G0 ; G1 ; G2 ???

 From:  Michael Gibson
290.2 In reply to 290.1 
Hi Bernard, I will try to break it down here.

The "G" in those terms refers to a math concept called "Geometric Continuity", which is a way of describing how two smooth curves meet up with one another.

G0 refers to "position continuity" which is a fancy way of saying whether two curves touch each other. Here is an example - the curves on the left are not G0 with each other because they do not touch. The 2 curves on the right have G0 continuity.

Another property of a curve is the tangent of a curve at any point. G1 means "tangent continuity" - that is that the 2 curves share the same tangent direction at their ends (in addition to being G0). The left curves here are only G0, the right curves have G1 continuity:

Ok, so that's all pretty straightforward up to that point. The next bit is a little bit less familiar. G0 means touching and G1 means shared tangent. So G1 means that the curves are smooth with regard to each other, right? Well, yes they are smooth but only to a certain amount. There are different qualities of a curve beyond just the tangent, there is also a quality called "curvature". Each point of a curve not only has a tangent direction, it also has a curvature value which is what radius of circle the curve resembles at that one point. So for instance when the curve is going around a tight bend, it resembles a small circle at that point. When a curve is broader and more shallow, it resembles a much larger radius circle at those points, all the way to having infinite curvature for a line segment.

When 2 curves share the same curvature at a point, then that is G2 continuity (which includes G1 and G0 continuity as well).

Here is kind of the classic example of G1 but not G2 continuity:

That is a circular arc on the left and a line on the right. They share the same tangent so they are G1. But the curvature is abruptly changed between them - the arc has a curvature of the circle that it is a part of, and the line has infinite curvature. This abrupt change has some consequences, most notably reflections and specular highlights will not cross in a smooth manner over this seam, you can see this in the real world on filleted parts, where the specular highlights will sort of bunch up near the edge of a fillet. When curvature is continuous between 2 pieces, then reflections and specular highlights will continue unbroken across those seams.

I have tried to minimize the use of these terms in MoI, but it is a bit difficult not to talk about them in blending.

The definition of G0, G1, G2, for surfaces is analagous to how it works for curves.

Please let me know if you would like additional information about any part of this.

- Michael