Cesàro Equation

An Cesàro equation is a natural equation which expresses a curve in terms of its arc length function s(t) and radius of curvature rho(t) (or equivalently, the curvature kappa(t)=1/rho(t)). Note that while the Cesàro equation is said to be intrinsic because it is invariant under transformations that preserve length and angle, it is not intrinsic to a curve because it depends on the starting point from which arc length is measured and hence on the parametrization (see the table below for examples).

The following table summarizes the Cesàro equations for certain parametrizations of a number of curves (cf. Lawrence 1972, p. 5 and Yates 1952, p. 126).

curve parametrization Cesàro equation

astroid x=acos^3t rho^2+4s^2=6as

y=asin^3t

cardioid x=a(1-cost)cost 9rho^2+s^2=8as

y=a(1-cost)sint

cardioid x=2a(1+cost)cost 9rho^2+s^2=64a^2

y=2a(1+cost)sint

catenary x=t a^2+s^2=arho

y=acosh(t/a)

circle x=acost rho=a

y=asint

circle involute x=a(tsint+cost) rho^2=2as

y=a(sint-tcost)

cycloid x=a(t-sint) rho^2+s^2=8as

y=a(1-cost)

cycloid x=a(t+sint) rho^2+s^2=16a^2

y=a(1+cost)

deltoid x=1/3a[2cost+cos(2t)] rho^2+9s^2=16as

y=1/3a[2sint-sin(2t)]

nephroid x=a[3cost-cos(3t)] 4rho^2+s^2=12as

y=a[3sint-sin(3t)]

tractrix x=a(t-tanht) rho^2+a^2=a^2e^(2s/a)

y=asecht

Tschirnhausen cubic x=a(1-3t^2) arho(8rho^2+108(4a^2+s^2))

y=at(3-t^2) =27(4a^2+s^2)^2+108a^2rho^2

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 4-5, 1972.Yates, R. C. "Intrinsic Equations." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 123-126, 1952.

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Cesàro Equation

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Weisstein, Eric W. "Cesàro Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CesaroEquation.html

Cesàro Equation

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