# CURVES

From time to time I’m having interesting talks with people who are mainly using Grasshopper and Rhino for their job. They are working in the field of architecture and/or design and even though they are usually quite interested in Houdini one of the first things I always hear is how bad the curve tools are compared to Rhino/Grasshopper. Well, that’s not true! Or let’s say, it’s just half the truth. Even though curves, and Nurbs in general, could get some love they are quite robust and not too bad to work with. (As long as we don’t need to fiddle around with trimmed surfaces). Houdini is not a CAD program and naturally it doesn’t have all the features which are available in Rhino/Grasshopper. So, when I asked what the main curve tools are that Grashopper users are missing the most in Houdini I got a list with the following five points:

• Curvature
• Curvature graph
• Curve deviation
• Curvature based resampling
• Approximate curves by circular arcs

Of course, there are many other things which are missing, for instance, better handles to create and modify curves but since the points above seem to be the most important features to Grasshopper users I’ll stick with those. However, curves are not too complicated and if you ever need to implement or generate “custom” curves there’s plenty of good information available online. For a start you could take a look at these simple example files on OdForce.
But now back to the missing feature list. Before we start, it’s important to note that there are mainly two different kinds of curves in Houdini. On one hand we have Nurbs and Bezier curves which are basically polynomial curves parameterized by a function. On the other hand we have polygonal curves which are usually just points connected by straight lines. If we like, we could call the two types smooth and discrete curves.

Curvature
Curvature is fairly easy to calculate if we understand what it actually means in the context of curves. Basically it tells us how fast a curve is changing direction when we travel along it. In more mathematical terms, it’s the magnitude of acceleration. So far so good but what exactly does this mean? Well, acceleration is the second derivative, or in other words, the normal of the curve. The first derivative is velocity or slope and this is nothing more than just the tangent. So, if we want to compute curvature it comes down to implementing the following three simple steps:

• Compute first derivative namely the tangent
• Compute second derivative (which is the normal) based on tangents
• Measure magnitude of normal which finally is curvature

A nice side effect of this procedure is that we can easily compute a local frame since we already have tangent and normal. We just need to compute the cross product of these two known vectors and we’ll get the third vector and hence an orthogonal local frame on our curve. To get the osculating circle at a given point, like it’s possible in Grasshopper, we need to calculate the radius which is just the inverse of curvature.
All this is relatively easy to implement in VEX, both for Nurbs and polygonal curves and it’s also quite fast. However, for discrete curves the easiest method is most probably the use of a slopeChop in CHOPS. On OdForce there is a rather old example file using this technique to compute the Frenet frame. It’s not exactly about curvature but it shows the general setup.

Curvature graph
Well, this is easy because it’s just the normals reversed and scaled by curvature and a multiplier by the user.

Curve deviation
This is easy too. We just need to use xyzdist() at sampling points on the first curve to get the distance to the second curve.

Curvature based resampling
This is again a job for xyzdist() which we could use to implement our own curve sampling algorithm in VEX. This could be done based on a minimal distance or maximal angle threshold by using a bit of basic trigonometry together with the xyzdist() and primuv() combo.

Approximate curves by circular arcs
This was the most important point on the list what makes totally sense for applications in the field of architecture and design. If someone is actually going to built something physically it might be a good idea to find a way how to approximate an arbitrary 3D curve by circular arcs as close as possible. So, how could we implement this in Houdini? Well, there are many different ways but the easiest would be to “brute-force” it. Since an arc is well defined by three points we could start with two close points at one end of the curve. Next we find the point in between construct an arc and estimate the maximal deviation to the curve. If it’s smaller than an user given threshold increase the distance between the points and repeat the steps until we get the largest arc within our tolerances. Then we just repeat the whole procedure until we finally get our arc approximated curve. VEX is pretty fast and hence this algorithm should work quite well performance-wise, even for very dense resampled curves.
However, I thought it might be better to choose a different and probably smarter method. The reason is mainly because the approximation should be as smooth as possible without large kinks along the curve. Instead of using the algorithm described above, which is not guaranteed to result in tangent arcs, we could use a method called biarc fitting. So, what is a biarc? A biarc is a pair of two arcs with the same tangent at the point at which they meet. And this is exactly what we need for a good and smooth approximation of the curve. The general algorithm is fairly easy to implement and works basically as follows:

• Get tangent at endpoints
• Sample curve and search for best connection point for biarc
• Estimate deviation between biarc and curve
• If deviation is larger than given tolerance split curve and repeat
• If deviation is smaller than given tolerance compute the biarc

Instead of using just the distance between biarc and curve I’m using various different measurands to estimate the error, such as max. distance, average distance, torsion angle and tangent angle. The images below show some results using the distance with different tolerance settings. Of course, there are some things we need to take care of, for instance, splitting curves at inflection points but generally biarc fitting is fairly easy to implement and works quite well for all sorts of curves.