Involute

Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment.Then wind the string up, keeping it always taut. The Locus of points traced out by the end of the string is theinvolute of the original curve, and the original curve is called the Evolute of its involute. Although a curve has aunique Evolute, it has infinitely many involutes corresponding to different choices of initial point. An involutecan also be thought of as any curve Orthogonal to all the Tangents to agiven curve.

The equation of the involute is

(1)

(2)

(3)

			(4)
			(5)

References

Cundy, H. and Rollett, A. ``Roulettes and Involutes.'' §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46-55, 1989.

Dixon, R. ``String Drawings.'' Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.

Gray, A. ``Involutes.'' §5.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 81-85, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40-42 and 202, 1972.

Lee, X. ``Involute.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Involute_dir/involute.html.

Lockwood, E. H. ``Evolutes and Involutes.'' Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.

Pappas, T. ``The Involute.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 187, 1989.

Yates, R. C. ``Involutes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 135-137, 1952.

• Determinant
• Determinant (Binary Quadratic Form)
• Determinant Expansion by Minors
• Determinant (Knot)
• Determinant Theorem
• Developable Surface
• Deviation
• Devil on Two Sticks
• Devil's Curve
• Devil's Staircase
• Diabolical Cube
• Diabolical Square
• Diabolo
• Diacaustic
• Diagonalization
• Diagonal Matrix
• Diagonal Metric
• Diagonal (Polygon)
• Diagonal (Polyhedron)
• Diagonal Ramsey Number
• Diagonal Slash
• Diagonal (Solidus)
• Diagonals Problem
• Diagram
• Diameter