Conic Section - Homogeneous Coordinates

Homogeneous Coordinates

In homogeneous coordinates a conic section can be represented as:

Or in matrix notation

The matrix is called the matrix of the conic section.

is called the determinant of the conic section. If Δ = 0 then the conic section is said to be degenerate; this means that the conic section is either a union of two straight lines, a repeated line, a point or the empty set.

For example, the conic section reduces to the union of two lines:

Similarly, a conic section sometimes reduces to a (single) repeated line:

is called the discriminant of the conic section. If δ = 0 then the conic section is a parabola, if δ < 0, it is an hyperbola and if δ > 0, it is an ellipse. A conic section is a circle if δ > 0 and A₁ = A₂ and B₁ = 0, it is an rectangular hyperbola if δ < 0 and A₁ = −A₂. It can be proven that in the complex projective plane CP2 two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection points and there is always one intersection point (possibilities: four distinct intersection points, two singular intersection points and one double intersection points, two double intersection points, one singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating.

Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is said to be tangent and it is called the tangent line. Because every straight line intersects a conic section twice, each conic section has two points at infinity (the intersection points with the line at infinity). If these points are real, the conic section must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic section has one double point at infinity it is a parabola. If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle (see circular points at infinity). If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated then it not a real conic section (its coefficients are complex).