Lab for Determinants and Conic Section Curves

The implicit equation for a 5 point conic.

The implicit equation for a 5 point conic. The general equation for a conic section is a quadratic equation in two variables involving six coefficients:

(11)    .  

The coefficients in (11) cannot all be zero.  If it were known a priori which coefficient is non zero, then each term can be divided by it to reduce the number of unknown coefficients to five.  Thus, five points    are sufficient to uniquely determine a conic.  Equation (11) will determine an: ellipse, hyperbola, parabola, circle or the degenerate cases of two intersecting lines or two parallel lines.

An alternate way to formulate the solution to (11) is to observe that the five additional equations must be satisfied:
   
(12)      for  i = 1,2,...,5.

Equations (11) and (12) form a homogeneous system of six equations in six unknowns.  

Since the solution vector    must be non zero, the determinant of the coefficient matrix must be zero, i.e.

(c) John H. Mathews