Shading Mask Algorithm for Radiation Calculations

Computational geometry - why not keep triangles in a fixed clockwise/counterclockwise order?

Simple: in 3D, there is no clockwise/couterclockwise order, you can look at a polygon from both sides.

(On a complete mesh, you can demand that all faces be oriented consistently, provided if forms an orientable manifold.)

Final remark: if such an order was possible, you can be sure that it would be in use everywhere.

How to impose C1 continuity while generating NURBS Surfaces?

There are a couple of ways to do this.

One thing you could do is just solve independently and then force adjacent patches to be C1, by putting the border control points at the midpoint of the two control points towards the interior of the two adjacent patches. In other words, if you have four control points, Q0, Q1, P0, and P1, where P points are on one surface and Q points are on another surface, and P0 = Q0, then set P0 = 0.5 * (Q1 + P1). If the knot width of the NURBS patches are different or you have more than 4 NURBS patches as a corner, then this equation will need to be different. I'm also assuming you have fully multiple end knots (Bezier end conditions).

Another thing you could do is solve a global system which includes the C1 condition as a linear equality constraint. This will give you an optimal result, but it could be much slower.

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