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A catacaustic is a curve that is the envelope of rays emanating from a specified point (or a point at infinite distance producing parallel rays) for a given mirror shape. The ...

A 3-cusped hypocycloid, also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner ...

The distance between two points is the length of the path connecting them. In the plane, the distance between points (x_1,y_1) and (x_2,y_2) is given by the Pythagorean ...

The volumes of any n n-dimensional solids can always be simultaneously bisected by a (n-1)-dimensional hyperplane. Proving the theorem for n=2 (where it is known as the ...

In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77): 1. The locus of the center of a variable circle, tangent to a fixed circle and ...

Number Theory

The Bernoulli numbers B_n are a sequence of signed rational numbers that can be defined by the exponential generating function x/(e^x-1)=sum_(n=0)^infty(B_nx^n)/(n!). (1) ...

An alternating multilinear form on a real vector space V is a multilinear form F:V tensor ... tensor V->R (1) such that ...

The process used to generate an expression for a covariant in the first degree of any one of the equivalent sets of coefficients for a curve.

An asymptotic direction at a point p of a regular surface M in R^3 is a direction in which the normal curvature of M vanishes. 1. There are no asymptotic directions at an ...