Ambiguity I - Collinear Points
- Ryan Hirst
- Conic Sections
So far, GGB's Conic tool and I agree. If the coefficient matrix has two free parameters, it would appear safe to conclude that the conic section is not uniquely defined. But this is not so. There are two cases that cause ambiguity: I. four or five points are collinear. II. two or more points coincide. Let me tackle case I, as it can be resolved with relative ease. Consider five points in space:
Now. Drag the points so that four points fall on the same line. The worksheet will help you line them up. I think that you will find that when points are moved continuously in time, it is not true that the point of intersection is unknown in the case of four collinear points. That, in fact, bringing the fourth point to meet three others on a line uniquely --and inescapably-- determines the point of intersection. Second, we have the case of five collinear points. I think you will find there can only be one algebraic solution.