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| [[Image:Intrinsic coordinates (Whewell equation).png|thumb|300px|Important quantities in the Whewell equation]]
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| The '''Whewell equation''' of a [[plane curve]] is an [[equation]] that relates the [[tangential angle]] (<math>\varphi</math>) with [[arclength]] (<math>s</math>), where the tangential angle is the 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 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.
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| 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.
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| The term is named after [[William Whewell]], who introduced the concept in 1849, in a paper in the [[Cambridge Philosophical Society|Cambridge 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.
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| ==Properties==
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| If the curve is given parametrically in terms of the arc length <math>s</math>, then <math>\varphi</math> is determined by
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| : <math>\frac {d \vec r}{ds} = \begin{pmatrix} dx/ds \\ dy/ds \end{pmatrix} = \begin{pmatrix} \cos \varphi \\ \sin \varphi \end{pmatrix} \quad \text {since} \quad \left | \frac {d \vec r}{ds} \right | = 1 ,</math>
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| which implies
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| : <math>\frac{dy}{dx} = \tan \varphi.</math>
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| Parametric equations for the curve can be obtained by integrating:
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| : <math>x = \int \cos \varphi \, ds</math>
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| : <math>y = \int \sin \varphi \, ds</math>
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| Since the [[curvature]] is defined by
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| : <math>\kappa = \frac{d\varphi}{ds},</math>
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| the [[Cesàro equation]] is easily obtained by differentiating the Whewell equation.
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| ==Examples==
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| {| class="wikitable" border="1"
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| ! Curve
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| ! Equation
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| | [[Line (mathematics)|Line]]
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| | <math>\varphi = c</math>
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| |-
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| | [[Circle]]
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| | <math>s = a\varphi</math>
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| |-
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| | [[Catenary]]
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| | <math>s = a\tan \varphi</math>
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| |}
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| ==References==
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| * Whewell, W. Of the Intrinsic Equation of a Curve, and its Application. Cambridge Philosophical Transactions, Vol. VIII, pp. 659-671, 1849. [http://books.google.com/books?id=2vsIAAAAIAAJ&pg=PA659 Google Books]
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| * 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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| * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=1–5 }}
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| * Yates, R. C.: ''A Handbook on Curves and Their Properties'', J. W. Edwards (1952), "Intrinsic Equations" p124-5
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| ==External links==
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| * {{MathWorld | title=Whewell Equation | urlname=WhewellEquation }}
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| [[Category:Curves]]
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