Continuity 101

I’d wager that 95% of CAD users don’t know what “Curvature” continuity really is in the mathematical sense, much less how it makes sense in the broader context of surfacing. Here we’ll look at the basic principles of “continuity”, and use these principles as the basis for a longer-term study on class a surfacing continuity. This video presentation isn’t just for geeks, it’s for real people who just want to understand what “G0″, “G1″, “G2″, “G3″, and “G4″ really mean in practical terms.

Continuity is something you?ll encounter a lot as a class-a surfacer. You?ll find it affects the aesthetics of the surface, your ability to build geometry from geometry, as well as to the weight and complexity of your model.

In this class-a we?ll be talking about G0, G1, G2, G3, and G4 continuity types, and exploring what they mean at a very basic level. This will be a very high-level overview, and in future tutorials we?ll delve more into the details of what these concepts mean in the context of class-a surfacing.

That brings me to ?G?. During the course of this class and elsewhere on class a surfacing dot com you?ll hear me talking about ?G0?, ?G1?, ?G2?, etc. ?G? in this case stands for ?Geometric Continuity?, and ?G0? stands for ?Geometric Continuity, Degree Zero.? You may also see similar notation substituting ?C? for ?G,? like ?C1?, ?C2?, etc. This notation has a slightly different meaning, but is mostly for programmers and mathematicians, so we will simply use the more commonly-used term, ?G.?

G0 refers to what we call ?point continuity,? meaning that the two curves in question touch each other. Mathematically it just means that if you solve the equations for each curve at a certain x and y, the solutions will be equal. Very simple.

G1 takes this a step further. It means that the two curves not only touch, but they go the same direction at the point where they touch. In calculus this would mean that the first derivative of the two curves is equal at the point where they touch, hence the name G1.

G2 builds on G1 by adding the stipulation that the curves not only go the same direction when they meet, but also have the same radius at that point. In calculus this would mean that both the first and second derivatives of the equations are equal at that point.

G3 continuity ads yet a third requirement to the continuity: planar acceleration. Curves that are G3 continuous touch, go the same direction, have the same radius, and that radius is accelerating at the same rate at a certain point. You calculus heads have already got this thing figured out: G3 continuous curves have equal third derivatives.

G4 continuity is very seldom used, but can be important in certain isolated cases. G4 continuous curves have all the same requirements as G3 curves, but their curvature acceleration is equal in three dimensions.

In review, we?ve got five levels of continuity commonly used in CAD. In G0, the curves touch. In G1, they touch and go the same direction. In G2, they touch, go the same direction, and their radii are equal at the contact point. In G3 their radii are equal and accelerating at the same rate, and in G4 the acceleration takes place in 3D space.

In this presentation, I?ve been using 2D curves to demonstrate different types of continuity, but you?ll be able to use the exact same principles and terminology when working with continuity between different surfaces.

But who cares? What difference does it make to you as a surfacer, designer, etc, when the result looks good enough to you? There are a few reasons.

Number one, what?s visible on screen is very low resolution, and does not give an accurate picture of the real curvature of a line. Using an appropriate level of curvature will help to ensure that you?re surfaces actually are as smooth as you expect them to be.

Secondly, glossy and/or reflective surfaces ?shine? differently on tangency-continuous blends than on curvature continuous ones. The difference is subtle, but can be important on surfaces of high aesthetic importance.

Finally, geometric degradation can cause problems when you build a model. I?ll be demonstrating this phenomenon in a future class, but for now suffice it to say that understanding continuity types will help you achieve the desired results when working in 3D space.

There are a few important notes before we end this class.

Firstly, continuity types only apply to appropriate curves. Curvature continuity is useless when one of the curves being compared is a line. Similarly, curvature plus acceleration is basically irrelevant if the curve in question has a constant radius, like an arc.

Class-A surfacing CAD packages allow the user to adjust the ?magnitude? or ?tension? of a blend curve, allowing you to control how much influence the parent curve has over the blend. This functionality can sometimes make a curve or surface look like it is curvature continuous, even when it is actually only tangent continuous. This trick is acceptable if the desired look is achieved, but does not help with the problem of continuity degradation, discussed in a future tutorial.

I hope this helps with a basic understanding of continuity types, what they are, and how they’re used in a theoretical sense. We’ll get into more practical examples soon. Until then, have fun!

about adam: Adam O'Hern is an industrial design consultant specializing in visual brand languages, and has designed products ranging from laptops to power tools, classroom toys to bathroom fixtures, and robots to lint rollers. He has published with 3DWorld Magazine, CGTuts+, and Luxology, and works with Josh Mings of SolidSmack.com on EngineerVsDesigner.com.