Advanced Computer Graphics Spring 2005

1
Advanced Computer Graphics
(Spring 2005)

• COMS 4162, Lecture 12 Spline Curves (review)
  Ravi Ramamoorthi

http//www.cs.columbia.edu/cs4162
Most material taken from COMS 4160 lectures 6 and
7
2
To Do / Motivation

• One of written questions in assignment deals with
  material in this (actually next) lecture
• Otherwise, mainly intended for completeness,
  since splines are very common modeling tool
• This lecture reviews spline curves (from 4160).
  Next lecture discusses extensions to NURBs and
  surfaces

3
Curved Surfaces

• Motivation
• Exact boundary representation for some objects
• More concise representation than polygonal mesh
• Easier to model with and specify for many
  man-made objects and machine parts (started with
  car bodies)

HB Figure 10.46
4
Curve Representations

• Function y f(x)
• Implicit f(x,y) 0
• Subdivision (x,y) as limit of recursive process
• Parametric x f(t), y g(t)

5
Curved Surface Representations

• Function z f(x,y)
• Implicit f(x,y,z) 0
• Subdivision (x,y,z) as limit of recursive
  process
• Parametric x f(s,t), y g(s,t), z h(s,t)

6
Parametric Surfaces

• Boundary defined by parametric function
• x f(u,v)
• y f(u,v)
• z f(u,v)
• Example (sphere)
• x sin(?) cos(?)
• y sin(?) sin(?)
• z cos(?)

7
Parametric Representations

• One function vs. many (defined piecewise)
• Continuity
• How specified?

8
Parametric Polynomial Curves

• A parametric polynomial curve of order n
• Advantages of polynomial curves
• Easy to compute
• Infinitely differentiable everywhere

9
(Cubic) Spline Constructions

• Bézier 4 control points
• B-splines approximating, C2, local control
• Hermite 2 points, 2 normals
• Natural splines interpolating, C2, no local
  control
• Catmull-Rom interpolating, C1, local control

Book chapter 11 covers some spline
constructions But material is hard to read,
doesnt follow format here. I recommend
following lecture.
10
Bezier Curve (with 4160-HW2 demo)

• Motivation Draw a smooth intuitive curve (or
  surface) given a few key user-specified control
  points
• hw2.exe

Control points (all that user specifies, edits)
Control polygon
Smooth Bezier curve (drawn automatically)
11
Bezier Curve (Desirable) properties

• Interpolates, is tangent to end points
• Curve within convex hull of control polygon

Control points (all that user specifies, edits)
hw2.exe
Control polygon
Smooth Bezier curve (drawn automatically)
12
deCasteljau Linear Bezier Curve

• Just a simple linear combination or interpolation
  (easy to code up, very numerically stable)

P1
F(1)
Linear (Degree 1, Order 2) F(0) P0, F(1)
P1 F(u) ?
F(u)
P0
F(0)
13
deCasteljau Quadratic Bezier Curve
Quadratic Degree 2, Order 3 F(0) P0, F(1)
P2 F(u) ?
F(u) (1-u)2 P0 2u(1-u) P1 u2 P2
14
Geometric interpretation Quadratic

u
1-u
1-u
u
u
15
Geometric Interpretation Cubic
16
Summary deCasteljau Algorithm
P1
P0
Linear Degree 1, Order 2 F(0) P0, F(1) P1
17
Idea of Blossoms/Polar Forms

• (Optional) Labeling trick for control points and
  intermediate deCasteljau points that makes thing
  intuitive
• E.g. quadratic Bezier curve F(u)
• Define auxiliary function f(u1,u2) number of
  args degree
• Points on curve simply have u1u2 so that F(u)
  f(u,u)
• And we can label control points and deCasteljau
  points not on curve with appropriate values of
  (u1,u2 )

18
Idea of Blossoms/Polar Forms

• Points on curve simply have u1u2 so that F(u)
  f(u,u)
• f is symmetric f(0,1) f(1,0)
• Only interpolate linearly between points with one
  arg different
• f(0,u) (1-u) f(0,0) u f(0,1) Here,
  interpolate f(0,0) and f(0,1)f(1,0)

00
01
11
1-u
u
1-u
u
0u
1u
1-u
u
uu
F(u) f(uu) (1-u)2 P0 2u(1-u) P1 u2 P2
19
Geometric interpretation Quadratic
0110

u
1u
1-u
1-u
u
uu
0u
u
00
11
20
Polar Forms Cubic Bezier Curve
001
011
111
000
21
Geometric Interpretation Cubic
22
Why Polar Forms?

• Simple mnemonic which points to interpolate and
  how in deCasteljau algorithm
• Easy to see how to subdivide Bezier curve (next)
  which is useful for drawing recursively
• Generalizes to arbitrary spline curves (just
  label control points correctly instead of 00 01
  11 for Bezier)
• Easy for many analyses (beyond scope of course)

23
Subdividing Bezier Curves

• Drawing Subdivide into halves (u ½) Demo
  hw2.exe
• Recursively draw each piece
• At some tolerance, draw control polygon
• Trivial for Bezier curves (from deCasteljau
  algorithm) hence widely used for drawing
• Why specific labels/ control points on
  left/right?
• How do they follow from deCasteljau?

24
Geometrically
0½1
001
011
½½½
0½½
½½1
00½
½11
000
111
25
Geometrically
0½1
001
011
½½½
0½½
½½1
00½
½11
000
111
26
Subdivision in deCasteljau diagram
001
011
111
000
These (interior) points dont appear in
subdivided curves at all
Right part of Bezier curve (uuu, 1uu, 11u,
111) Always right edge of deCasteljau pyramid
Left part of Bezier curve (000, 00u, 0uu,
uuu) Always left edge of deCasteljau pyramid
27
Properties (brief discussion)

• Demo
• Interpolation End-points, but approximates
  others
• Single piece, moving one point affects whole
  curve (no local control as in B-splines later)
• Invariant to translations, rotations, scales etc.
  That is, translating all control points
  translates entire curve
• Easily subdivided into parts for drawing. Hence,
  Bezier curves easiest for drawing (4160-HW2)

hw2.exe
28
Bezier Disadvantages

• Single piece, no local control (move a control
  point, whole curve changes) hw2.exe
• Complex shapes can be very high degree,
  difficult
• In practice, combine many Bezier curve segments
• But only position continuous at join since Bezier
  curves interpolate end-points (which match at
  segment boundaries)
• Unpleasant derivative (slope) discontinuities at
  end-points
• Can you see why this is an issue?

29
Piecewise Polynomial Curves

• Idea
• Use different polynomial functionsfor different
  parts of the curve
• Advantage
• Flexibility
• Local control
• Issue
• Smoothness at joints (continuity)

30
Continuity

• Continuity Ck indicates adjacent curves have the
  same kth derivative at their joints

31
C0 Continuity

• Adjacent curves share
• Same endpoints Qi(1) Qi1(0)
• Aside discontinuous curves sometimes called C-1

32
C1 Continuity

• Adjacent curves share
• Same endpoints Qi(1) Qi1(0)
• Same derivatives Qi (1) Qi1 (0)

33
C2 Continuity

• Adjacent curves share
• Same endpoints Qi(1) Qi1(0)
• Same derivatives Qi (1) Qi1 (0)
• Same second derivatives Qi (1) Qi1 (0)

34
(Cubic) Spline Constructions

• Bézier 4 control points
• B-splines approximating, C2, local control
• Hermite 2 points, 2 normals
• Natural splines interpolating, C2, no local
  control
• Catmull-Rom interpolating, C1, local control

Book chapter 11 covers some spline
constructions But material is hard to read,
doesnt follow format here. I recommend
following lecture.
35
B-Splines

• Cubic B-splines have C2 continuity, local control
• 4 segments / control point, 4 control
  points/segment
• Knots where two segments join Knotvector
• Knotvector uniform/non-uniform (we only consider
  uniform cubic B-splines, not general NURBS)

Demo hw2.exe
36
Polar Forms Cubic Bspline Curve

• Labeling little different from in Bezier curve
• No interpolation of end-points like in Bezier
• Advantage of polar forms easy to generalize

Uniform knot vector -2, -1, 0, 1, 2 ,3 Labels
correspond to this
37
deCasteljau Cubic B-Splines

• Easy to generalize using
  polar-form labels
• Impossible remember
  
  without

1 0 1
0 1 2
1 2 3
-2 1 0
-2 -1 0
1 2 3
-1 0 1
0 1 2
?
?
?
?
?
?
-1 0 u
0 1 u
1 2 u
38
deCasteljau Cubic B-Splines

• Easy to generalize using
  polar-form labels
• Impossible remember
  
  without

1 0 1
0 1 2
1 2 3
-2 1 0
-2 -1 0
1 2 3
-1 0 1
0 1 2
(2u)/3
(1-u)/3
(1u)/3
(2-u)/3
u/3
1-u/3
-1 0 u
0 1 u
1 2 u
?
?
?
?
1 u u
0 u u
39
deCasteljau Cubic B-Splines

• Easy to generalize using
  polar-form labels
• Impossible remember
  
  without

1 0 1
0 1 2
1 2 3
-2 1 0
-2 -1 0
1 2 3
-1 0 1
0 1 2
(2u)/3
(1-u)/3
(1u)/3
(2-u)/3
u/3
1-u/3
-1 0 u
0 1 u
1 2 u
(1u)/2
(1-u)/2
1-u/2
u/2
1 u u
0 u u
1-u
u
u u u
40
Explicit Formula (derive as exercise)
P0
P2
P1
P3
-2 -1 0
1 2 3
-1 0 1
0 1 2
(2u)/3
(1-u)/3
(1u)/3
(2-u)/3
u/3
1-u/3
-1 0 u
0 1 u
1 2 u
(1u)/2
(1-u)/2
1-u/2
u/2
1 u u
0 u u
1-u
u
u u u