二次曲线的全部切线构成对偶空间中的二次曲线

hbghlyj

Exercise 3.6
8. Prove that the set of tangent lines to a nonsingular conic in $P(V)$ is a conic in the dual space $P(V')$.

证明
$P(V)$中的二次曲线在适当的坐标系下可写成
$$C:ax^2 + by^2 + cz^2 = 0$$
$C$在点$[p:q:r]$的切线为$apx + bqy + crz = 0$.
所以,对于$[X:Y:Z]∈P(V')$,直线$Xx+Yy+Zz=0$与$C$相切当且仅当存在点$[p:q:r]∈C$使得$X=ap,Y=bq,Z=cr$.
解得$[p:q:r]=[\frac Xa:\frac Yb:\frac Zc]$,那么$[p:q:r]∈C$即为$[\frac Xa:\frac Yb:\frac Zc]∈C$,即$a(X/a)^2 + b(Y/b)^2 + c(Z/c)^2 = 0$
所以$$\Set{[X:Y:Z]∈P(V')|\frac1aX^2+\frac1bY^2 +\frac1cZ^2 = 0}$$
是$P(V')$的一条二次曲线.

hbghlyj

Blowing up the Veronese Surface
二次曲线的对偶曲线由它的切线组成.
直接联立$ax^2+by^2+cxy+dxz+eyz+fz^2$和$Xx+Yy+Zz=0$消$x$,然后令$\Delta_y=0$:

1. Collect[(FactorList@Discriminant[Resultant[a x^2+b y^2+c x y+d x+e y+f,X x+Y y+Z,x],y])[[3,1]],{X,Y,Z}]

Copy the Code

\[X^2 \left(e^2-4 b f\right)+Y^2 \left(d^2-4 a f\right)+Z^2 \left(c^2-4 a b\right)+XY (4 c f-2 d e)+ZX (4 b d-2 c e)+Y Z (4 a e-2 c d)\]