Mathematical basics of Bezier surfaces (patches) continuity

[mralexandermiller]

Hello, everybody! I'm learning surface modelling with Alias and for some purposes I need to understand how Alias does align surfaces with various degree of continuity (G1, G2, G3). I know that we can understand Bezier CURVES continuity mathematically as satisfying the following conditions: for G0 two aligning curves should share the same point; for G1 the first-degree derivative of both curves should be equal in the common point; for G2 the SECOND-degree derivate of the both curves should be equal the the common point and so on. I understand how to perform this for Bezier curves. But the question is that I do NOT understand the conditions for G1, G2 and G3 continuity when we are talking about Bezier SURFACES. I didn't find any information about that in the net. If I could understand this conditions, I could create some equation systems for more "profound" CV coordinates' manipulations. So, resuming all above written: for Bezier curves we work with that curves' derivatives, but what should we work with, when aligning Bezier surfaces mathematically.

Thank you very much for your answers. With hope, Alex Miller. 

[rauscht · Accepted solution]

Hi,

there are quite a number of excellent textbooks out there explaining the CAGD background and also the needed mathematical concepts of C^0, C^1, C^2 as well as G^0, G^1, G^2 continuity for curves and surfaces. To explain all the details in depth in the context of this forum probably is a bit off scope.

I am trying to give a brief summary though - excuse me in case I fail due to that briefness.

For curves, you have described the situation almost correct:

• C^0 = G^0 means the curves share the same point (positional continuity)
• C^1 means C^0 plus both curves share the same tangent, i.e. the tangent's direction and their length. A check for C^1 would compare the curves derivatives in the point under investigation
• G^1 means C^0 plus the requirement that the two curve tangents share the same direction. Note that we drop the requirement of having the same tangent's length
• C^2 means C^1 plus both curve curvature values at the common point are identical. There are formulas to compute the curvature value of a space curve which allows to compute its curvature value for a given curve parameter value. This is how one would check if this requirement is met.
• G^2 means G^1 plus both curvature values are identical for that common point. While this seems to be the same as C^2 continuity - it isn't. Remember G^1 only asks for the same tangential direction.

For two surfaces S_1 and S_2, the continuity conditions are somewhat similar. One obvious difference though is that for surfaces we are looking at the situation of a shared edge E (rather than a single point).

• C^0 = G^0 means that the edges of surfaces S_1 and S_2 are positional identical
• G^1 means G^0 plus the requirement that the tangential planes of S_1 are coplanar to the ones of S_2 for each according point of the shared edge E. The tangent plane of a surface S at a certain point p = S( u_0, v_0 ) is defined to be the affine linear space created by the vectors dS / du (u_0, v_0 ) and dS / dv (u_0, v_0 ). These two vectors are partial derivatives of S in direction of parameter u and v resp. 
• C^1 means C^0 plus the requirement that the partial derivatives  dS / du (u_0, v_0 ) and dS / dv (u_0, v_0 ) must be identical for S_^and S_2 along the common edge E. Again: this implies G^1 continuity, but is a far stronger requirement.
• G^2 means G^1 plus for each point p of the shared edge E the following must be true:
  the curve curvature value of any curve which runs from S_1 to S_2 through p is equal to the curve curvature of the curve running through p from S_2 to S_1 in the same (tangential) direction.

There so much more where one might want to look into specific details here. Hopefully, this is already helping you getting an understanding on curve and surface continuity in CAGD.

Thomas

[mralexandermiller]

Thank you very much for your answer!