- Whewell equation
The Whewell equation of a
plane curveis an equationthat relates the tangential angle() with arclength(), 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 equationof 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 equationfor the same curve.
The term is named after
William Whewell, who introduced the concept in 1849, in a paper in the Cambridge Philosophical Transactions.
If the curve is given parametrically in terms of the arc length , then is determined by
Parametric equations for the curve can be obtained by integrating:
Cesàro equationis easily obtained by differentiating the Whewell equation.
* 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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