Intersection between conic and line in homogeneous space

Question

In homogeneous space (so 3 coordinates for each point) I have:

• A conic C, defined by a symmetric 3x3 matrix of real values. The conic actually should have only imaginary points (don't know if this is important).
• A line l, defined by a vector of 3 real values

How do I find the intersection of the two?

I think I shoud be able to find the intersection (I expect two complex solutions), but I'm having troubles doing it.

Solving it with the classic pen&paper leads me to a solution X=(x, y, 1) with x and y complex such that, when I try to verify that the point belongs to C (by cheching if XCX' = 0, where the ' stands for transposed), it seems that it does NOT belong to C.

For those who know computer vision stuff: C is actually the image of the absolute conic, estimated from a picture, while l is a vanishing line of a plane. I'm trying to intersect the two in order to find the circular points, and then do a metric rectification of the plane in the image.

Answer 1

First find the pole (point) of the line ${\bf L}=\pmatrix{a \\ b \\ c}$ using the conic $\mathtt{C}=\left[ \matrix{A & C & D \\ C & B & E \\ D & E& F} \right]$. This is found using the inverse of the conic

$$ \mathbf{P} = \mathtt{C}^{-1} \mathbf{L} = \pmatrix{u \\ v \\ w}$$

Since we will use the inverse again, set

$$\mathtt{C}^{-1}=\left[ \matrix{a & c & d \\ c & b & e \\ d & e& f} \right]$$

The pencil of lines through $\bf P$ is parametrically defined as

$$\mathbf{T}(\psi) = \pmatrix{-w \sin \psi \\ w \cos \psi \\ u \sin \psi - v \cos \psi}$$

Now if you find a line $\mathbf{T}$ that is tangent to the conic, it will have $\mathbf{T}^\top \mathtt{C}^{-1} \mathbf{T}=0$ and the tangent point $\mathbf{Q}=\mathtt{C}^{-1} \mathbf{T}$ lies on $\mathbf{L}$. So $\mathbf{Q}$ is an intersection point.

To solve $\mathbf{T}^\top \mathtt{C}^{-1} \mathbf{T}=0$ for $\psi$ involves solving an equation of the form $$K_0 + K_1 \sin(2 \psi) + K_2 \cos(2 \psi)=0$$ with $$\begin{align} K_0 & = a w^2-2 d u w + f u^2 \\ K_1 & = -2 ( c w^2-w (d v+e u)+f u v)\\ K_2 &= w^2 (b-a)+2 w (d u-e v) + f (v^2-u^2) \end{align}$$

There are two solutions to the above trig equation

$$ \psi = \begin{cases} \frac{1}{2} \left( \tan^{-1} \left( \frac{K_1}{K_2} \right) - \sin^{-1} \left( \frac{2 K_0 + K_2}{\sqrt{K_1^2+K_2^2}} \right) -\frac{\pi}{2}\right) & \mbox{solution 1}\\ \frac{1}{2} \left( \tan^{-1} \left( \frac{K_1}{K_2} \right) + \sin^{-1} \left( \frac{2 K_0 + K_2}{\sqrt{K_1^2+K_2^2}} \right) +\frac{\pi}{2}\right) & \mbox{solution 2} \end{cases} $$

In the end the two intersection points are defined by $\mathbf{Q} = \mathtt{C}^{-1} \mathbf{T}(\psi) $

$$ \mathbf{Q} = \pmatrix{ (c w-d v) \cos \psi + (d u-a w) \sin \psi \\ (b w-e v) \cos \psi + (e u-c w) \sin \psi \\ (e w-f v) \cos \psi + (f u-d w) \sin \psi } $$

Example

Conic $\mathtt{C} = \left[ \matrix{1 & -\tfrac{7}{6} & -3 \\ -\tfrac{7}{6} & 4 & 1 \\ -3 & 1 & \tfrac{13}{6} } \right] $ and line $\mathbf{L} = \pmatrix{3 \\ -2 \\ -7}$. The polar point is $$\mathbf{P} = \mathtt{C}^{-1} \mathbf{L} = \pmatrix{u \\ v \\ w} = \pmatrix{ \tfrac{11208}{5245} \\ \tfrac{1134}{5245} \\ -\tfrac{390}{1049} } $$

The tangent lines are thus $$\mathbf{T}(\psi) = \pmatrix{ \tfrac{390}{1049} \sin\psi \\ -\tfrac{390}{1049} \cos\psi \\ \tfrac{11208}{5245} \sin\psi - \frac{1134}{5245} \cos\psi }$$

The tangency equation $\mathbf{T}^\top \mathtt{C}^{-1} \mathbf{T}=0$ simplifis to the following:

$$ 1.26602894762909 \cos^2 \psi+0.330351717237625 \cos\psi \sin \psi-1.24876309636214=0 $$

with solution

$$ \begin{cases} \psi = 0.299923260411840 & \mbox{solution 1} \\ \psi =-0.0446792696983973 & \mbox{solution 2} \end{cases} $$

The intersection points are

$$ \begin{align} \mathbf{Q} &= \pmatrix{ -0.231100412938826 \\ -0.141550789771816 \\-0.0585999513246922} & \mathbf{Q} &= \pmatrix{ 0.136962490550640 \\-0.0727785333232919 \\0.0794920768997864 } \end{align} $$

Answer 2

The matrix

$$L=\hat l=\begin{pmatrix}0&-l_3&l_2\\l_3&0&-l_1\\-l_2&l_1&0\end{pmatrix}$$

can be used to describe a cross product with $l$: $Lg=l\times g$.

Now consider $D = L^T\cdot C\cdot L$. It is a degenerate conic which you best interpret dually as a pair of points, namely the points of intersection. A line $g$ is tangent to that conic if its intersection with $l$, which can be computed as $l\times g=Lg$, lies on $C$, i.e. if $(l\times g)^TC(l\times g)=0$.

All you have to do is decompose that conic $D$ into its components. You already have $\operatorname{rank}(D)=2$. Now consider $P=D+\lambda L$. For some suitable $\lambda$, this matrix will have rank $1$. Simply look at any $2\times 2$ subdeterminant, and choose $\lambda$ in such a way that it becomes zero. You will have two possible choices which only differ by sign. Either one will do.

Once you have this, $P=pq^T$ is a matrix of rank $1$. You can choose any non-zero column and call that $p$, and any non-zero row will be $q$ (up to scalar multiples). So I'd look for the greatest absolute value in that matrix and choose its row and column as the two points of intersection.