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Consider the family of ellipses (x^2)/(c^2)+(y^2)/((1-c)^2)-1=0 (1) for c in [0,1]. The partial derivative with respect to c is -(2x^2)/(c^3)+(2y^2)/((1-c)^3)=0 (2) ...

Two nonsuperimposable objects which are mirror images of each other are called enantiomers. The term enantiomer is synonymous with enantiomorph. For example, the letters b ...

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 = ...

A three-dimensional shadow of a four-dimensional Klein bottle.

Let phi be a map. Then phi is expansive if the statement that the distance d(phi^nx,phi^ny)<delta for all n in Z implies that x=y. Equivalently, phi is expansive if the ...

If P(x,y) and P(x^',y^') are two points on an ellipse (x^2)/(a^2)+(y^2)/(b^2)=1, (1) with eccentric angles phi and phi^' such that tanphitanphi^'=b/a (2) and A=P(a,0) and ...

A set of curves whose equations are of the same form but which have different values assigned to one or more parameters in the equations. Families of curves arise, for ...

A plane curve of the form y=x^n. For n>0, the curve is a generalized parabola; for n<0 it is a generalized hyperbola.

The closed plane curve that crosses itself once and consists of one lobe on each side of the intersection. It can be viewed as a circle with a half twist. The fundamental ...