NURBS: Curves and Surfaces

Among others, Sverchok supports a special kind of curves and surfaces, called NURBS (Non-Uniform Rational B-Splines). NURBS objects are widely used in 3D graphics, and, especially, in CAD software.

Blender itself has support of NURBS objects, but it is limited at the moment; NURBS support in Sverchok is wider.

At the moment, Sverchok supports two implementations of NURBS mathematics:

  • Geomdl, a.k.a. NURBS-Python, library. It is relatively widely known and used, so we consider it as better tested. However, to use this implementation, you have to install Geomdl library into your Blender installation. Please refer to Dependencies wiki page for installation instructions.
  • Built-in (native) implementation in Sverchok itself. In many cases, it is faster than Geomdl implementation, and it does not require any additional dependencies.
  • FreeCAD libraries. FreeCAD uses OpenCASCADE kernel, which is implemented in C++, so it is faster than Geomdl implementation; it is also considered to be more widely tested compared to Sverchok built-in implementation. However, it is in general slower comparing to built-in implementation, when you want to evaluate a lot of points on one curve or surface. Please refer to Dependencies wiki page for installation instructions.

Terminology

While widely used, for some reason, NURBS do not seem to have a “standard” terminology: different software and different books use different terms for the same things. Sverchok tries to not invent a 15th standard, and to follow the terminology used in “The NURBS Book”. Here we are listing some common terms for reference.

  • Control Point: a point in 3D space, one of several points that define common shape of the curve or surface. In general, NURBS curve / surface does not go through it’s control points, apart of some special cases. NURBS surfaces always have a rectangular grid of control points (m x n).
  • Control Polygon (for a curve): a polygon composed from control points of a curve, connected with edges.
  • Control Net (for a surface): a grid-like mesh, composed from control points of a surface, connected with edges.
  • Weight: a number, usually positive, which defines how strong a given control point “attracts” the curve or surface. Sverchok can work with negative weights usually, but there is a lot of software that can’t, so usually you want to avoid use of negative weights.
  • Degree: an integer number, from 1 to however you want, which is the degree of polynomials used in mathematical definition of NURBS curves. NURBS surfaces have two degrees: one along U direction, and another along V direction.
  • Non-rational B-Spline, or simply B-Spline: a special kind of NURBS objects, that does not allow to specify weights; it is the same as a NURBS object with all weights set to 1.
  • Rational B-Spline, or NURBS: a general kind of objects, that allows to specify different weights for different control points. Mathematically, such objects are defined as a relation of two polynomial expressions, hence the name “rational”.
  • Knot vector: a non-descending sequence of numbers, which define ranges of curve/surface parameters, to which different control points have some relation. It is hard to explain in a few words how exactly the knot vector affects the shape of curve of surface, so for details please refer to the literature. Each number in the knot vector is called knot. Knot vector of a NURBS curve with degree p and n control points always must contain exactly p + n + 1 knots. NURBS surfaces have two knot vectors: one for U parameter and one for V.
  • Knot multiplicity. A number of times one knot is repeated in the knot vector. For example, if we have a knot vector 0 1 2 2 2 3 4 5, then we say that knot 2 has multiplicity of 3.
  • Uniform knot vector: a knot vectors, values in which increase always by the same amount (uniformly). For example, 0 1 2 3 4 5. Sometimes this kind of knot vectors is called “periodic” (for reasons that we have no place to explain here). Uniform knot vectors are not very often used, because curves with such knot vectors do not pass through their first and last control points (in general).
  • Clamped knot vector. A special kind of knot vector, which has multiplicity of it’s first and last knots equal to p +1, where p is the degree of the curve (or surface, along one of directions). For example, if we are talking about a curve with degree equal to 2, then 0 0 0 1 2 3 4 5 5 5 is a clamped knot vector. Curves with such knot vectors have a good property: they always pass through their first and last control points. This is a reason why clamped knot vectors are widely used.
  • Clamped Uniform knot vector. Knot vector which is “almost uniform”, but clamped; i.e. it has multiplicity of first and last knots equal to p+1, but all other knots are ascending evenly. For example, 0 0 1 2 3 4 4 is a clamped uniform knot vector.
  • Non-Uniform knot vector. Most general form of knot vectors. Knot vector, in which distance between neighbour knots is different in different places. For example, 0 0 0.5 0.6 1.5 1.7 1.9 2 2 2 is a non-uniform (and clamped) knot vector.

Workflow

Some software products have a “full NURBS workflow”, which means that all curves / surfaces it is operating with is always NURBS, and whatever complex things you do with those objects you will always have NURBS again. Sverchok does not have a goal to have “full NURBS workflow”, at least at the moment. Blender is, first of all, a mesh editing software, so, it is very probable, that the most widely used workflow always will be to manipulate with NURBS curves / objects for some time, together with other types of curves / objects, but then convert them to mesh and apply some nodes that manipulate with mesh, to receive a mesh in the end.

Some nodes have “NURBS output” parameter in their settings; when this parameter is enabled, they output NURBS objects, otherwise a generic curve or surface is generated.

There are some number of nodes, that can be called NURBS-transparent; such nodes have a property: when they receive NURBS on the input, they will always output NURBS. Examples of NURBS-transparent nodes are “Ruled Surface” and “Surface of Revolution”. Some nodes are NURBS-transparent only when you enable some setting in them (see documentation of specific nodes). Number of NURBS-transparent nodes will probably grow, but there is no guarantee that some time all curve / surface processing nodes will become NURBS-transparent (and there is no such goal at the moment).

“NURBS-transparent” nodes also automatically convert several special kinds of curves into NURBS, when such curves are passed to inputs of such nodes. We call types of curves, that can be automatically (and exactly, not approximately) converted to NURBS, “NURBS-like”. Examples of NURBS-like curves are:

  • Bezier curves
  • Cubic splines
  • Polylines
  • Circular arcs
  • Line segments

Sverchok has “NURBS to JSON” and “JSON to NURBS” nodes, which allow to save NURBS objects in JSON format and read NURBS from it; such JSON format can be used with rw3dm utlity to convert it from / to 3dm files. Later there can appear nodes that will export NURBS objects to other widely-used formats.

So, with some restrictions, it is possible to prepare complex scenes built from NURBS objects only, to export them to other CAD software for further processing or manufacturing. This is, however, not a primary workflow at the moment.

Blender NURBS compatibility

Blender’s internal NURBS support is currently limited in two aspects:

  • It’s Python API for manipulating NURBS objects is very poor;
  • It does not allow to specify an arbitrary knot vector for curve or surface; only two special kinds of knot vectors are supported: “clamped uniform knot vectors” and “uniform knot vectors”.

So, Sverchok has limited features in interacting with Blender’s native NURBS objects:

  • NURBS In node can bring arbitrary Blender’s NURBS curves or surfaces from scene to Sverchok space;
  • NURBS Curve Out and NURBS Surface Out nodes allow to generate Blender’s NURBS objects in scene, but without possibility to specify arbitrary knot vectors.