## Mathematical Principles

The following chapter describes some basics about splines and how to use them in TinySpline. At first, we start with a mathematical examination of NURBS, B-Splines, Bézier curves, lines, and points. The given code examples are using the C interface.

Let $$\supset$$ be the superset relation so that we can write $$A \supset B$$ in case of two sets $$A, B$$ with $$A$$ being the superset of $$B$$. Furthermore, we define the following sets of splines:

• $$\text{NURBS}$$
The set of all NURBS curves.
• $$\text{BSPLINES}$$
The set of all B-Spline curves.
• $$\text{BEZIERS}$$
The set of all Bézier curves.
• $$\text{LINES}$$
The set of all lines.
• $$\text{POINTS}$$
The set of all points.

Then, the following equation holds:

$$\text{NURBS} \supset \text{BSPLINES} \supset \text{BEZIERS} \supset \text{LINES} \supset \text{POINTS}.$$

To put it differently, we can say that every point is a line, every line is a Bézier curve, every Bézier curve is a B-Spline, and ever B-Spline is a NURBS. In order to understand this relation, we will start with a (simple) definition of B-Splines that is more code- than maths-related.

A B-Spline $$b$$ consists of four attributes:

1. The degree of $$b$$ describing the (general) smoothness of $$b$$.
2. The control points of $$b$$ used to shape $$b$$.
3. The knot vector of $$b$$ (simply knots) used to define the continuity of $$b$$ at certain points (not the control points!).
4. The dimension of $$b$$ defining how many components each control point has (for instance, (x, y) in 2D).

A B-Spline $$b$$ of degree $$p$$ with $$m$$ control points has $$n = m+p+1$$ knots. The knot vector $$knots$$ must be in ascending order (monotonically increasing). That is, $$u_i \leq u_{i+1}$$ for all knot values $$u$$ in $$knots$$ and $$0 \leq i < n-1$$. The domain of $$b$$ is given by $$u_p$$ and $$u_{n-p}$$. That means, one can retrieve points laying on $$b$$ by evaluating $$b$$ at any knot value $$u$$ with $$u_p \leq u \leq u_{n-p}$$. The evaluation of $$b$$ at $$u$$ is denoted by $$b(u)$$.

The following example illustrates the domain of B-Splines. Let's say $$b$$ is a B-Spline of degree 3 (cubic B-Spline) with the following knot vector:

$$knots = [4, 5, 6, 10, 11, 11, 18, 19, 20]$$

This implies that $$p = 3$$, $$n = 9$$, and $$m = n-p-1 = 9-3-1 = 5$$. The domain of $$b$$ is $$[6, 18]$$ so that $$b$$ can be evaluated at any knot value $$u$$ with $$6 \leq u \leq 18$$, for instance, $$6, 6,1316, 11, 17,999999, 18$$ and so on. As you may have noticed, the knot value $$11$$ occurs twice in $$knots$$. Repeating knot values is permitted in principle, but is limited to the degree of a B-Spline (as we will see in the following definition).

Given a B-Spline $$b$$ of degree $$p$$ with knot vector $$knots$$, the multiplicity $$s(u)$$ of any $$u \in knots$$ is limited by the order ($$p+1$$) of $$b$$. That is, $$\forall u \in knots: s(u) \leq p+1$$. Furthermore, $$b$$ is $$C^{p-s(u)}$$ continuous at $$u$$ so that increasing $$s(u)$$ by $$1$$ decreases $$b$$'s continuity at $$u$$ by $$1$$.

That said, you will notice that B-Splines are discontinuous ($$C^{-1}$$ continuous) at certain points if the corresponding knot values have multiplicity $$p+1$$. Evaluating a B-Spline $$b$$ at $$u$$ with $$s(u) = p+1$$ almost always returns two points $$q, r$$ rather than one, though $$q$$ and $$r$$ may be the same point. As if all this wasn't complicated enough, there is a special case in which evaluating $$b$$ always returns a single point regardless of $$b$$'s continuity at this point. It occurs when the evaluated point is the very first or rather very last point of a B-Spline (you can think of a B-Spline's ending points).

To be continued