Defining a path along a Bézier Surface after it has been projected onto a 2D plane A Bézier Surface with labelled control points
The path in question
Given a Quadratic Bézier surface defined by 9 control points, it undergoes arbitrary translation, rotation, scaling and projection into 2D space. The control points are then defined purely by (x,y) values.
The two-dimensional area occupied by the projected surface is bounded by five curves. Four of these curves result from the Bézier edge curves of the surface. The fifth curve results when the surface occludes one of the Bézier edge curves - it is highlighted (very poorly) in purple in the second picture.
How can this fifth curve, highlighted in purple, be defined in terms of the 9 control points?
(Sorry for the poor formulation of the question, any advice would be appreciated)
 A: The fifth curve you described is a "silhouette" curve. Suppose you first apply the translation, rotation, and scaling (but not the projection) to the original surface, $F$ to get a surface $G$. This is easy -- the control points of $G$ are obtained just by applying these transforms to the control points of $F$. 
Suppose the projection is in the direction of a unit vector $W$. Now we have to calculate the silhouette curve(s) of $G$ when viewed from the direction $W$. Let $N(u,v)$ denote the surface normal of $G$ at the point with parameter values $(u,v)$. The silhouette curve is the locus of points where $W \cdot N(u,v)=0$. If $G^u$ and $G^v$ denote the partial derivatives of $G$, then it is enough to find points where $W \cdot (G^u \times G^v) = 0$. 
In general, calculating silhouette curves requires numerical methods. But biquadratic surfaces are fairly simple, so it's conceivable that some special case analytic method is possible. I doubt it, though.
All decent 3D CAD systems (Catia, NX, SolidWorks, etc.) have functions to calculate silhouette curves, and you will find some discussion in the CAD literature. For example, this book.
