Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
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A term applied to a curve C′ that cuts at right angles all tangents of a curve C (see illustration). Each curve C has infinitely many involutes and the distance between corresponding points of any two involutes is constant. If xi = xi (s), with i = 1, 2, 3, are parametric equations of C, with parameter s arc length on C, all involutes C′ of C have parametric equations Xi = x i (s) + (k − s) x′ i (s), with i = 1, 2, 3, where x′ i = , with i = 1, 2, 3, and k denotes an arbitrary constant. Let a length of string be coincident with a curve C, with one end fastened at a point P0 of C. If the string is unwound, remaining taut, the other end of the string traces an involute C ′ of C. By varying the length of the string, all involutes of C are obtained. See also: Analytic geometry
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