what curve type the dual conic matrix of parabola matrix

Question

when $A_Q$ is a matrix of one conic type , I get its dual conic matrix $A_Q^*$ by the $inv(A_Q)$ or $adjoint(A_Q)$ using Matlab. what type is $A_Q^*$ according to the sign $ det(A^*_Q(1:2,1:2))$ , it is not always same type with its dual conic or not other fixed relationship. I dont know the relationship.

Answer 1

It depends on whether the origin is inside or outside the conic. Specifically,

• if the origin is inside the conic, the dual conic is an ellipse
• if the origin is on the conic, the dual conic is a parabola
• if the origin is outside the conic the dual conic is a hyperbola

Easiest way to see this is to think of the dual conic as the locus of poles of tangents to the original conic with respect to the unit circle (see polar reciprocal). If the origin is outside of the conic, two tangents will go through the origin, and the corresponding poles will be at infinity (and you will get a hyperbola). If the origin is on the conic, one tangents will go through the origin, and one pole will be infinite (and you will get a parabola). If the origin is inside the conic, no tangents will go through the origin, and all poles will be finite (and you will get an ellipse).

Note that the polar reciprocal is the reflection in the origin of the dual conic obtained via inversion of the quadratic form. But the same logic holds.