A Missing Piece: Early Elementary Plane Rotations by Bob Palais
Notices of the AMS February 2014, Doceamus
, Web Supplement

The x-y equations of conic sections are often derived by intersecting tilted planes with the standard right circular cone x2 +y2 =z2. The standard form is messy, and neglects the fact that x-y-coordinates in the tilted plane will not be the same as the x-y-coordinates of the original x-y-z space due to the tilt. The equations can be derived more easily, correctly, and in a cleaner standard form by instead rotating the cone using the rotation formula, shifting the vertex along the rotated axis, then intersecting with the x-y-plane, z=0, in other words, by omitting any terms involving z! The details are carried out here, and an animation resulting from the process is shown below.

The elementary rotation formula in the x-y-plane makes it possible to treat quadratics with a cross term ax2+2bxy+cy2=1, and connect the type of conic section with the sign of the discriminant of ax2+2bx+c, with the determinant of the corresponding matrix, and eventually, with determining whether a critical point of a function f(x,y) is an extreme or saddle point.


Cosine Subtraction

Conic Sections