Line and plane tangent to a quadric surface

Question

I'm stuck with what is seemingly a simple exercise or problem of linear algebra. Notice that the problem is purely geometrical in $\Bbb R^3$ (no derivatives, differential calculus).

Let $\mathbf{x} = \mathbf{a}$ be a point on the quadric $\mathbf{x}^TA\mathbf{x} = 1$. Show that the line $\mathbf{x} = \mathbf{a} + s \mathbf{\lambda}$ (where $s$ a parameter, $\mathbf{\lambda}$ the direction cosine) is tangent to the quadric if $\mathbf{\lambda}^T A \mathbf{a} = 0$. Hence show that the tangent plane at this point is $\mathbf{a}^TA\mathbf{x} = 1$.

Any suggestion on how should I approach this problem?

Also, would anyone please recommend a good introductory book on quadrics?

Answer 1

The line intersects the quadric at two points whose parameter values $s$ satisfy the equation $(\mathbf a + s\lambda)^T A (\mathbf a + s\lambda)=1$. If you expand this out, you’ll get a quadratic equation that gives you the two relevant values of $s$. If the line is actually tangent to the quadric, then the two intersection points will be coincident, so the quadratic will have equal roots. This will immediately give you the desired condition for tangency. In fact the quadratic “$b^2 = 4ac$“ condition that we all learned in high-school becomes: $$ 4(\lambda^TA \mathbf a)^2 = 4(\lambda^TA \lambda)(1 - \mathbf a^TA \mathbf a) $$ The second term on the right is zero because the point $\mathbf a$ lies on the quadric, so $\lambda^TA \mathbf a=0$.

As far as I know, no good books about quadrics have been written in the past 50 years or so. The best references I know are old books about 3D coordinate geometry. Look for authors like Sommerville, Snyder and Sisam, and Salmon.

Sommerville, Analytical Geometry of Three Dimensions.

Snyder & Sisam, Analytical Geometry of Space

Salmon, A Treatise on the Analytic Geometry of Three Dimensions

Answer 2

Do a Taylor expansion around $a$ in the direction $\lambda$:

$$(a+s\lambda)^TA(a+s\lambda)=\underbrace{a^TAa}_1+s\underbrace{2\lambda^TAa}_{f(a,\lambda)}+s^2\lambda^TA\lambda$$

The first order term $f(a,\lambda)$ (which in fact the $\stackrel{\longrightarrow}{\operatorname{grad}}(f).\lambda$) expresses the "linear part". Being equal to $0$ expresses that $\lambda$ belongs to the tangent plane

Therefore, as

$$\lambda \ \text{ belongs to the tangent plane at point } a \ \iff \ f(a,\lambda)=0$$

Changing $\lambda$ into $x$ gives the looked-for equation of the tangent plane:

$$f(a,x)=0 \ \iff \ 2x^TAa=0 \ \iff \ a^TAx=0$$

(due to the symmetry of matrix $A$).