# Whewell equation

﻿
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

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

which implies

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

Parametric equations for the curve can be obtained by integrating:

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

Since

: $kappa = frac\left\{dvarphi\right\}\left\{ds\right\},$

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

*

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Cesàro equation — In geometry, the Cesàro equation of a plane curve is an equation relating curvature (κ) to arc length (s). It may also be given as an equation relating the radius of curvature (R) to arc length. (These are equivalent because R = 1 / κ.) Two… …   Wikipedia

• Intrinsic equation — In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve s intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve.… …   Wikipedia

• Construction of an equation — Construction Con*struc tion, n. [L. constructio: cf. F. construction.] 1. The process or art of constructing; the act of building; erection; the act of devising and forming; fabrication; composition. [1913 Webster] 2. The form or manner of… …   The Collaborative International Dictionary of English

• Catenary — This article is about the mathematical curve. For other uses, see Catenary (disambiguation). Chainette redirects here. For the wine grape also known as Chainette, see Cinsaut. A hanging chain forms a catenary …   Wikipedia

• Evolute — In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve. The original curve is an involute of its evolute. (Compare and… …   Wikipedia

• List of mathematics articles (W) — NOTOC Wad Wadge hierarchy Wagstaff prime Wald test Wald Wolfowitz runs test Wald s equation Waldhausen category Wall Sun Sun prime Wallenius noncentral hypergeometric distribution Wallis product Wallman compactification Wallpaper group Walrasian… …   Wikipedia

• Coordinate system — For geographical coordinates on Wikipedia, see Wikipedia:WikiProject Geographical coordinates. In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other… …   Wikipedia

• List of curve topics — This is a list of curve topics in mathematics. See also curve, list of curves, and list of differential geometry topics.*acnode *algebraic curve *arc *asymptote *asymptotic curve *Barbier s theorem *barycentric… …   Wikipedia

• List of curves topics — This is a list of curve topics in mathematics. See also curve, list of curves, and list of differential geometry topics. acnode algebraic curve arc asymptote asymptotic curve Barbier s theorem barycentric[1] Bézier curve Bézout s theorem Birch… …   Wikipedia

• nature, philosophy of — Introduction       the discipline that investigates substantive issues regarding the actual features of nature as a reality. The discussion here is divided into two parts: the philosophy of physics and the philosophy of biology.       In this… …   Universalium