CMPUT 498 Curves and Surfaces

1
CMPUT 498Curves and Surfaces

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

2
Curves

http//cougar.kniggets.org/cs219/mp1/curves/curves
.html
3
Conics
4
Parametric Cubic Curves

• A curve
• Locus of a point moving with one degree of
  freedom
• Torsion
• how much a space curve deviates from a plane
  how much it attempts to escape the osculating
  plane
• Arc length
• length measured along a curve
• Characterisation of all plane curves
• torsion 0
• Characterisation of all straight lines
• curvature 0

5
Parametric Cubic Curves

• Equations
• explicit
• y f(x)
• implicit
• f(x,y) 0

6
Parametric Cubic Curves

• Why use parametric cubic curves?
• why curves (rather than polylines)?
• reduce of points
• interactive manipulation is easier
• why parametric (as opposed to y,zf(x))?
• arbitrary curves are easily representable
• rotational invariance
• why parametric (rather than implicit)?
• simplicity and efficiency

7
Parametric Cubic Curves
8
Parametric Cubic Curvesmatrix form
T
and
9
Continuity

• Continuity between two parametric cubic curves
• Geometric continuity
• G0 the two curves are connected
• G1 the two tangents have the same direction
• Parametric continuity
• C0 the two curves are connected
• C1 the two tangents are equal

10
Continuity

• Parametric continuity generally implies geometric
  continuity. An exception is a pair of null vectors

11
Continuity
connecting curves with C0, C1 , and C2 continuity
12
The basis matrix

• Q(t) T . C
• C M . G
• Q(t) T . M . G

expanding for x(t)
13
A line segment
geometric constraints
blending functions
14
Hermite CurvesCharles Hermite (1822-1901)

• MH Hermite basis matrix
• GH geometric constraint vector
• stores the first and last points and the tangent
  at these points

(Hermite derivation on board)
15
Hermite curves
16
Hermite basis matrix
17
Hermite blending functions
18
Varying the magnitude of the tangent vector
19
Varying the direction of the tangent vector
20
Obtaining geometric continuity G1
for parametric continuity C1, k 1
21
Evaluation
22
Bézier curvesPierre Bézier (1910-1999)

• UNISURF system at Renault (1970s)
• http//www.cs.unc.edu/mantler/old/Bezier.html
• http//www.cs.princeton.edu/min/cs426/classes/bez
  ier.html

23
Bézier basis functions
24
Bézier basis functions
25
Bézier curvesproperties

• The convex hull property
• Partition of unity
• Invariance under affine transformations

26
The Bernstein polynomialsn3
27
The Bernstein polynomials
28
Bézier curvesgeometric construction
29
Uniform Nonrational B-Splines

• Drafting splines
• metal sheets wrapped in rubber
• B-splines
• local control local effect of moving one control
  point
• adjacent set of independently defined splines
• Cubic B-splines
• cubic Basis functions
• use a parameter t
• 0 lt t lt 1
• Usually use t in 3,4, 4,5, 5,6,
• There is a knot at 4, 5, 6,
• Uniform
• knots (t parameter) equally spaced

30
Uniform Nonrational B-Splines
http//page.inf.fu-berlin.de/vratisla/Bildverarbe
itung/Bspline/Bspline.html
http//i33www.ira.uka.de/applets/mocca/html/noplug
in/curves.html
31
Uniform Nonrational B-Splines
32
Uniform Nonrational B-Splines
33
Uniform Nonrational B-Splines

• The B-spline basis matrix
• The B-spline geometry vector
• Ti

34
Uniform Nonrational B-Splines
35
Uniform Nonrational B-Splines
36
Uniform Nonrational B-Splinesblending functions
37
Uniform Nonrational B-Splinesmultiple control
points
38
Surfaces

39
Quadric Surfaces
40
Parametric bicubic surfaces
41
Bézier surfaces
42
Hermite surfaces
43
Adjacent Bézier patches
44
B-spline surfaces
45
Reference
Chap. 11
46
(No Transcript)
47
(No Transcript)
48
p. 479
49
(No Transcript)
50
(No Transcript)
51
(No Transcript)
52
(No Transcript)
53
(No Transcript)
54
(No Transcript)