Whewell equation

The Whewell equation of a plane curve is an equation that relates the tangential angle ($varphi$) with arclength ($s$), 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 an intrinsic equation of the curve, or, less precisely, the intrinsic 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 a Cesàro 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.

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

External links

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