steiner-conic.md

The Steiner Conic is an alternative definition of a conic: Given two pencils P(A), P(B) of lines at two points A, B (all lines containing A and B resp.) and a projective but not perspective mapping π of P(A) onto P(B), then the intersection points of corresponding lines form a non-degenerate projective conic section.

In the above figure,

Therefore, by given lines cdec'd'e' or points ABCDE, and by given an arbitrary line f passing through A, we can construct line f' or point F by these steps:

1. 
2. 
3. 
4. 

This is the construction of Steiner point conic. Analogously, we can construct the Steiner line conic according to the principle of duality.

In this page, we simplify the projective mapping as the invariant of cross-ratio, i.e. .

Steiner conic → Quadratic curve

The Steiner conic follows the quadratic curve equation .

Here we get the equation about point by the relation .

According to the duality, this process also shows that the Steiner line conic follows the quadratic curve (the envelope of a set of straight lines) equation.

Here we get the same result by a rule that five points determine a conic:

According to the duality, these two processes also shows that five straight lines determine a line conic.

Quadratic curve → Steiner conic

Conversely, any quadratic curve is a Steiner conic. We only need to prove that for any 6 points ABCDEF on a quadratic curve, .

WLOG, we can put A onto origin and AB onto y-axis, and denote the quadratic curve as . For any line passing through A, we can get the other intersection of the line and the quadratic curve .

Here is the proof process.

To prove the dual fact, we can put line A onto x-axis and point AB onto origin, and denote the quadratic curve (the envelope of a set of straight lines) as . For any point (x,0,z) (z=0 means the point at infinity) lying on A, we can get the other tangent line .

Here is the proof process.

In the following proofs, we use the cross-ratio relation instead of the quadratic curve equation.

Pascal's theorem and Brianchon's theorem

The proof of Pascal's theorem on a Steiner conic is much simpler than on a quadratic curve, because only incidence relations of points and straight lines should be considered, just like Desargues's theorem and Pappus's theorem.

Here is the computational proof.

According to the duality, this process also proves Brianchon's theorem.

Braikenridge-Maclaurin theorem

Similarly, the proof of Braikenridge-Maclaurin theorem (which is the converse of Pascal's theorem) on a Steiner conic is also much simpler.

Here is the computational proof.

This process also proves its dual theorem (which is also the converse of Brianchon's theorem).

Braikenridge-Maclaurin theorem provides another construction of a conic: by given points ABCDE and an arbitrary line f passing through A, we can construct point F by , where line GHJ is a Pascal line.

Braikenridge-Maclaurin construction (additional 6 lines and 3 intersections) is simpler than Steiner construction (additional 10 lines and 7 intersections, note that c" is not necessary).

Projective mapping

Let's define the projective mapping of two point sets A₁A₂... and B₁B₂... on a conic Γ as , where P and Q are two arbitrary points on Γ.

Here proves that all lie on one straight line p (the projective axis).

Involution

A projective mapping is an involution if and only if all A_iB_i meet at the same point P. We call this involution a perspective mapping, and P is the perspective center.

Here and here are the computational proofs.

Note that in general. Let's look at , which is not infinity in general, but .

Pole and Polar

In an involution on a conic Γ, the perspective center P is the pole of the projective axis p, and p is the polar of P, with respect to Γ. In other words, for any line passing through P, intersecting p at Q and intersecting Γ at M and N, .

Here is the computational proof.