TheWhewell equation of aplane curve is anequation that relates thetangential angle (φ) witharc length (s), where the tangential angle is the angle between the tangent to the curve at some point and thex-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 thex-axis, so this is anintrinsic equation of the curve, or, less precisely,the intrinsic equation. If one curve is obtained from another curve bytranslation then their Whewell equations will be the same.

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 thecurvature. Thus, taking the derivative of the Whewell equation yields aCesàro equation for the same curve.

The concept is named afterWilliam Whewell, who introduced it in 1849, in a paper in theCambridge Philosophical Transactions. 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.

If a point${\displaystyle \overrightarrow{r}=(x,y)}$ on the curve is givenparametrically in terms of the arc length,${\displaystyle s\mapsto \overrightarrow{r},}$ then the tangential angleφ is determined by

$${\displaystyle \frac{d\overrightarrow{r}}{ds}=\left(\begin{array}{c}\frac{dx}{ds}\\ \frac{dy}{ds}\end{array}\right)=\left(\begin{array}{c}\cos \phi \\ \sin \phi \end{array}\right)\text{since}\left|\frac{d\overrightarrow{r}}{ds}\right|=1,}$$

which implies$${\displaystyle \frac{dy}{dx}=\tan \phi .}$$

Parametric equations for the curve can be obtained by integrating:

Since thecurvature is defined by$${\displaystyle \kappa =\frac{d\phi }{ds},}$$

theCesàro equation is easily obtained by differentiating the Whewell equation.

Curve	Equation
Line	${\displaystyle \phi =c}$
Circle	${\displaystyle s=a\phi }$
Logarithmic Spiral	${\displaystyle s=\frac{a{e}^{\phi \tan \alpha }}{\sin \alpha }}$
Catenary	${\displaystyle s=a\tan \phi }$
Tautochrone	${\displaystyle s=a\sin \phi }$

• Whewell, W. Of the Intrinsic Equation of a Curve, and its Application. Cambridge Philosophical Transactions, Vol. VIII, pp. 659-671, 1849.Google Books
• 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.
• J. Dennis Lawrence (1972).A catalog of special plane curves. Dover Publications. pp. 1–5.ISBN 0-486-60288-5.
• Yates, R. C.:A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Intrinsic Equations" p124-5

• Weisstein, Eric W."Whewell Equation".MathWorld.

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