Conics 101

Some of you might not know what I’m talking about when I ramble on about “conics” in CAD. In this short video, I’ll explain what conics are, and the basic mathematical and practical ideas that underpin their use in CAD. Sound like a party? Oh yeah, baby. It’s geek-out time.

We?ll be looking at how conics can help you make cleaner, lighter surfaces, more reliable surfaces.

Anybody who knows my friend Don, knows that Don loves conics. As a super-start class-A surfacer, Don knows that conics make his geometry lighter, cleaner, and more reliable. We at Class-A love conics too, and this tutorial will be the first in a series explaining why we surfacers love these shapes so much, and how we use them to create our Class-A Surfaces.

What is a Conic? Simple: a conic is the intersection of a cone and a plane. This fact means that it is a simple, predictable curve. It is always planar, always convex, and always second-order.

Conic sections include ellipses, parabolas, hyperbolas, and technically even circles and arcs, though circles aren?t usually thought of as ?conics?. These shapes are surprisingly powerful, and often under-utilized by surfacing beginners.

There are many advantages to the use of conics in your CAD models. For one thing, a conic is a second-order curve: the math is very light-weight, and will help keep your models small and easy to regenerate. Because they are always planar, conic sections create very clean swept surfaces when used with a clean ?spine,? which we?ll go over in detail in a future tutorial. Thirdly, conics can appear attenuated, and give you the power to control their apparent ?sharpness?, and even to simulate the look of G2 continuity. But, as I mentioned before, unlike G2, a conic will never ?ogee?, or ?wobble.? A conic?s curvature acceleration will always be smooth, and positive. Lastly, there are many ways to define conics; probably many more than you?ve considered so far.

The most common use for conics in Class-A Surfacing is as a ?blend? between curves or surfaces. Usually this is accomplished by providing two tangency surfaces, contact points, and what?s called a ?rho? value to define how sharp the curve is. In other cases, it is not uncommon to use a third contact point instead of the rho value.

But what does that mean? What is a Rho value, and what does it really do? If a conic is an intersection between a plane and a cone, the ?Rho? value is the angle of the plane relative to the cone. By steepening the angle of the plane, the resulting curve will appear ?sharper?. By flattening the plane, the curve will appear ?rounder?. A Rho value is not expressed as an angle, but as a ratio between zero and one. A Rho value of .5 is a fairly ?round? curve, and a Rho value of .9 is a very ?sharp? curve.

So in practical use, the most common example of a conic in Class-A Surfacing is as a ?blend? between curves or surfaces. As I mentioned before, this has many advantages over spline-based G1 and G2 blends.

For example, while conic sections can look very much like G2 curves, then will always remain perfectly convex as shown above. The G2 curve can be adjusted to correct the inflection, but this requires manual manipulation of the G2 ?tension?, and can easily be overlooked if the inflection is very visually subtle. With a conic, there can be no doubt that the curvature never inverts.

Conic sections with a Rho value of about .65 or higher will usually appear to be ?smooth? visually. To the human eye, the difference between G2 and a conic in these cases is virtually non-existent. With very careful manipulation, a conic with a perfectly-calculated Rho value can even actually become mathematically G2 continuous in certain situations. This isn?t common at all, but I just wanted to point out that it can be done in a pinch. Typically you must remember that conic blends are tangent continuous, not curvature continuous, and that therefor geometric continuity degradation can occur.

So this ends our quick introduction to the world of conics! In the next series of tutorials we?ll be exploring how these beauties work in actual practice, using real CAD tools. Thanks for watching, and happy surfacing!

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.