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For two lines in the plane with endpoints (x_1,x_2) and (x_3,x_4), the angle between them is given by costheta=((x_2-x_1)·(x_4-x_3))/(|x_2-x_1||x_4-x_3|). (1) The angle theta ...

The plane determined by the points x_1, x_2, and x_3 and the line passing through the points x_4 and x_5 intersect in a point which can be determined by solving the four ...

A curve on a surface whose tangents are always in the direction of principal curvature. The equation of the lines of curvature can be written |g_(11) g_(12) g_(22); b_(11) ...

A Maurer rose is a plot of a "walk" along an n- (or 2n-) leafed rose curve in steps of a fixed number d degrees, including all cosets.

The generalization of the Schönflies theorem to n dimensions. A smoothly embedded n-hypersphere in an (n+1)-hypersphere separates the (n+1)-hypersphere into two components, ...

Let a triangle have side lengths a, b, and c with opposite angles A, B, and C. Then (b+c)/a = (cos[1/2(B-C)])/(sin(1/2A)) (1) (c+a)/b = (cos[1/2(C-A)])/(sin(1/2B)) (2) ...

A conic section on which the midpoints of the sides of any complete quadrangle lie. The three diagonal points P, Q, and R also lie on this conic.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

The plane spanned by the normal vector N and the binormal vector B.

A point which lies on at least one ordinary line is called an ordinary point, or sometimes a regular point.