Automotive a-class explained
“NURBS” and transition theory
NURBS
Non-uniform rational basis splines
nurbs and mesh
There are two types of 3d data that dominate today; mesh and nurbs. There are many other formats, but these two are the most common.
Mesh was the first kind of 3d data used in computer graphics, it is easy to work with and quick to render. A mesh consists of a set of connected points.
Two points connected define a line, and three points define a plane. So a mesh is seen as a lot of connected triangular* surfaces.
* I know, I know quads and n-gons “exist”. It is a much more practical way to work. But when you get down to brass tacks they are still made of triangles.
Nurbs are a little more complicated. They have their origin in drafting (making technical drawings) especially those of boat builders. Drafters needed to have a precise and reproducible way of making complex curves. The answer was a flexible “ruler” called a spline held in place with weights.
In the 60’s engineers at Renault and Citroën adapted polynomial functions into a something very akin to the old spline tool. They came up with slightly different solutions, but functionally they do the same thing. The Renault engineer named it “Bézier curves” (after himself). The Citroën engineer stuck with the mathematical name for them; “non-uniform rational basis splines” or NURBS for short.
Old-school spline ruler
Bezier curve
In this digital version the weights are replaced by points that “pull” on the spline. Many industries recognized this utility quickly and it was adapted into many other fields. The font in text you are reading right now is defined by the same function these engineers came up with in the 60’s.
If we ignore the computing power needed, transitioning this concept from curves to surfaces was relatively minor. It was basically just stepping up one dimension.
Length x width = area.
An apt analogy when comparing mesh to nurbs is to compare pixel and vector graphics. You can approximate something curved, but the closer the approximation you need, the more data you will need to describe it.
In the picture to the right the mesh surface is defined with 50 points and is quite jagged. The nurbs surface next to it is defined with 12 points and is perfectly curved no matter what scale.
nurbs and mesh
Transition theory
Beyond the tangent
Quick word of warning. In this section I will be talking about surface quality. If you are blissfully unaware how atrocious the surface quality of most consumer products are.. you might want to skip this part.
G0 “Position”
The two edges are touching. G0 is the lowest level of tolerance when surfacing. If the surfaces aren’t touching (within tolerance) you won’t be able to treat them as a solid object and so they can’t be produced.
What’s with the weird stripes?
Reflecting a straight line, like the horizon or fluorescent lamps in a garage, is a worst case scenario for highlights.
These images are rendered with an evaluation tool commonly called a zebra shader. You can imagine it like the surfaces are completely reflective and they are reflecting a wall with black and white stripes on it so it is a very good way of scrutinizing the highlights of the surfaces.
People who do “paintless dent repair” use a light panel with stripes for the same reason. Check the link below for an artisan in action.
To demonstrate the different transitions I will be adding a rounding “fillet” to the edge from first picture.
G1 “Tangent”
The surfaces have the same angle where they meet. This transition is often circular, it is relatively quick and easy to make and is (too) often seen as good enough.
Unfortunately a tangent transition look pretty bad even in matte or grained materials. They cause what looks like darker lines at the transition, commonly called tangent lines.
This is what I was talking about earlier. Since it is so commonplace I get a little annoyed by random things on a day to day basis; plastic bottles, appliances, electronics, peoples glasses, phones, trains, furniture.. I’ve even seen it on a building.
If you enjoy pet peeves like these I would also recommend you look into “kerning”
So how do you fix tan-lines?
G2 “Curvature”
The surfaces have the same bend (curvature) in the transition
This is typically the transition we strive for in the automotive industry. It looks a lot smoother than a tangent transition. It can however still show a little bit of a tangent line if the surfaces the fillet builds on are too flat.
G3 “Torsion”
The surfaces have the same curvature change in the transition
This transition is pretty much “invisible” it is mostly used for very visible edges, large transitions and flat surfaces. It is also a pain to build with. I try to remember what Renoire said “The pain passes but the beauty remains.”
