5.6.5.1 Implicitization

We can eliminate to form the resultant , then solve for and use the inversion algorithm to obtain .

Example 5.6.2. Let us consider an ellipse and a circle

as in Fig. 5.16.

Figure 5.16: Ellipse and circle intersection

First we eliminate from these two equations. This leads to

which has two real roots and . These lead to and , respectively.

However there are possible numerical problems at the tangential intersection point . Let us assume that due to error , we have

hence

This implies that is imaginary and that no real roots exist. This would have as a consequence missing an intersection solution, leading to a robustness problem.