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Flat polygons embedded in three-space can be transformed into a congruent planar polygon as follows. First, translate the starting vertex to (0, 0, 0) by subtracting it from ...

The special orthogonal group SO_n(q) is the subgroup of the elements of general orthogonal group GO_n(q) with determinant 1. SO_3 (often written SO(3)) is the rotation group ...

A dozen dozen, also called a gross. 144 is a square number and a sum-product number.

The involute of the astroid is a hypocycloid involute for n=4. Surprisingly, it is another astroid scaled by a factor (n-2)/n=2/4=1/2 and rotated 1/(2·4)=1/8 of a turn. For ...

The polar angle on a sphere measured from the north pole instead of the equator. The angle phi in spherical coordinates is the colatitude. It is related to the latitude delta ...

A complex magnification is a map of the form z|->az, where a is a positive real number, which corresponds to magnification about the origin of points in the complex plane by ...

A complex translation is a map of the form z|->z+a, where a is a fixed complex number, which corresponds to translation (shifting) of points the complex plane by the complex ...

The cubic groups are the point groups T_h and O_h together with their pure rotation subgroups T_d, T, and O (Cotton 1990, pp. 433-434).

The rotation vector of the trihedron of a curve with curvature kappa!=0 when a point moves along a curve with unit speed. It is given by D=tauT+kappaB, (1) where tau is the ...

A curve which can be turned continuously inside an equilateral triangle. There are an infinite number of delta curves, but the simplest are the circle and lens-shaped ...