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Let C_1, C_2, C_3, and C_4 be four circles of general position through a point P. Let P_(ij) be the second intersection of the circles C_i and C_j. Let C_(ijk) be the circle ...

The following table gives the centers of the first Yff circles triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100. X_n center ...

The evolute of a hyperbola with parametric equations x = acosht (1) y = bsinht (2) is x_e = ((a^2+b^2))/acosh^3t (3) y_e = -((a^2+b^2))/bsinh^3t, (4) which is similar to a ...

A curve y(x) is osculating to f(x) at x_0 if it is tangent at x_0 and has the same curvature there. Osculating curves therefore satisfy y^((k))(x_0)=f^((k))(x_0) for k=0, 1, ...

Given a parabola with parametric equations x = at^2 (1) y = at, (2) the evolute is given by x_e = 1/2a(1+6t^2) (3) y_e = -4at^3. (4) Eliminating x and y gives the implicit ...

The south pole is the point on a sphere with minimum z-coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is ...

By the definition of the functions of trigonometry, the sine of pi is equal to the y-coordinate of the point with polar coordinates (r,theta)=(1,pi), giving sinpi=0. ...

A real-valued univariate function f=f(x) has a jump discontinuity at a point x_0 in its domain provided that lim_(x->x_0-)f(x)=L_1<infty (1) and lim_(x->x_0+)f(x)=L_2<infty ...

The Yff circles are the two triplets of congruent circle in which each circle is tangent to two sides of a reference triangle. In each case, the triplets intersect pairwise ...

Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The ...