Why aren't regular quadratic and cubic splines used much in games?

Question

After investigating a bit about splines and reading a bunch of docs, I have noticed regular quadratic and cubic splines are not used very much in games.

Splines using quadratic/cubic Bézier curves, Catmull-Rom, B-splines, and other spline interpolation methods seem to be used widely though.

I can be totally wrong here but from books and tutorials I have seen that for regular cubic/quadratic you have to solve systems including first derivative for continuity and second derivative to match curvature. It seems that the most used splines (Bézier, Catmull-Rom, etc...) implicitly bring these features with fewer calculations.

Is this the reason that regular quadratic and cubic spline interpolation are not often used in game programming?

Answer 1

A nice place to start (without the mathematical heaviness associated with the Differential Geometry of Curves and Surfaces) would be this. The author discusses the uses and practical advantages or disadvantages of these curve primitives with use cases taken from a game designer's perspective (or from a game editor developer's perspective if you will).

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This source seems to discuss the essentials: types of curves and their properties (Hermite, Bezier, Catmull-Rom, Cubic Splines), how to evaluate them.

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Performance wise, this Dr. Dobbs article discusses what can be done to evaluate Bezier curves efficiently. Many others can be found as the field is really mature and the applications are widespread.

Finally, this is a digital book that discusses Bezier and Spline curves in minute detail. If you are already familiar with these concepts, you can skip to the chapters that explain what algorithms are used to evaluate them.

Answer 2

From the question its not entirely clear what "regular" means in "regular quadratic and cubic splines are not used much." If you mean standard cubic equations, such as ax^3 + bx^2 + c, the reason they are not widely used is there is no direct relationship between coefficients and spatial control points. Given a set of points one would have to solve a system of equations for the coefficients a,b,c. This is called curve fitting: it reduces dimensionality, is slow to compute, but is widely used in the sciences/statistics to approximate data with a lower-degree curve.

Bezier Cubic, Catmull-Rom and B-Splines are used primarily because they have local control (moving controls moves the curve in an expected way), piecewise connectivity (they can be placed end-to-end for arbitrary number of controls), C1 or C2 continuity (smoothness) and may or may not interpolate their controls. The piecewise aspect is especially useful because it means you can have arbitrarily many data points (controls) and the curve will pass thru or near all of them. These features define the specialization of splines. Yet under-the-hood each of them does have a blending of standard cubic polynomials.

Answer 3

Graphics accelerator cards are not equipped to deal with them. Polygon-related operations like rasterization itself are taken care of quickly and automatically. The other operations require discrete points to operate on, like frustrum culling or even view projections.

These operations don't work at all with this kind of pipeline. They only work properly with a ray-tracing-like render algorithm. I say like to stave off any super pedantic people out there that might correct me and talk about other algorithms like photon-tracing that are basically the same thing.

When rendering triangles as in OGL you basically take discrete object data and transform them onto the screen. When you take these other more abstract graphics representations you don't have a discreet object at all. When you do ray tracing this doesn't really matter. When you do rendering like in OGL first you have to transform all of these into discrete objects (which generally look terrible compared to the "real" object). Or else use some kind of ray tracing with compute shaders or something and somehow put those results in among the results of your discreet rendering.

So while you can do it, it's mostly not practical to do so in OGL at least for a gaming application. Maybe it can be used here and there for certain things like allowing users to create content, but you won't see it used for data representation as the norm any time soon.