Chapter 8.6, Problem 1PP

A spline usually refers to a curve that passes through specified points. A B-spline, however, usually does not pass through its control points. A single segment has the parametric form

$x\left(t\right)=\frac{1}{6}\left[{\left(1-t\right)}^3{p}_0+\left(3t^3-6t^2+4\right)p_1+\left(-3t^3+3t^2+3t+1\right)p_2+t^3{p}_3\right]$    (14)

for 0 ≤ t ≤ 1. where p₀, p₁, p₂, and p₃ are the control points. When t varies from 0 to 1, x(t) creates a short curve that lies close to $\overline{p_1{p}_2}$. Basic algebra shows that the B-spline formula can also be written as

$x\left(t\right)=\frac{1}{6}\left[{\left(1-t\right)}^3{p}_0+\left(3t{\left(1-t\right)}^2-3t+4\right)p_1+\left(3t^2\left(1-t\right)+3t+1\right)p_2+t^3{p}_3\right]$    (15)

This shows the similarity with the Bézier curve. Except for the 1/6 factor at the front, the p₀ and p₃ terms are the same. The p₁ component has been increased by – 3t + 4 and the p₂ component has been increased by 3t + 1. These components move the curve closer to $\overline{p_1{p}_2}$ than the Bézier curve. The 1/6 factor is necessary to keep the sum of the coefficients equal to 1. Figure 10 compares a B-spline with a Bézier curve that has the same control points.

FIGURE 10 A B-spline segment and a Bézier curve.

1. Show that the B-spline does not begin at p₀, but x(0) is in conv {p₀, p₁, p₂}. Assuming that p₀, p₁, and p₂ are affinely independent, find the affine coordinates of x(0) with respect to {p₀, p₁, p₂}.