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A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point. When dealing with topological spaces, a ...

Homogeneous coordinates (x_1,x_2,x_3) of a finite point (x,y) in the plane are any three numbers for which (x_1)/(x_3)=x (1) (x_2)/(x_3)=y. (2) Coordinates (x_1,x_2,0) for ...

The solid cut from a horizontal cylinder of length L and radius R by a single plane oriented parallel to the cylinder's axis of symmetry (i.e., a portion of a horizontal ...

Find a closed plane curve of a given perimeter which encloses the greatest area. The solution is a circle. If the class of curves to be considered is limited to smooth ...

The isoperimetric quotient of a closed curve is defined as the ratio of the curve area to the area of a circle (A=pir_A^2) with same perimeter (p=2pir_p) as the curve, Q = ...

A surface of revolution defined by Kepler. It consists of less than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of ...

The level set of a differentiable function f:R^n->R corresponding to a real value c is the set of points {(x_1,...,x_n) in R^n:f(x_1,...,x_n)=c}. For example, the level set ...

Line segment picking is the process of picking line segments at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a ...

For triangles in the plane, AD·BE·CF=BD·CE·AF. (1) For spherical triangles, sinAD·sinBE·sinCF=sinBD·sinCE·sinAF. (2) This can be generalized to n-gons P=[V_1,...,V_n], where ...

The radius rho of the midsphere of a polyhedron, also called the interradius. Let P be a point on the original polyhedron and P^' the corresponding point P on the dual. Then ...