Spline: Here is a good video
linear spline: connecting two points
quadratic spline: two linear interpolation on 3 points, generates 2 points, and interpolate the 2 points
qubit spline: most common, break down to 4 basis equation. Spline will pass through first and last point. Middle two points are tangent.
Continuity:
C_0: position is continuous
C_1: velocity is continuous
Every points in the middle of a spline is C^\infty, you can construct any continuity by putting constraints on piecewise qubit splines to ensure anywhere from not continuos to C^\infty
Bounces: 1: light 2: direct lighting, soft shadow, specular, diffuse 3: color bleeding 4: caustics, refracted object
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