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.

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    $\begingroup$ 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

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).

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