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(t) = 1/6[(1 - t)^3p0 + (3t^3 - 6t^2 + 4)p1 + (-3t^3 + 3t^2 + 3t + 1)p2 + t^3p3] (14) for 0 ≤ t ≤ 1, where p0, p1

Question

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

[latex] x(t) = \frac{1}{6}[(1 - t)^{3}p_{0} + (3t^{3} - 6t^{2} + 4)p_{1} + (-3t^{3} + 3t^{2} + 3t + 1)p_{2} + t^{3}p_{3}] [/latex] (14)

for 0 ≤ t ≤ 1, where [latex] p_{0} , p_{1} , p_{2} [/latex], and [latex] p_{3} [/latex] are the control points. When t varies from 0 to 1, x(t) creates a short curve that lies close to [latex] \overline{p_{1}p_{2}} [/latex]. Basic algebra shows that the B-spline formula can also be written as

[latex] x(t) = \frac{1}{6}[(1 - t)^{3}p_{0} + (3t(1 - t)^{2} - 3t + 4)p_{1} + (3t^{2}(1 - t) + 3t + 1)p_{2} + t^{3}p_{3}] [/latex] (15)

This shows the similarity with the Bézier curve. Except for the 1/6 factor at the front, the [latex] p_{0} [/latex] and [latex] p_{3} [/latex] terms are the same. The [latex] p_{1} [/latex] component has been increased by -3t + 4 and the [latex] p_{2} [/latex] component has been increased by 3t + 1. These components move the curve closer to [latex] \overline{p_{1}p_{2}} [/latex] 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.

1. Show that the B-spline does not begin at [latex] p_{0} [/latex], but x{0} is in conv {[latex] p_{0} , p_{1} , p_{2} [/latex]}. Assuming that [latex] p_{0} , p_{1} [/latex], and [latex] p_{2} [/latex] are affinely independent, find the affine coordinates of x(0) with respect to {[latex] p_{0} , p_{1} , p_{2} [/latex]}.

2. Show that the B-spline does not end at [latex] p_{3} [/latex], but x(1) is in conv {[latex] p_{1} , p_{2} , p_{3} [/latex]}. Assuming that [latex] p_{1} , p_{2} [/latex], and [latex] p_{3} [/latex] are affinely independent, find the affine coordinates of x(1) with respect to {[latex] p_{1} , p_{2} , p_{3} [/latex]}.

Accepted Answer

1. From equation (14) with [latex] t = 0 , x(0) eq p_{0} [/latex] because

[latex] x(0) =\frac{1}{6}[p_{0} + 4p_{1} + p_{2}] = \frac{1}{6}p_{0} + \frac{2}{3}p_{1} + \frac{1}{6}p_{2} [/latex] .

The coefficients are nonnegative and sum to 1, so x(0) is in conv {[latex] p_{0} , p_{1} , p_{2} [/latex]}, and the affine coordinates with respect to {[latex] p_{0} , p_{1} , p_{2} [/latex]} are {[latex] \frac{1}{6} , \frac{2}{3} , \frac{1}{6} [/latex]}.

2. From equation (14) with [latex] t = 1 , x(1) eq p_{3} [/latex] because

[latex] x(1) =\frac{1}{6}[p_{1} + 4p_{2} + p_{3}] = \frac{1}{6}p_{1} + \frac{2}{3}p_{2} + \frac{1}{6}p_{3} [/latex] .

The coefficients are nonnegative and sum to 1, so x(1) is in conv {[latex] p_{1} , p_{2} , p_{3} [/latex]}, and the affine coordinates with respect to {[latex] p_{1} , p_{2} , p_{3} [/latex]} are {[latex] \frac{1}{6} , \frac{2}{3} , \frac{1}{6} [/latex]}.

Question 8.6.PP.1: A spline usually refers to a curve that passes through speci...

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