I'd like a little help understanding Levin's pencil method. Here is what I have gleaned so far.
Given two quadric surfaces $\mathbf{Q}$ and $\mathbf{Q}'$, if an intersection exists, there is a parameterization surface $\mathbf{P}$ that is the pencil of the two quadrics that also contains the intersection.
To find this parameterization surface, one can find the roots of the equation $|\alpha \mathbf{Q} + \mathbf{Q}'| = 0$. Then, $\mathbf{P}=\alpha \mathbf{Q} + \mathbf{Q}'$
In my particular case, say that I have determined that the parameterization surface is a hyperbolic paraboloid.
This surface must be rotated to canonical form using a transformation $\mathbf{T}$ such that $uv=w$. If my parameters are $(s,t)$, then the mapping of parameters to the canonical space is $u=t,v=s,w=st$.
Once the surface is in canonical form, we can select values for the parameter $t$, and solve for $s$ using
$$\pi_2=Ct^2 +2Et + B $$ $$\rho_2 = 2[Ft^2 +(D+J)t+H]$$ $$\sigma_2 = At^2 + 2Gt + K $$ $$ \pi_2 s^2 + \rho_2 s + \sigma_2 = 0 $$
This will allow us to find the parameters $s$ and $t$, trace the curve in $uvw$ space, and then transform this curve using $\mathbf{T}^{-1}$ to bring the curve back into $xyz$ space.
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My questions are as follows:
- Where do the coefficients used in the equations for $\pi_2 $ , $\sigma_2 $, and $\rho_2 $ come from? Are these taken from the discriminant form of the parameterization surface $\mathbf{P}$?
- How do I figure out the range for the parameter $t$? Is it $-\infty$ to $+\infty$? If so, presumably there is a finite interval where the intersection exists in real space?
- To determine $\mathbf{T}$, I came across this post on math.stackexchange. The initial transformation $\mathbf{T}_1$ starts with an Eigendecomposition that results in two non-zero eigenvalues and a rotation matrix in $SO3$. It's not clear to me how to find that specific Eigendecomposition...
Any help is greatly appreciated!
