1. Geometric Modelling 2 INFO410 & INFO350 S2 2015 Jack Pinches
INFORMATION
SCIENCE

2. Previously in Geometric Modelling…
Geometry Information
Topological Information
Geometric Model Techniques:
Wireframe Modelling
Surface Modelling
Solid Modelling (CSG Example)

3. How to model curves and free-form surfaces?
Primitives:
Lines
Triangles
Primitives are mathematically exact, resolution independent, unaffected by changes in position/scale/orientation.
Could use primitives for arbitrarily shaped objects, but at the cost of the above properties.

4. Outline
Previous material
Parametric Curves
Bézier curves
B – Splines
NURBS
Free-Form Surface Modelling

5. Computational fluid dynamics Physical simulation in applied mechanics
Modelling Curves
Manufacturing
Car
Aircraft
Boats
Medical Imaging
Molecular Modelling
Computational fluid dynamics
Physical simulation in applied mechanics
Oceanography
Shape reconstruction
Weather analysis
Computer animation
Architecture
Art

6. Modelling Curves: Parametric Curves
Bezier
B – Spline
NURBS
Beta – Spline
Cubic Splines
Parametric Curves: parametric equations of a curve express the coordinates as functions of a variable – called a parameter. t often denotes the parameter.

7. Modelling Curves: Bézier Curves
1960s research: Paul de Casteljau and Pierre Bézier
Curve defined by:
P(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝐵 𝑖, 𝑛 (𝑡)
for 0 ≤ t ≤ 1
Where 𝐵 𝑛, 𝑖 are the Bernstein base functions, of degree n
For each t in put into the equation, every Pi will contribute.
e.g. have 25 points, move P1, the entire curve will be modified.
Bernestein polynomial known since 1912
Renault and Citeron.

8. Modelling Curves: Bézier Curves
Bézier curve is tangent to the first and last sections of the control polygon
Curve order is equal to number of vertices of control polygon
Curve is contained in the convex hull of control points

9. Modelling Curves: Bézier Curves
Problem: Lack of local control
P(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝐵 𝑖, 𝑛 (𝑡)
For each 𝑡, 𝑃 𝑖 will contribute.
Example:
25 control points, move P2, entire curve is modified.
For each t in put into the equation, every Pi will contribute.
e.g. have 25 points, move P1, the entire curve will be modified.
Poses a problem in CAD/animation
Bernstein problem: high number of points can be computationally intensive.

10. Modelling Curves: Bézier Curves
Piecewise Bézier Curve:
Join small curves
More local control
Called “B-Splines”

11. Modelling Curves: B – Spline
Basis Spline
1946: Curve fitting for experimental data
1963: Used in CAD systems by J. Ferguson (Boeing)
𝑃 𝑖 are points of the control polygon 𝑁 𝑖,𝑘 are the B-Spline base functions
an order k, and a set of knots { 𝑡 0 , 𝑡 1 ,…, 𝑡 𝑛+𝑘 }
𝑃(𝑡)= 𝑖=0 𝑛 𝑃 𝑖 𝑁 𝑖,𝑘 (𝑡)
Knots tell us the parameter values where the curves pieced together are joined.

12. Modelling Curves: B – Spline
Multiple Bézier curves
Low degree to reduce complexity
Similar properties
Increased local control
Curve doesn’t have to go to end points

13. Modelling Curves: NURBS
Non - Uniform Rational B-spline
Commonly used in CAD, CAM, CAE
3D modelling and animation software
Control points: directly connected, connected by a link

14. Modelling Curves: NURBS Curve Definition
NURBS Curve defined by four things:
Degree
Control points
Knot vector
Evaluation rule
NURBS curves are generalization on Bsplines/Bezier curves.

15. Modelling Curves: NURBS Curve Definition
Degree:
Positive whole number, often 1-5
Linear, quadratic, cubic, quantic
Order of curve: degree + 1
1 = linear
2 = circles (quadratic)
3-5 are most free form curves
Order = degree+1

16. Modelling Curves: NURBS Curve Definition
Control Points:
At least degree + 1 number of points
Change curve shape
Points are weighted
Shape formed: Control Polygon
Connecting lines to make the points easier to see/their relationship to curve.
Desirable property: control points make very localised changes, little/no effect on parts of the curve further away.

17. Modelling Curves: NURBS Curve Definition
Knot Vector: The knot vector is a sequence of parameter values that determines where and how the control points affect the NURBS curve.
This slide displays the answer I gave in the Q&A for the presentation.

18. Modelling Curves: NURBS Curve Definition
Knots (Knot Vector):
List of (degree+N-1) numbers, N is number of control points
Several conditions
Example:
3 degree NURBS curve with 11 control points
0,0,0,1,2,2,2,3,7,7,9,9,9
0,0,0,1,2,2,2,2,7,7,9,9,9
Vector has nothing to do with direction as usually meant by vector.
Conditions: can’t repeat a number more times than the degree
- Can keep the numbers the same, or get larger as go down list

19. Modelling Curves: NURBS Curve Definition
Evaluation Rule:
Mathematical formula, takes a number and assigns a point
“You can think of the evaluation rule as a black box that eats a parameter and produces a point location. The degree, knots, and control points determine how the black box works.”

20. Modelling Curves:
The development of curve modelling is progressive. The new methods were often designed to over come weaknesses present in other methods.
B-Splines created to achieve more control of curve shape than Bézier is an example. (Slide 12)

21. Freeform Surface Modelling
Modelling in CAD:
Create curves, mesh surface
Create surface, manipulate control points
Model car bodies, boat hulls, aircraft
Most CAD software uses NURBS

22. Freeform Surface Modelling
Continuity between surfaces
Require continuous rate of change
High quality, higher degree
Context dependent
Continuity: how smoothly they connect to one another.

23. Potential Exam Questions

24. Exam Questions
How do Bézier curves and B-Splines differ? Including reasons why B-Splines was developed.

25. Resources
General:
NURBS:
Bezier and B-Splines:
Free-Form Surface Modelling:
Title page image: