Intersection of a conic (only 5 points known) and a line

Question

I have 5 points of a conic. And a given line. I have to construct from these the intersection(s) of the conic and the line. From 5 points I can construct more points of the conics. But I am not sure how to construct the points of the intersection of the conic (given by 5 points only) and the line. Could you help me construct their intersection?

Answer 1

Let's assume your five points lie in the $xy$-plane. Then we can express them as $P_i = (x_i, y_i)$, $i=1,2,3,4,5$. Then the equation of the conic passing through them is: $$ \left| \matrix{ x^2 & y^2 & xy & x & y & 1 \\ x_1^2 & y_1^2 & x_1y_1 & x_1 & y_1 & 1 \\ x_2^2 & y_2^2 & x_2y_2 & x_2 & y_2 & 1 \\ x_3^2 & y_3^2 & x_3y_3 & x_3 & y_3 & 1 \\ x_4^2 & y_4^2 & x_4y_4 & x_4 & y_4 & 1 \\ x_5^2 & y_5^2 & x_5y_5 & x_5 & y_5 & 1 } \right| = 0 $$

Now suppose our line passes through the point $(a,b)$ and its direction is given by the vector $(u,v)$. Then its parametric equations can be written: $$ x = a + tu \\ y = b + tv $$ The intersection of the line with the ellipse occurs at $t$ values that satisfy $$ \left| \matrix{ (a + tu)^2 & (b + tv)^2 & (a + tu)(b + tv) & a + tu & b + tv & 1 \\ x_1^2 & y_1^2 & x_1y_1 & x_1 & y_1 & 1 \\ x_2^2 & y_2^2 & x_2y_2 & x_2 & y_2 & 1 \\ x_3^2 & y_3^2 & x_3y_3 & x_3 & y_3 & 1 \\ x_4^2 & y_4^2 & x_4y_4 & x_4 & y_4 & 1 \\ x_5^2 & y_5^2 & x_5y_5 & x_5 & y_5 & 1 } \right| = 0 $$

This is a quadratic equation in $t$, which you can easily solve to get the $t$ values at the intersections (if any). Then you substitute these $t$ values back into the line equation to get the points of intersection.

Answer 2

Projective geometry textbooks from the 19th and early 20th centuries are the best place to find recipes for constructions like this.

The basic idea is to construct an involution on the line, the fixed points of which are the points of intersection of the line and conic.

One such construction is given in Cremona, Projective Geometry, Art. 212 (pg 211). You'll notice it's more work than finding more points on the conic, but it'll expand your toolkit of constructions.

(Here's a cleaner copy)

The excerpted diagram below shows the five points $O,O',A,B,C$ that define the conic, along with the line $s$. The points $M,N$ are the points of intersection of the line and conic, and are constructed via a reverse and forward stereographic projection to and from a circle. How cool is that?