A Missing Piece: Early Elementary Plane Rotations by Bob Palais 

The xy equations of conic sections are often derived by intersecting tilted planes with the standard right circular cone x^{2} +y^{2} =z^{2}. The standard form is messy, and neglects the fact that xycoordinates in the tilted plane will not be the same as the xycoordinates of the original xyz 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 xyplane, 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 xyplane makes it possible to treat quadratics with a cross term ax^{2}+2bxy+cy^{2}=1, and connect the type of conic section with the sign of the discriminant of ax^{2}+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.
