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If a one-parameter family of curves has index N and class M, the number tangent to a curve of order n_1 and class m_1 in general position is m_1N+n_1M.

An ellipse intersects a circle in 0, 1, 2, 3, or 4 points. The points of intersection of a circle of center (x_0,y_0) and radius r with an ellipse of semi-major and ...

The orthotomic of the unit circle represented by x = cost (1) y = sint (2) with a source at (x,y) is x_o = xcos(2t)-ysin(2t)+2sint (3) y_o = -xsin(2t)-ycos(2t)+2cost. (4)

Given two closed curves, the circuminscribed curve is simultaneously inscribed in the outer one and circumscribed on the inner one.

For the parametric representation x = (2t^2)/(1+t^2) (1) y = (2t^3)/(1+t^2), (2) the catacaustic of this curve from the radiant point (8a,0) is given by x = ...

A complete metric is a metric in which every Cauchy sequence is convergent. A topological space with a complete metric is called a complete metric space.

A concho-spiral, also known as a conchospiral, is a space curve with parametric equations r = mu^ua (1) theta = u (2) z = mu^uc, (3) where mu, a, and c are fixed parameters. ...

The envelope of refracted rays for a given curve.

A proof based on a dissection which shows the formula for the area of a plane figure or of the volume of a solid. Dozens of different dissection proofs are known for the ...

The Doppler spiral is the curve obtained from an Archimedes' spiral which is translated horizontally with speed k. It has parametric equations x(t) = a(tcost+kt) (1) y(t) = ...