G2 curvature intuition (kinda tutorial)

G0-G1-G2 curves/surface connection is the topic you have to understand from the very beginning of modeling practice in Plasticity.

There are 3 things you have to keep in mind when you relate end points of 2 curves:

1) position

2) tangent line

3) curvature

G0 is when these points have only common position (just connected)

G1 is 2 of them: position+tangent (connected and aligned)

G2 is all 3 at the same time: position+tangent+curvature (connected, aligned and ... what?)

Nikita gives sufficient explanation to understand general idea: G0 - is not smooth, G1 - is smooth with tangents alignment, G2 - is even more smooth.

Any way you can check smoothness of surface with black/white zebra.

However I wasn't satisfied and would like to be more precise.

Curvature of a smooth line at a point is the reciprocal of tangent circle's radius at this point. Namely:

Curvature = 1 / (radius of tangent circle)

G2 is when connected curves at their end points have equal tangent and curvature or (what is the same thing) their tangent circles coincide (or reflected).

For surfaces the definition is almost the same: just replace circle with sphere.

This definition also explains why mirrored surface with only G1 on the edge is enough to make G2: mirror gives the same surface with the same curvature

You can test it in Plasticity (with a small error) by drawing a circle from 3 points very cloth to an end point. With G2 bridge you'll get almost coinciding circles. In contrast G1 produces circles with different size in most cases.

In attached video there is an example how it works.

Luckily you don't need to bother yourself with these technical details to model high quality stuff. There is a Curvature Toggle for curves end Zebra shader for surfaces to check curvature.

But sometime it's interesting to know what is "behind the scene" :)

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G2 curvature intuition (kinda tutorial)