1. 3D Modeling, Graphics, and Animation
J. Rossignac CS4451&CS6491
11/27/2018
Curves and surfaces 4451
3D Modeling, Graphics, and Animation
Prof. Jarek Rossignac
College of Computing
Georgia Institute of Technology
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002
Subdivision of curves & surfaces

2. Jarek Rossignac, CoC, GT, ©Copyright 2002
Lecture Objectives
Learn (by heart) the Split&Tweak and the 4-point subdivision rules for polygons
Be able to discuss their advantages and drawbacks
Be able to design a trivial algorithm for these schemes
Understand how to construct good bounds (in 2D and in 3D) around the final curve from one of its approximations
Understand how to use these subdivisions to define tessellations (quads or triangles) of smooth surfaces in 3D
Be able to design the surface subdivision schemes
Understand the benefits of having a parametric polynomial form for the curves and surfaces
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3. Jarek Rossignac, CoC, GT, ©Copyright 2002
Motivation
Smooth curves and surfaces are used for aesthetic, manufacturing, and analysis applications where discontinuities due to triangulated approximations would create misleading artifacts.
Designers like to specify such curves by adjusting a control polygon with only a few vertices
How to go from a crude control polygon (black) to a smooth curve (red)?
Suggestions?
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

4. Split&tweak B-spline subdivision
Given a control polygon, for example (a,b,c,d), repeat the following sequence of two steps, until all consecutive 4-tuples of control points are nearly co-linear.
1. “Split”: insert a new control point in the middle of each edge (2,4,6,8)
2. “Tweak”: move the old control points half-way towards the average of their new neighbors (1,3,5,7)
The control polygon converges rapidly to a smooth curve, which happens to be a cubic B-spline curve. a 2 b 4 c 6 d 8 1 3 5 7
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

5. My first Split&Tweak in Scheme
(define w 200) (define s 10)
(define (draw-p-to-p p1 p2 col) (draw-solid-line (make-posn (+ (* (/ w s) (posn-x p1)) w) (- w (* (/ w s) (posn-y p1)))) (make-posn (+ (* (/ w s) (posn-x p2)) w) (- w (* (/ w s) (posn-y p2)))) col))
(define (draw-port col) (begin (start (* 2 w) (* 2 w)) ))
(define (last w)(cond [ (empty? (rest w)) (first w)][else (last (rest w))] ))
(define (mid a b) (make-posn (/ (+ (posn-x a) (posn-x b)) 2) (/ (+ (posn-y a) (posn-y b)) 2)))
(define (refine q r) (cond [(empty? (rest q)) (cons (first q) (cons (mid (first r) (first q)) empty))] [true (cons (first q) (cons (mid (first q) (first (rest q))) (refine (rest q) r))) ]))
(define (subdivide q n color) (cond [(= n 0) (draws q color)] [else (subdivide (tweaks (tweakh (refine q q ))) (- n 1) color)]))
(define (tweaks q) (cons (first q) (tweak (rest q))))
(define (tweak q) (cond [(or (empty? (rest q)) (empty? (rest (rest q)))) q] [else (cons (first q) (cons (mid (first (rest q)) (mid (first q) (first (rest (rest q))) ) (tweak (rest (rest q) ))) )] ))
(define (tweakh r) (cons (mid (first r) (mid (first (rest r)) (last r))) (rest r)))
(define (draws q color) (draw q q color))
(define (draw q r color) (cond [(empty? (rest q)) (draw-p-to-p (first r) (last r) color) ] [else (and (draw-p-to-p (first q) (first (rest q)) color) (draw (rest q) r color) )]))
(define p (list (make-posn -8 8) (make-posn 8 8) (make-posn 8 -8) ))
(draw-port 'yellow)
(draws p 'black) (subdivide p 1 'red) (subdivide p 2 'green) (subdivide p 4 'blue)
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Jarek Rossignac, CoC, GT, ©Copyright 2002

6. Examples from my Split&Tweak program
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Jarek Rossignac, CoC, GT, ©Copyright 2002

7. Yang’s version in MatLab
function result = Sub_divide (P, color, width);
[m, N] = size (P);
P = [P, P(:,1)]; % make it closed loop
result = zeros (m, 2*N);
even = [2:2:2*N];
odd = [1:2:2*N-1];prev = [1:N]; next = [2:N+1]; result(:,even) = 0.5 * (P(:,prev) + P(:,next));
prev = [2*N, 2:2:2*N-2]; next = [2:2:2*N];
result(:,odd) = 0.5 * (result(:,prev) + result(:,next));
result(:,odd) = 0.5 * (result(:,odd) + P(:,1:N));
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Jarek Rossignac, CoC, GT, ©Copyright 2002

8. Implement your own version
Try for the most elegant and shortest (not the fastest)
Start with an array V of n vertices of a closed polygon P in 2D
Write a procedure SHOW(V) that renders P
Make another array M of n midpoints of edges of P
Make a third array A, where each vertex of V is moved by 1/4 towards its neighbors in V
Display the polygons of V, M and A to verify your program
Make an array S of 2n vertices by interleaving A and M
{A[0],M[0],A[1],M[1],A[2],M[2]…A[n],M[n]}
Display polygon of S and of V in different colors to check
Make a procedure BSPLINE(V) that takes V and returns S
SHOW(BSPLINE(BSPLINE(BSPINE(V))))…
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Jarek Rossignac, CoC, GT, ©Copyright 2002

9. Jarek Rossignac, CoC, GT, ©Copyright 2002
Inspiration
Check previous year projects with examples and sourcew code
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Jarek Rossignac, CoC, GT, ©Copyright 2002

10. Can the limit curve self-intersect?
Can it self-intersect when P does not?
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Jarek Rossignac, CoC, GT, ©Copyright 2002

11. Where do the control vertices converge to?
The original control vertex b converges to b’=(a+4b+c)/6
The tangent at b’ to the B-spline curve is parallel to ac b b’ c a
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

12. Local control for B-spline curves
Trace the original vertices during subdivision
Do they move along straight lines?
They subdivide the final curve into spans
Each span is influenced by 4 control vertices
Each control vertex influences 4 spans
Each span converges to a cubic Bezier curve
The coordinates of C(t) are cubic polynomials in t b’ a b d e d’
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Jarek Rossignac, CoC, GT, ©Copyright 2002

13. Each span is a cubic polynomial
A polynomial formulation is available for each span
C(t)= B0(t)P0+ B1(t)P1+ B2(t)P2+ B3(t)P3
Bi are cubic polynomials in t
Pi are control points easily derived from a, b, d, e b P1 P0 c a b’ a b d e d’
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14. Jarek Rossignac, CoC, GT, ©Copyright 2002
Convex hull
Each span lies in the convex hull of 4 original vertices
The two from which its end-points evolved
Their immediate neighbors on each side
The convex hull of 4 points in 3D?
A tetrahedron
The convex hull of 4 points in 2D?
Covered by 3 triangles: ABD, ABE, BDE B’ A B D E D’
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Jarek Rossignac, CoC, GT, ©Copyright 2002

15. How can we use the convex hull property?
Display uncertainty (region guaranteed to contain the limit curve)
Shade triangles (in 2D) or tetrahedra (in 3D)
Decide when to stop the subdivision
All triangles or tetrahedra are nearly linear
Quickly decide when two B-spline curves in 2D are not intersecting
The convex hulls do not overlap
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Jarek Rossignac, CoC, GT, ©Copyright 2002

16. Jarek Rossignac, CoC, GT, ©Copyright 2002
Open curves
If you want to subdivide an open polygon
Trace the original vertices through the subdivision process
They split the curve into spans (bent versions of the original edges)
v0 should not influence the head (beginning of the curve)
vn–1 should not influence the tail (end) v0 v1 v2
vn-2
vn-1
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Jarek Rossignac, CoC, GT, ©Copyright 2002

17. Jarek Rossignac, CoC, GT, ©Copyright 2002
4-point subdivision
The B-spline subdivision produces a curve that does not interpolate the vertices of the original polygon P
To obtain an interpolating subdivision rule, we must leave the original vertices of P where they are, and thus must displace the mid-points.
By how much?
C(t)=at3+bt2+ct+d, and thus C(0)=d
A=C(-3)=-27a+9b-3c+d
B=C(-1)=-a+b-c+d
D=C(1)=a+b+c+d
E=C(3)=27a+9b+3c+d
D=M+(M-M’)/8,
with 2M=B+D and 2M’=A+E
Offset is 1/8 of ||MM’||
C(0)
C(-1)
C(1) B M D
C(-3)
C(3) A M’ E
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

18. Implement your own 4-point subdivision
Start with an array V of n vertices of a closed polygon P in 2D
Make another array M of n midpoints of edges of P
Make a third array A, where each vertex of M is moved by 1/4 away from its neighbors in M
Display the polygons of V, M and A to verify your program
Make an array S of 2n vertices by interleaving V and A
{V[0],A[0],V[1],A[1],…V[n],A[n]}
Display polygon of S and of V in different colors to check.
Make a procedure FOURPTS(V) that takes V and returns S
SHOW(FOURPTS(FOURPTS(FOURPTS(V))))…
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

19. Compare both approaches
Which one interpolates the original vertices?
Which one produces the smoothest curve?
When is this important?
Can both have cusps (non smooth points)?
How can you force the B-spline curve to go through a point?
Which one has a convex-hull property?
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

20. Cubic B-spline and four-points
From James Koch
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21. Jarek Rossignac, CoC, GT, ©Copyright 2002
Which one is smoother?
By Stevie Strickland
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

22. Making a sharp corner in a B-spline curve
By Stevie Strickland
11/27/2018
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23. Jarek Rossignac, CoC, GT, ©Copyright 2002
A compromise
B-spline subdivision rounds the corners and thus shrinks a convex shape
4-points subdivision bulges the edges out and hence grows a convex shape
Both produce curves that are relatively far from the initial control polygon
We may want a compromise
Closer to the polygon
Bulging less
Cuttling less
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

24. Jarek Rossignac, CoC, GT, ©Copyright 2002
Jarek’s subdivision
Split each edge as before (in both schemes)
Tweak old vertices by half of the B-spline displacement and the new vertices by half of the 4-point displacement
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

25. Jarek’s subdivision curve
Bezier
Jarek
4-points
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Jarek Rossignac, CoC, GT, ©Copyright 2002

26. Jarek Rossignac, CoC, GT, ©Copyright 2002
Implementing Jarek’s subdivision
1 - For each vertex vi, compute the average ai of its neighbors and store a displacement vector Di=(ai–vi)/8.
2 - For each edge (vi,vi+1), insert a new vertex mi=(vi+vi+1)/2, and displace it by –(Di+Di+1)/2.
3 - Displace each old vertex vi by Di.
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

27. Three subdivision schemes
B-spline (uniform cubic B-spline), most popular:
Introduces new vertices at edge mid-points (split)
Tucks in the old vertices by 1/2 the displacement towards the average of their new neighbors (tweak)
Very smooth result: piecewise cubic parametric formulation
Does not interpolate the original vertices
Shrinks convex regions
4-points:
Leaves the old vertices unchanged
Bulges the edges by displacing their mid-point away from the average of the neighbors (split & tweak)
Interpolates the original vertices
Grows convex regions
Jarek
Combines half of the tweaks of the other two
Final curve is closer to the original polygon
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

28. Jarek Rossignac, CoC, GT, ©Copyright 2002
What about curves in 3D?
Nothing was said about the dimension of the space.
The procedures work for any dimension and in any space.
You can apply the subdivision to the x coordinates first, then to the y coordinates, then to the z, or to the 3D points.
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

29. Jarek Rossignac, CoC, GT, ©Copyright 2002
Draw a surface
Take m polygons of n vertices each.
Subdivide each one k times: to get m polygons of n2k vertices
Transpose: make n2k new polygons of m vertices each:
The first one is made of the first vertices of each one of the m polygons
The second one is made of the second vertices of the m polygons…
Subdivide each one of the n2k new polygons k times: to get n2k polygons of m2k vertices each
Draw each polygon
Transpose again
transpose …
subdivide
transpose
subdivide
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

30. Example of a B-spine subdivision
By James Koch (notice that the surface does not even go through the control loops
11/27/2018
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31. Example of a four-point subdivision
By Justin Kennedy
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Jarek Rossignac, CoC, GT, ©Copyright 2002

32. Example of a mixed subdivision
By Jesse Shieh
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

33. B-spline, 4-points and mixed
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Jarek Rossignac, CoC, GT, ©Copyright 2002

34. How to produce a tessellation of the surface
Perform the subdivide-transpose-subdivide-transpose
Capture all quads, each bounded by four edge produced by the previous process.
Display them as quads or triangulate each one
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Jarek Rossignac, CoC, GT, ©Copyright 2002

35. Four-point torus from 3 triangles
By Ignacio Llamas
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

36. More complex torus with normals
By Peter Presti
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

37. Jarek Rossignac, CoC, GT, ©Copyright 2002
What you must remember
Names and subdivision rules for all 3 schemes
How to implement them
How to compute the convex hull for B-spline spans
How to generate 3D surfaces by using these schemes
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002

38. Examples of simple questions for quiz
Name the three curve subdivision schemes discussed in class?
What are the two steps of the cubic B-spline subdivision?
Does a B-spline subdivision interpolate the original vertices?
What is the difference between the implementations of the B-spline and the 4-point subdivision schemes?
What is a span in a polygon produced by B-spline subdivision.
Provide a formulation of a pointset that contains such a span.
Provide a detailed pseudocode for Jarek’s subdivision scheme.
Explain how you can use the curve subdivision to produce surfaces.
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Jarek Rossignac, CoC, GT, ©Copyright 2002

39. Research questions to think about
Jarek’s subdivision is a 50/50 compromise between the B-spline and the 4-point schemes. How would you implement a t/(1-t) compromise?
Design a simple algorithm that would implement any of the split&tweak subdivisions step in a single pass over the vertices. (You may keep a small buffer of the last few vertices, but assume no random access.)
Suppose that you move a single vertex of the original control polygon. How many spans of the final curve are affected? Is the answer the same for all three subdivision schemes?
Are these two formulations of Jarek’s subdivision equivalent?
Tweak old vertices by half of the B-spline displacement and the new vertices by half of the 4-point displacement
For each vertex vi, compute the average ai of its neighbors and store a displacement vector Di=(ai–vi)/8. Then, for each edge (vi,vi+1), insert a new vertex mi=(vi+vi+1)/2, and displace it by –(Di+Di+1)/2. Finally, displace each old vertex vi by Di.
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Jarek Rossignac, CoC, GT, ©Copyright 2002

40. Jarek Rossignac, CoC, GT, ©Copyright 2002
Project 1
In pairs. Design & program together! Report individually.
January 13: Phase 1A (making and drawing polygons)
Create/draw/save polygons in 2D using OpenGL
January 20: Phase 1B (split&tweak subdivision)
Implement a weighted average of B-spline and 4-points
January 27: Phase 1C (Studies span of control, area change)
See how much of the curve is affected by moving one vertex
Compute change in area change between the original and the three subdivision curves
February 3: Phase 1D (simplification and error measure)
Explore different criteria for deleting vertices in simplification
Measure the resulting error
Start with previous year projects (source code and discussions):
Most of the grade is in the report, not the code!
Plan your work: sequence, testing, figures
11/27/2018
Jarek Rossignac, CoC, GT, ©Copyright 2002