Conics

In Euclidean geometry, the second-order conic sections (ellipses, parabolas, and hyperbolas) are important phenomena, beyond the first-order curves such as lines and planes. Ellipses, parabolas, and hyperbolas lose their distinction in projective geometry because they are all projectively equivalent, that is, any form can be projected into any other form. Collectively, these curves are referred to as conics, with no distinction between the different forms.

Just as a circle in Euclidean geometry is defined as a locus of points with a constant distance from the center, so a conic in projective geometry is defined as a locus of points with a constant cross ratio to four fixed points, no three of which are collinear. Note that in both cases the shape of the curve is defined with respect to an invariant of the particular geometry, distance in the case of Euclidean, and cross ratio in the case of projective.

The equation of a conic is given by:

$$\ensuremath{{\bf p}} ^TC\ensuremath{{\bf p}} = 0,$$

or

c₁₁X² + c₂₂Y² + c₃₃W² + 2c₁₂XY + 2c₁₃XW + 2c₂₃YW=0,

where $\ensuremath{{\bf p}}$ is a $3 \times 1$ vector and C is a symmetric $3 \times 3$ matrix.

Since points and lines are dual concepts, it is not surprising that a conic is a self-dual figure. That is, it can be considered as a locus of points (as we have just done), or it can be considered as an envelope of tangent lines (the set of lines that are tangent to the conic). The equation for the envelope of lines is $\ensuremath{{\bf u}} ^T\vert C\vert C^{-1}\ensuremath{{\bf u}}$.