Finding the points where the conic is tangent to the quadrilateral

By projecting the square from the plane \(PQR\) in the previous picture, we can construct the points \(W, X, Y, Z\) in the \(xy\) plane. In order to construct the conic which is the projection of the circle onto the \(xy\) plane, we would first like to find the points \(A, B, C, D\) where the conic touches the quadrilateral \(WXYZ\).

First, consider the problem of finding the points of tangency for a circle inscribed in the square \(W' X' Y' Z'\). If we can solve this problem in a way that is invariant under projective transformations, then we can use the same solution for the projection of the circle onto a conic inscribed in a quadrilateral.

We know that straight lines project to straight lines and intersections of lines project to intersections of lines. Therefore, since the centre \(O'\) of the circle is the intersection of the diagonals of the square, then this will project to the intersection point \(O\) of the diagonals of the quadrilateral.

Now draw a line through the centre \(O'\) of the circle which is parallel to a pair of opposite sides of the square. This will intersect the square at two points of tangency between the square and the circle. For example, in the picture on the right the line through \(O'\) parallel to \(W' X'\) and \(Y' Z'\) intersects the square at \(B'\) and \(D'\). The other two points of tangency can be found in the same way, except now using a line through \(O'\) parallel to the other pair of opposite sides of the square.

The diagonals \(W'Y'\) and \(X' Z'\) project to the diagonals of the quadrilateral \(WY\) and \(XZ\), therefore the centre \(O'\) of the circle projects to the intersection \(O = WY \cap XZ\).
Parallel lines do not necessarily project to parallel lines, however they do project to lines passing through a common point (the projection of the point at infinity corresponding to the original set of parallel lines). Define the points \(P_1 = XY \cap WZ\) and \(P_2 = WX \cap YZ\). Consider first the pair of opposite sides of the square \(W'Z'\) and \(X'Y'\). These project to \(WZ\) and \(XY\). The line through the centre of the square parallel to this pair of sides will project to the line \(OP_1\) (coloured green in the picture below), and therefore we can find two points \(A\) and \(C\) of tangency between the conic and the quadrilateral \(WXYZ\) as the projection of the points \(A'\) and \(C'\) of tangency between the circle and the square. The same idea with the other pair of opposite sides of the square \(W'X'\) and \(Y'Z'\) (which now project to lines passing through \(P_2\)) allows us to find the other two points of tangency \(B\) and \(D\).