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 of TinySpline, though you don't need to be a C expert to follow this tutorial.
###About Splines
Let > be the superset relation so that we can
write A > B in case of two sets A, B with A being the superset of B.
Furthermore, we define the following sets of splines:
#NURBS#- The set of all NURBS curves.#BSPLINES#- The set of all B-Spline curves.#BEZIERS#- The set of all Bézier curves.#LINES#- The set of all lines.#POINTS#- The set of all points.
Then, the following equation holds:
#NURBS# > #BSPLINES# > #BEZIERS# > #LINES# > #POINTS#
To put it differently, we can say that every point is a line, ever 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!), and (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 <= u_{i+1} for all knot values u in knots and 0 <= i < n-1. The domain of b is
given by u_p and u_{n-p}. That means, you can retrieve points laying on b by
evaluating b at any knot value u with u_p <= u <= 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 is 3, n is 9, and m = n-p-1 = 9-3-1 is 5. The domain of b is
[6, 18] so that you can evaluate b at any knot value u between 6 and 18, for
instance, 6, 6,1316, 9, 10, 11, 12, 13,333, 13,6, 17,999999, 18 and so
on, but not at values less than 6 or greater than 18. As you may have noticed, the knot
value 11 occurs twice in knots. Repeating knot values is permitted in principle but,
however, 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, s(u) <= p+1 for
all knot values u in knots. 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' (q = r) and, thus, can be seen as a
single 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).