Class A: Geometric Continuity 101: "class A surfacing | classasurfacing | class A surface| Digital Sculpting" Post by: adamohern on February 09, 2009, 12:07:47 PM
...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.
Digital Sculpting - Class A Surfacing
Knowledge and Definitions - Surface and Curve Continuities
Hello and welcome to class-a, a learning and support resource for industrial class-a surfacing. My name is Adam, and today’s class-a is about Geometric Continuity, and what it means to you as a surfacer.
Continuity is something you’ll encounter every day 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.
So here we can see that curves L and M are G0 continuous at point p. They touch. Good, easy.
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.
You can see here that curve L is G1 continuous with Curve M, since they are moving the same direction.
But they only have to go the same direction at the point where the meet. All of the above curves are tangent with curve L: they all move in the same direction at point p. What they do after point P is irrelevant, so long as the direction is the same at that point.
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.
Here the blue curves are G2 continuous with curve A because they share not only its direction, but it’s radius at point p. The radius can change thereafter, as in curve C, but at point p the radii are equal.
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.
Here you can see that curve R shares not only the same radius as curve at at point p, but it is accelerating at the same rate. Curve C has the same condition at point p, but then gradually tapers off in another direction. Both curves are said to be “G3 continuous” with curve A.
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.
As you can see in this 3D curve, a G4 continuous extension curve would continue the direction, radius, and 3D acceleration of the parent curve. This means that the curve will continue to travel at the same rate of change in 3D space.
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.
Have fun surfacing!
Title: Re: class-a: Geometric Continuity 101; overview
Post by: Kevin De Smet on June 24, 2009, 04:30:43 AM
Hey - I love the videos, only the audio seems to be out-of-sync with the video.
I've been there my friend I've been there, and I don't know an easy way to stop that from happening when recording videos :(
I would much appreciate it if you could fix this, because the material is so excellent, but hard to follow as is with the audio.
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