- Whewell equation
The

**Whewell equation**of aplane curve is anequation that relates thetangential angle ($varphi$) witharclength ($s$), where the tangential angle is angle between the tangent to the curve and the x-axis and the arc length is the distance along the curve from a fixed point. These quantities are do not depend on the coordinate system used except for the choice of the direction of the x-axis, so this is anintrinsic equation of the curve, or, less precisely, theintrinsic equation . If a curve is obtained from another by translation then their Whewell equations will be the same.When the relation is a function, so that tangential angle is given as a function of arclength, certain properties become easy to manipulate. In particular, the derivative of the tangential angle with respect to arclength is equal to the

curvature . Thus, taking the derivative of the Whewell equation yields aCesàro equation for the same curve.The term is named after

William Whewell , who introduced the concept in 1849, in a paper in the Cambridge Philosophical Transactions.**Properties**If the curve is given parametrically in terms of the arc length $s$, then $varphi$ is determined by

: $left(frac\{dx\}\{ds\},\; frac\{dy\}\{ds\}\; ight)\; =\; (cos\; varphi,\; sin\; varphi),$

which implies

: $frac\{dy\}\{dx\}\; =\; an\; varphi.$

Parametric equations for the curve can be obtained by integrating:

: $x\; =\; int\; cos\; varphi\; ,\; ds$: $y\; =\; int\; sin\; varphi\; ,\; ds$

Since

: $kappa\; =\; frac\{dvarphi\}\{ds\},$

the

Cesàro equation is easily obtained by differentiating the Whewell equation.**Examples****References*** Whewell, W. Of the Intrinsic Equation of a Curve, and its Application. Cambridge Philosophical Transactions, Vol. VIII, pp. 659-671, 1849.

* 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.

*

* Yates, R. C.: "A Handbook on Curves and Their Properties", J. W. Edwards (1952), "Intrinsic Equations" p124-5

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