0
$\begingroup$

I have a complex surface (real dimension 2) in $\mathbb{C}^2$ with coordinates $(z,w)$ given explicitely: for any $\xi \in \mathbb{C}$ I know points $w(\xi)$ of intersection of surface with complex line $z = \xi$. I have to draw it's projection on fixed 3d plane. Please help me with algorithm.

$\endgroup$
  • 1
    $\begingroup$ A slightly friendlier way of asking for help -- maybe including a description of what you have already tried -- would probably get you more answers. $\endgroup$ – Wolfgang Bangerth Aug 1 '12 at 15:10
2
$\begingroup$

Since you have an explicit parametric representation, the easiest way is direct rasterization of a rectilinear patch. You don't say what kind of projection you want, so for concreteness say we want an orthographic (orthogonal) projection onto a 3D hyperplane defined by a linear function $A : \mathbb{R}^3 \to \mathbb{R}^4 = \mathbb{C}^2$. Given a point $y \in \mathbb{R}^4$, the projection is defined by minimizing

$$|y - Ax|^2 = |y|^2-2y^TAx+x^TA^TAx$$

over $x \in \mathbb{R}^3$. The minimum is attained at

$$x = (A^T A)^{-1} A^T y$$

Now pick a 2D grid of points $z_i \in \mathbb{C}=\mathbb{R}^2$, map them to 4D with your function, and project them back to 3D with the above formula. The result can be passed to a suitable plotting function (I don't know Matlab so I don't know which).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.