ECS 178 -- Geometric Modeling

Unit 5 -- The B-Spline Curve

B-Spline curves generalize B├ęzier Curves

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Unit 5 -- The B-spline Curve

Section Description Video Notes Exercises

The B-Spline Curve

Messy

The Geometric Definition of the B-spline Curve:

We define the geometric definition of the B-spline curve. This is clearly an extension of the geometric definition for the Bézier curve.

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Bezier Knots

A Bézier Curve is a B-Spline Curve:

What would be the knots for a Bézier curve?.

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Analytic Def

The Analytic Definition of the B-spline Curve:

We can specify the blending functions in a recursive way, similar to the Bernstein polynomials for Bézier curves. Unfortunately, since B-splines are piecewise Bézier curves, these functions are somewhat messy.

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Pyramid

Thinking in Pyramids

It is useful to use a pyramid structure to visualize the B-spline algorithms.

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Sum

The DeBoor-Cox Calculation

We use the DeBoor-Cox calculation to show that the geometric and analytic definitions of B-spline curves are equivalent. This is an interesting computation, as the technique usually doesn't work in general. This time it does.

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C2

Properties of B-Spline Curves:

Lots of properties for B-spline curves. They are similar to those for Bézier curves, but more complex because we are piecing things together.

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On Knots

Knots are slippery things.

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The Catmull-Rom Spline

A look at an interpolating spline for comparison.

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B-Spline Curves and OpenGL

NURBS

How do we define B-Spline Curves in OpenGL?

    Exercise1
         

 
 

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