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The inverse curve for a parabola given by x = at^2 (1) y = 2at (2) with inversion center (x_0,y_0) and inversion radius k is x = x_0+(k(at^2-x_0))/((at^2+x_0)^2+(2at-y_0)^2) ...

The involute of a parabola x = at^2 (1) y = at (2) is given by x_i = -(atsinh^(-1)(2t))/(2sqrt(4t^2+1)) (3) y_i = a(1/2t-(sinh^(-1)(2t))/(4sqrt(4t^2+1))). (4) Defining ...

For a parabola oriented vertically and opening upwards, the vertex is the point where the curve reaches a minimum.

A cyclide formed by inversion of a standard torus when inversion sphere is tangent to the torus.

A quadratic surface given by the equation x^2+2rz=0.

A parabolic cyclide formed by inversion of a horn torus when the inversion sphere is tangent to the torus.

A parabolic cyclide formed by inversion of a ring torus when the inversion sphere is tangent to the torus.

A parabolic cyclide formed by inversion of a spindle torus when the inversion sphere is tangent to the torus.

A paracompact space is a T2-space such that every open cover has a locally finite open refinement. Paracompactness is a very common property that topological spaces satisfy. ...

Rings produced by cutting a strip that has been given m half twists and been re-attached into n equal strips (Ball and Coxeter 1987, pp. 127-128).