Algebraic Geometry - Computer Graphics and Geometric Modeling - page 710

Now multiply equation (10.45) by Z and use equation (10.46) to get the desired result.

Additional facts about multiplicities and singular points are postponed to the end
of the next section after we have discussed Bézout's theorem. We finish this section
with an interesting fact about the intersection of lines with a curve.

10.6.13. Theorem. Let C be a plane curve order n and let p be a point not on C .
Then (in P 2 ( C )) there are at most n(n - 1) lines L i through p so that for any other line
L through p , L intersects C in exactly n distinct points.

Proof. The proof will follow the one given in [BriK81]. By a change of coordinates,
we may assume that p = [1,0,0] and C is the set of zeros of the equation F(X,Y,Z) = 0
for some homogeneous polynomial F of degree n. Let L y denote the line with equation
X = 0. Since every line through p intersects L y in a unique point, let us parameterize
those lines by their intersection point, that is L s,t will denote the line through p and the
point [0,s,t] ΠL y . By (the complex version of) Theorem 3.4.1.4 every point on L s,t - p
has a unique representation [r,s,t], r ΠC . It follows that the intersections of L s,t and the
curve C are defined by the zeros of the equation F(r,s,t) = 0. Note that F(r,s,t) is not iden-
tically zero because L s,t is not a component of C . In fact, it will have degree n in r. If we
fix s and t and think of F as a polynomial in r, then by Corollary 10.4.5 F will have mul-
tiple roots for r if and only if its resultant is zero. But the resultant of F and ∂F/∂r is a
homogeneous polynomial in s and t of degree n(n - 1). This can be turned into a poly-
nomial in s/t or t/s of the same degree and hence has at most n(n - 1) zeros. Only the
lines corresponding to those zeros will intersect C with a multiplicity higher than 1.

For example, the case n = 2 in Theorem 10.6.13 says that at most two lines through
a point not on a quadratic curve in C 2 are tangent to that curve. The case
n = 3 says that at most 6 = 3 · 2 lines are tangent to a cubic. Figure 10.11 shows a cubic
that actually has six such tangents.

10.7 Intersections of Plane Curves

We now come to one of the basic results in the theory of intersections of plane curves,
namely, Bézout's theorem. This theorem is important in other areas of algebraic

C

L 1
L 2
L 3

p

L 4
L 5
L 6

Figure 10.11. A cubic with six concurrent

tangents.