Cubic B-Splines

May 22, 2020

Roger Ngo

Cubic B-splines are different than Hermite, and Bezier curves in a couple of ways.

The curves generated by cubic B-Spline formulation do not pass through the control points -- only near them.
The curves generated are an interpolation of segments of $t$ , and not points at $t$ .

Unlike Hermite, and Bezier curves, where they can generate equivalent curves, the cubic B-spline is different.

This discussion is limited to uniform B-splines.

A uniform B-spline consists of control points $P_0 ... P_n$ , and is defined by the following conditional expression:

\gamma(t) = \sum^{n}_{i = 0} P_ib_3(t - i) \\<br>b_3(t) = \begin{cases} <br>\frac{1}{6}{t}^3, 0 \le t \le 1 \\<br>\frac{1}{6}(-3(t - 1)^3 + 3(t - 1)^2 + 3(t - 1) + 1), 1 \le t \le 2 \\<br>\frac{1}{6}(3(t - 2)^3 - 6(t - 2)^2 + 4), 2 \le t \le 3 \\<br>\frac{1}{6}(-(t - 3)^3 + 3(t - 3)^2 - 3(t - 3) + 1), 3 \le t \le 4 \\<br>0, otherwise<br>\end{cases}

We are interested in interpolating a single segment for a B-spline. For example, given four control points, $P_0 = (0.2, 0), P_1 = (0.2, 0.2), P_2 = (0.5, 0.2), P_3 = (0.5, 0)$ , we can interpolate a single segment which comes close to point $P_1$ , and $P_2$ .

Segment

Controlling the other two endpoints, $P_0$ , and $P_2$ controls the curvature of the segment.

We can compute a B-spline quickly by using the matrix representation, given $\bold{T} = \begin{bmatrix} 1 & t & t^2 & t^3 \end{bmatrix}$

\gamma(t) = \bold{G}\bold{M}\bold{T}

Where $\bold{G}$ represents the collection of control points for the particular segment to be interpolated.

\bold{G} = \begin{bmatrix} P_j & P_{j+1} & P_{j + 2} & P_{j + 3} \end{bmatrix}

$j$ in this case, presents the segment to be interpolated. So, if we want to interpolate the 2nd segment, we would the set of points would be: $P_2, P_3, P_4, P_5$ .

\bold{M} = \frac{1}{6} \begin{bmatrix} <br>0 & 0 & 0 & 1 \\<br>1 & 3 & 3 & -3 \\<br>4 & 0 & -6 & 3 \\<br>1 & -3 & 3 & -1<br>\end{bmatrix}

The expression for the first segment looks like:

t = 0..1, \Delta{t} = 0.01<br>\\~\\<br>\gamma(t) = \begin{bmatrix} P_0 & P_1 & P_2 & P_3 \end{bmatrix} \frac{1}{6} \begin{bmatrix} <br>0 & 0 & 0 & 1 \\<br>1 & 3 & 3 & -3 \\<br>4 & 0 & -6 & 3 \\<br>1 & -3 & 3 & -1<br>\end{bmatrix} \begin{bmatrix} 1 & t & t^2 & t^3 \end{bmatrix}

The below example interpolates many segments with the points:

X	Y
-0.5	0
-0.5	0.5
-0.3	0.5
-0.3	0.3
-0.1	0.1
-0.1	0.5
0	0
0.3	-0.5
0.5	0
0	0.6
-0.1	-0.6
0.8	0.8

With Control

Without Control

We are able to generate beautiful curves this way!

Limitations

With a finite set of control points, you cannot make a B-spline traverse a unit circle. We will need to use rational B-splines to achieve this.