CesĂ ro equation
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815:Tautochrone
106:If a point
73:translation
33:plane curve
938:Categories
859:References
555:Since the
102:Properties
49:arc length
925:MathWorld
841:φ
838:
798:φ
795:
752:α
749:
739:α
736:
730:φ
685:φ
642:φ
625:Equation
579:φ
567:κ
557:curvature
527:φ
524:
518:∫
491:φ
488:
482:∫
443:φ
440:
365:→
331:φ
328:
318:φ
315:
221:→
177:→
168:↦
120:→
81:curvature
772:Catenary
615:Examples
37:equation
47:) with
944:Curves
890:
665:Circle
622:Curve
35:is an
344:since
31:of a
888:ISBN
631:Line
607:the
27:The
884:1–5
835:sin
792:tan
746:sin
733:tan
521:sin
485:cos
437:tan
325:sin
312:cos
69:the
940::
922:.
886:.
928:.
896:.
832:a
829:=
826:s
789:a
786:=
783:s
726:e
722:a
716:=
713:s
682:a
679:=
676:s
648:c
645:=
593:,
587:s
584:d
576:d
570:=
537:.
534:s
531:d
515:=
508:y
501:,
498:s
495:d
479:=
472:x
446:.
434:=
428:x
425:d
420:y
417:d
392:,
389:1
386:=
382:|
376:s
373:d
362:r
356:d
350:|
337:)
306:(
301:=
296:)
287:s
284:d
279:y
276:d
263:s
260:d
255:x
252:d
243:(
238:=
232:s
229:d
218:r
212:d
197:φ
183:,
174:r
165:s
141:)
138:y
135:,
132:x
129:(
126:=
117:r
61:x
57:x
53:s
51:(
45:φ
43:(
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