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Relates evolutes to single paths in the calculus of variations. Proved in the general case by Darboux and Zermelo in 1894 and Kneser in 1898. It states: "When a single ...

A 1-cusped epicycloid has b=a, so n=1. The radius measured from the center of the large circle for a 1-cusped epicycloid is given by epicycloid equation (◇) with n=1 so r^2 = ...

The evolute of the epicycloid x = (a+b)cost-bcos[((a+b)/b)t] (1) y = (a+b)sint-bsin[((a+b)/b)t] (2) is another epicycloid given by x = a/(a+2b){(a+b)cost+bcos[((a+b)/b)t]} ...

The involute of the epicycloid x = (a+b)cost-bcos[((a+b)/b)t] (1) y = (a+b)sint-bsin[((a+b)/b)t] (2) is another epicycloid given by x = (a+2b)/a{(a+b)cost+bcos[((a+b)/b)t]} ...

The radial curve of an epicycloid is shown above for an epicycloid with four cusps. Although it is claimed to be a rose curve by Lawrence (1972), it is not.

The epispiral is a plane curve with polar equation r=asec(ntheta). There are n sections if n is odd and 2n if n is even. A slightly more symmetric version considers instead ...

The inverse curve of the epispiral r=asec(ntheta) with inversion center at the origin and inversion radius k is the rose curve r=(kcos(ntheta))/a.

The parametric equations of the evolute of an epitrochoid specified by circle radii a and b with offset h are x = ...

A proof of a formula on limits based on the epsilon-delta definition. An example is the following proof that every linear function f(x)=ax+b (a,b in R,a!=0) is continuous at ...

An equation is a mathematical expression stating that two or more quantities are the same as one another, also called an equality, formula, or identity.