3. B-spline curves

An easy way of making a controlled-design curve with many control points is to use B-spline curves. The ones we shall discuss are called relaxed uniform cubic B-spline curves. You start by specifying a control polygon of points $B_0, B_1,\dots B_n$, and you end by getting a curve like the one in Figure .

Figure: A relaxed uniform cubic B-spline curve

Here is the method, if done by hand: Divide each leg of the control polygon in thirds by marking two ``division'' points. At each $B_i$ except the first and last, draw the line segment between the two nearest ``division'' points, and call the midpoint $S_i$. Then you have made an A-frame with $B_i$ at the apex, as shown. For completeness, let $S_0 = B_0$ and $S_n = B_n$. See if you can locate the four A-frames in Figure .

Figure: A-frames for a relaxed cubic B-spline

Finally, sketch a cubic Bézier curve from each point $S_i$ to the next, using as Bézier control points the four points $S_i$, two ``division'' points, and $S_{i+1}$, as in Figure .

Figure: Construction of a relaxed cubic B-spline

As you see, the points of gluing meet the A-frame condition automatically and at the ends the second derivative is zero. Therefore you obtain a relaxed cubic spline curve.

The method as performed on a computer is the same; we merely need to find the Bézier control points in terms of the original B-spline control points:

The ``division'' points on the line segment from $B_{i-1}$ to $B_i$ are ${\frac 2 3} B_{i-1} + {\frac 1 3} B_i$ and ${\frac 1 3} B_{i-1} + {\frac 2 3} B_i$. Also, $S_i$ is the average of the ends of its ``cross-segment'', so that

$S_i = {\frac 1 2}({\frac 1 3} B_{i-1} + {\frac 2 3} B_i) + {\frac 1 2}( {\frac 2 3} B_i + {\frac 1 3} B_{i+1})$ $=$ ${\frac 1 6} B_{i-1} + {\frac 2 3} B_i + {\frac 1 6} B_{i+1}$, for $i = 1,\dots, n-1$.

To summarize the computer method:

Given B-spline control points $B_0,\dots, B_n$, calculate $S_i$ $=$ ${\frac 1 6} B_{i-1} + {\frac 2 3} B_i + {\frac 1 6} B_{i+1}$, for $i = 1,\dots, n-1$, and let $S_0 = B_0$, $S_n = B_n$. There are $n$ Bézier curves to plot; curve #$i$ has control points $S_{i-1}$, ${\frac 2 3} B_{i-1} + {\frac 1 3} B_i$, ${\frac 1 3} B_{i-1} + {\frac 2 3} B_i$, and $S_i$. On curve #$i$, you can plot points on the curve for, say, $t = 0, .05, .10,\dots, .95,1$.

Finally, let's consider the situation mathematically. Let $p_i (t)$ be the $i$th Bézier curve ( $0 \leq t \leq 1$). These $n$ curves can be combined into a single curve $P(t)$ for $0 \leq t \leq n$ by letting

$P(t) = p_1 (t)$ for $0 \leq t \leq 1$,

$P(t) = p_2 (t-1)$ for $1 \leq t \leq 2$, etc. In general,

$P(t) = p_i (t- (i-1))$ for $i-1 \leq t \leq i$, where $i = 1,\dots, n$.

Then $P(t)$ is a relaxed cubic spline curve. $P(t)$ is called a uniform spline curve because its domain $0 \leq t \leq n$ was made from intervals all of length 1. Non-uniform curves will be considered in Section 10.

An important virtue of B-spline curves is that the influence of individual control points is local. In fact, any one point on the curve is influenced by at most four of the B-spline control points. The reason is that for Bézier curve #$i$, all four control points can be computed from a knowledge of $B_{i-2}$, $B_{i-1}$, $B_i$, and $B_{i+1}$. Similarly, control point $B_i$ influences only four Bézier-curve segments: the two that join at $S_i$ and the two additional ones joined to those. The local effect can be illustrated by changing a single control point. In Figure , two choices of the middle control point are indicated, along with the corresponding B-spline curves. Dots on the curves indicate some gluing points $S_i$.

Figure: The effect of varying one control point

What if you don't want relaxed end conditions? In that case, you can just use less of the curve, say the part from $S_1$ to $S_{n-1}$, i.e., $1 \leq t \leq n$. $B_0$ and $B_n$ can still be used as control points to affect the shape of the part of the curve that you are using.