Conics as homogeneous quadratic geometry

Try

The affine unit circle x²+y²=1 becomes the projective conic X²+Y²−Z²=0. The point [4:3:5] lies on it because 16+9−25=0. Its tangent has equation 4X+3Y−5Z=0.

Writing a conic homogeneously makes its relation to projective transformations transparent. “Ellipse,” “parabola” and “hyperbola” describe how a real conic sits relative to a chosen infinity line; those affine names need not survive changing the chart.

Predict

Predict whether multiplying the entire quadratic equation by a nonzero scalar changes the conic. It does not. Predict whether a singular quadratic form necessarily describes a smooth conic. It may describe intersecting lines or a degenerate locus, so nonsingularity matters.

Predict whether every real nonsingular quadratic equation has real points. A positive definite form such as X²+Y²+Z² has no nonzero real zero. Nonempty real conics must be distinguished from this empty case.

Experiment

Use the standard circle conic to inspect tangency.

Point p	pᵀQp for Q=diag(1,1,−1)	Polar equation
[1:0:1]	0	X−Z=0, tangent
[4:3:5]	0	4X+3Y−5Z=0, tangent
[0:0:1]	−1	Z=0, polar but not a tangent at p

The same polar formula exists for points off the conic. Only when p lies on the conic does it give the tangent at that point. Verify that distinction algebraically rather than inferring it from the word “polar.”

Understand

Let Q be a nonsingular symmetric 3-by-3 matrix and define the conic by xᵀQx=0. Under an invertible coordinate substitution x=My, the equation becomes yᵀMᵀQMy=0. Thus the transformed matrix is MᵀQM, and its determinant remains nonzero.

At a conic point p, the differential of xᵀQx is the nonzero linear form 2pᵀQ, because Qp≠0. The tangent line is therefore pᵀQx=0. It contains p because pᵀQp=0. For any nonzero point p, the same equation defines its polar line. Symmetry gives reciprocity: p lies on q’s polar exactly when q lies on p’s polar.

A concrete parametrization of the standard real conic is [1−t²:2t:1+t²]. Expanding verifies the equation. Finite t covers every point except [−1:0:1]; that point is recovered at parameter infinity. This is a projective parametrization, not an assertion that one finite coordinate covers a compact circle globally.

Apply

Repair “Every tangent is perpendicular to the radius.” That is a Euclidean-circle description using a chosen metric and center. The projective tangent equation survives transformations that have no reason to preserve either perpendicularity or a center.

For optional depth, intersecting a nonsingular conic with the selected infinity line gives zero, one double, or two real ideal points in the familiar ellipse, parabola and hyperbola cases. This classification assumes a nonempty real affine conic and the appropriate nonsingularity conditions. A change of infinity line can change the affine type without changing the projective conic.

We have proved the transformation and tangent mechanisms; a full classification of all quadratic forms follows from linear algebra and is not re-proved here.

Continue

Use a homogeneous equation, a representative-independent calculation and an explicit chart whenever discussing a conic. Continue to the hyperbolic upper half-plane for a metric geometry whose lines are not ordinary affine lines.