Lab for Determinants and Conic Section Curves

Implicit equation for a standard ellipse.

Implicit equation for a standard ellipse. The standard form for an ellipse involves five coefficients:

(9)    .  

The coefficients in (9) 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 four.  Thus, four points    are sufficient to uniquely determine the standard formula for an ellipse.

An alternate way to formulate the solution to (9) is to observe that the four additional equations must be satisfied:
   
(10)      for  i = 1,2,3,4.

Equations (9) and (10) form a homogeneous system of five equations in five unknowns.  

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

(c) John H. Mathews