361: 17 curves and surfaces - simon fraser universitytorsten/teaching/cmpt361/lecture... · •...

© Machiraju/Zhang/Möller© Machiraju/Zhang/Möller

Curves and Surfaces

CMPT 361Introduction to Computer Graphics

Torsten Möller

© Machiraju/Zhang/Möller

Rendering Pipeline

Hardware

Modelling Transform Visibility

Illumination +Shading

ColorPerception,Interaction

Texture/Realism


Reading• Angel – Chapter 10• Foley et al. - Chapter 11

3


Today• Polynomials• Parametric representation• Fairness vs. smoothness• Parametric vs. geometric continuity• Hermite spline• Bezier spline• B-Spline• Surfaces• Subdivision 4

© Machiraju/Zhang/Möller [Zorin 01]

Filling a gap …• So far, we have focused on lines, flat

polygons, and simple objects, e.g., spheres• Missing: freeform curves and surfaces• Although smooth curves and surfaces are

converted to polygonal curves and meshes when rendered, they still provide a good option for modeling

• We follow the text loosely

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Why curves and surfaces?• Natural to use for modeling of smooth shapes,

e.g., – Body of an automobile– Shape of cartoon characters (Shrek)– Motion curves in animation, etc.

• Smoothness can often be guaranteed analytically

• Compact (analytical) representation• Theory of smooth curves and surfaces, e.g.,

from calculus and differential geometry, is well-developed 6

http://www.shrek2.com/


Polynomial curves and surfaces• In computer graphics, we prefer curves and

surfaces represented by polynomials– Approximation power: Can approximate any

continuous function to any accuracy (Weierstrass’s Theorem)

– Can offer local control for shape design through the use of piecewise polynomials

– All derivatives and integrals are available (infinitely smooth) and easy to compute

– Compact representation– Efficient evaluation – e.g., Horner’s rule 7


Horner’s rule• Consider a cubic polynomial

b(t) = at3 + bt2 + ct + d• Straightforward way to evaluate b(t) takes 6

multiplications and 3 additions• Horner’s Rule:

b(t) = ((a * t + b) * t + c) * t + d• It takes 3 additions and 3 multiplications. In

general, n +’s and n *’s for polynomial with degree = n.

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Curve & surface representation• Explicit: y = f(x), z = f(x, y)• Implicit (level-set): f(x,y) = 0, f(x, y, z) = 0• Parametric:

– 2D planar curve segment: (x(t), y(t)), t ∈[0, 1]

– 3D space curve segment: (x(t), y(t), z(t)), t ∈[0, 1]

– 3D parametric surface patch: (x(u, v), y(u, v), z(u, v)), u, v ∈[0, 1]

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Basis Functions

http://terpconnect.umd.edu/~petersd/interp.html

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Piecewise polynomial curves• Consider the problem of polynomial

interpolation:

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High-degree interpolation

Piece-wise cubic interpolation


Fairness vs. smoothness• Smoothness of curves and surfaces:

– Local property: often achieved by design– Related to existence and continuity of various

derivatives,– e.g., f(x) = 3x100 – 9x2 + … + 1 is infinitely smooth,

but is it “visually pleasing”?• Fairness (often appears in CAGD literature)

– Global property: often achieved by some form of energy minimization

– Related to the “energy” of a curve or surface– e.g., f(x) = 3x100 – 9x2 + … + 1 has high bending

energy, and is not very visually pleasing 13


Parametric cubic curves• Let us focus on parametric curves for now• Generalization to surfaces is quite straightforward• Questions: what degree (of the polynomials) to

use?– Degree 0 – 2 (constant, linear, or quadratic): often has

too little flexibility– High-degree: unnecessarily complex and easy to

introduce undesirable wiggles (most objects we want to model using curves and surfaces are somewhat fair) – fairness vs. smoothness

– Most commonly used in graphics: parametric cubic (degree-3) curves and surfaces 14


Parametric cubic segment• Consider a single piece:

x(t) = a3t3 + a2t2 + a1t + a0

y(t) = b3t3 + b2t2 + b1t + b0

z(t) = c3t3 + c2t2 + c1t + c0

• In matrix form:

• T is said to be the monomial basis15


Derivatives and continuity• 1st-order derivative of (x(t), y(t)): (x’(t), y’(t)) –

tangent

• 2nd-order derivative: (x’’(t), y’’(t)) – related to curvature

• Parametric continuity of a curve (smoothness of motion):– C0 continuous: curve is joined or connected– C1: requires C0 & 1st-order derivative is continuous– C2: requires C0 & C1 & 2nd-order derivative is cont.– Cn: requires C0 & … & Cn–1 & nth derivative cont. 16


Continuity of piecewise polynomials

• A single polynomial segment is always C∞ • But we mostly deal with piecewise polynomial

curves• Key: what happens at the joints between segments

– C0: curve segments are connected– C1: C0 & 1st-order derivatives agree at joints– C2: C0 & C1 & 2nd-order derivative agree at joints, etc.

• If Ci is not possible to enforce, can relax a bit– “Visual” smoothness: direction of tangents stays the

same but magnitude (speed) may change

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Geometric continuity• geometric continuity

– G0 cont.: curve segments are connected (same as C0)– G1: G0 & 1st-order derivatives are proportional at joints. – Note:

• Proportional = same direction but may have different magnitudes

• Weaker than C1

– G2: G1 & 2nd-order derivative proportional at joints• Example: p(t) = (3t, t3) and q(t) = (4t+3, 2t2+4t+1)

with t ∈ [0, 1] for each. Is this C0, G1, and/or C1?– p(1)=q(0)=(3,1), so G0; p’(1)=(3,3) and q’(0)=(4,4), so

G1 not C1 18


On to curve design• We want to design piecewise cubic polynomial curves

that satisfy certain design constraints, e.g.,– Curve should pass through certain points– Curve should have some given derivatives at specific points– Curve should be smooth: G1, C1, C2, or …– Curve must be contained in certain area, or has at most this

length, etc.• Need proper basis functions to facilitate design process• These basis functions or blending functions blend

together the individual contributions of the control points

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Basis of cubic polynomials• Monomial basis: {1, t, t2, t3}, is only one

possibility • Recall basis:

x(t) = P1b1(t) + P2b2(t) + P3b3(t) + P4b4(t)• From design point of view, want P1, P2, P3, and P4

to represent observable quantities, e.g.,– Position: for interpolation– Derivatives: to control direction and smoothness, etc.

• i.e., they are control points• Control points are blended by the basis functions;

they control the shape of the curve 20


Ex. 1: Hermite curves• Defined by two points (P1 and P4) and two

tangents (R1 and R4)• Aim: Achieve C1 or G1 continuity• Want cubic curve x(t),

t ∈ [0, 1], such that (y and z are similar)– x(0) = P1

– x(1) = P4

– x’(0) = R1

– x’(1) = R4 21

Let us note that the control “points” P1, P4, R1, and R4 are all observable quantities and they control the shape of the curve

R1

R4

P4

P1


Cubic Hermite curves• x(t) = TA = a3t3 + a2t2 + a1t + a0, where T =

[t3 t2 t 1] and A = [a3 a2 a1 a0]T. We want

• So G = BA and thus A = B–1G• It follows that x(t) = TA = TB–1G = HG

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Hermite curves• How to interpret this: x(t) = TA = TB–1G = HG

– G is a vector of observables or control points– H is vector of cubic Hermite bases (blending

polynomials)

– For any G, use H to blend four control points to get curve x(t)

– The matrix Mhermite = B–1 is really a change of basis matrix: it changes the monomial basis T into the Hermite basis H

– Hermite curves are completely determined by Mhermite 23


The cubic Hermite matrix

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Piecewise Hermite curves• Can obviously enforce C1 or G1 continuity

at the joints

• Each segment parameterized over [0, 1], as usual

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R1

R4 = kR’1

P4 = P’1

P1

P’4

R’4


Ex. 2: Cubic Bézier curve• Defined by four control points P0, P1, P2, and P3

– x(0) = P0

– x(1) = P3

– x’(0) = 3(P1 – P0)– x’(1) = 3(P3 – P2)

• Convex hull property: Bézier curve lies within the convex hull of the four control points – good control

• Convex hull of a set of points on the plane: tightest convex polygon enclosing the set

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Convex hull property• A cubic curve satisfies the convex hull property if

it lies within the convex hull of its four control points

• Convex hull property is satisfied if and only if the basis polynomials B1(t), B2(t), B3(t), B4(t) satisfy:– 1. 0 ≤ B1(t), B2(t), B3(t), B4(t) ≤ 1 for t ∈ [0, 1], and– 2. B1(t) + B2(t) + B3(t) + B4(t) = 1

• Then each point of the curve is a convex combination of the control points

• The basis Bi(t) form a partition of unity28


The cubic Bézier matrix

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– B0(t) = (1 – t)3, B1(t) = 3t(1 – t)2, – B2(t) = 3t2(1 – t), B3(t) = t3

• Well known as the Bernstein Polynomials of degree 3

• Bernstein polynomials of degree n• We have• Partition of unity easy to see: Σi Bi(t) = [t + (1 – t)]n


Bézier basis polynomials

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Piecewise Bézier curves• How to ensure C1 or G1 continuity for a

piecewise Bézier curve?• Each segment is parameterized over [0, 1]

as usual

http://www.cs.princeton.edu/~min/cs426/jar/bezier.html

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P0

P1

P2

P3

P4

P5

P6


Ex 3. Cubic B-Spline curves• Most popular choice in computer graphics• They are C2 continuous – this beats Hermite

and Bézier curves in terms of smoothness• The theory of B-splines is very rich• NURBS: nonuniform rational B-splines

– Each component is rational: x(t)/w(t), etc.– Can be used to specify circles, etc.

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Cubic B-Spline curves• Each cubic B-spline segment is specified by

four control points• Has the convex hull property• The cubic B-spline matrix:

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Piecewise cubic B-splines• Two consecutive segments share three control

points• m control points → m – 3 segments• Each segment parameterized over [0, 1] as usual• Exercise: what if control points repeat?

• http://www.cs.technion.ac.il/~cs234325/Applets/applets/bspline/GermanApplet.html 34

C1

C2

C3

C4

C5

P1

P3

P4

P5

P6

P8

P7

P2


bilinear interpolation

From curves to surfaces• One easy way: sweep a curve whose control

points also trace out some curves, e.g., bilinear interpolation

• Fit the simplest surface between four points

• Sweep a straight line and each point on the line traces a straight line

• Example of a ruled surface• Also an example of the more general class of

tensor-product surfaces 35


• The curve to sweep:• Control point ai goes through a curve

• The resulting surface is a tensor-product surface

• Surface is controlled by the grid of control points Pij

Tensor-product (TP) surfaces

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• The Bi(u)Bj(v)’s form a basis for bi-cubic polynomials

• They also form a partition of unity — the patch lies within the convex hull of the 16 control points Pij

• Patch can be seen as generated by sweeping a Bézier curve where each point on the curve sweeps out a Bézier curve

• The four corner vertices are interpolated (easy to see?)

Ex. 1: TP cubic Bézier patch

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Cubic Bézier surface patch

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In matrix form:[Angel 02]

http://www.math.psu.edu/dlittle/java/parametricequations/beziersurfaces/index.html


Ex. 2: TP cubic B-spline surfaces

• Analogues extension of B-spline curves

• N(u) and N(v) are the cubic B-spline bases

• 16 control points per bi-cubic B-spline patch

• Satisfy the convex hull property

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Smoothness of Bézier surface• G1 continuity: make 2-sets of 4 control

points on either side of an edge collinear• C1 continuity: collinear and equally spaced

40Collinear control points near patch boundary


Smoothness of B-spline surface

• C2 continuity is achieved if adjacent patches share control points

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deCasteljau Recursion• We can use the convex hull property of

Bezier curves to obtain an efficient recursive method that does not require any function evaluations– Uses only the values at the control points

• Based on the idea that “any polynomial and any part of a polynomial is a Bezier polynomial for properly chosen control data”

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• p0, p1, p2, p3 determine a cubic Bezier polynomial and its convex hull


Splitting a Cubic Bezier

Consider left half l(u) and right half r(u)44


l(u) and r(u)• Since l(u) and r(u) are Bezier curves, we

should be able to find two sets of control points {l0, l1, l2, l3} and {r0, r1, r2, r3} that determine them

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Convex Hulls• {l0, l1, l2, l3} and {r0, r1, r2, r3} each have a convex hull that

that is closer to p(u) than the convex hull of {p0, p1, p2, p3} This is known as the variation diminishing property.

• The polyline from l0 to l3 (= r0) to r3 is an approximation to p(u). Repeating recursively we get better approximations.

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Equations• Start with Bezier equations p(u)=uTMBp• l(u) must interpolate p(0) and p(1/2)

• Matching slopes, taking into account that l(u) and r(u) only go over half the distance as p(u)

• Symmetric equations hold for r(u)47

l(0) = l0 = p0l(1) = l3 = p(1/2) = 1/8( p0 +3 p1 +3 p2 + p3 )

l’(0) = 3(l1 - l0) = p’(0) = 3/2(p1 - p0 )l’(1) = 3(l3 – l2) = p’(1/2) = 3/8(- p0 - p1+ p2 + p3)


Efficient Form

Requires only shifts and adds!

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l0 = p0r3 = p3

l1 = ½(p0 + p1)r1 = ½(p2 + p3)

l2 = ½(l1 + ½( p1 + p2))r1 = ½(r2 + ½( p1 + p2))

l3 = r0 = ½(l2 + r1)


Every Curve is a Bezier Curve• We can render a given polynomial using the

recursive method if we find control points for its representation as a Bezier curve

• Suppose that p(u) is given as an interpolating curve with control points q

• There exist Bezier control points p such that

• Equating and solving, we find p=MB-1MI49

p(u)=uTMIq

p(u)=uTMBp


Matrices

Interpolating to Bezier

B-Spline to Bezier

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Example

Bezier Interpolating B Spline

• These three curves were all generated from the same original data using Bezier recursion by converting all control point data to Bezier control points

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Surfaces• Can apply the recursive method to surfaces if we

recall that for a Bezier patch curves of constant u (or v) are Bezier curves in u (or v)

• First subdivide in u – Process creates new points – Some of the original points are discarded

– 52original and kept new

original and discarded


Second Subdivision

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16 final points for1 of 4 patches created


Normals• For rendering we need the normals if we

want to shade– Can compute from parametric equations

– Can use vertices of corner points to determine– OpenGL can compute automatically

54


Utah Teapot• Most famous data set in computer graphics• Widely available as a list of 306 3D vertices

and the indices that define 32 Bezier patches

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Subdivision Schemes• In order to draw a curve we need to find

points on it:

• first idea - go step by step, i.e.:

56


Subdivision Schemes• Same for surfaces:

• How can we find those new points?• What about triangles/arbitrary polygons?

57


Subdivision Schemes (2)

58


Loop• Guaranteed to be smooth everywhere except

at extraordinary vertices• Face Split• Triangular

meshes• approximating

59


Loop (2)

60


Loop - Boundary• Subdivision Mask for

Boundary Conditions

61

Edge Rule Vertex Rule


Modified Butterfly• Face Split• Triangular

meshes• interpolating

62


Catmull-Clark• Face Split• Quad. Meshes

(extents toarbitrary)

• approximating• Masks for odd vertices

63


Catmull-Clark (2)• Face Split• Quad. Meshes

(extents toarbitrary)

• approximating• Masks for even vertices

64


Catmull-Clark (2)

65

FACE

EDGE

VERTEX


Kobbelt• Face Split• Quad. Meshes• interpolating

66


Kobbelt (2)• Mask for face vertex

adjacent to anextraordinary vertex.

• Two steps– The edge vertices– The face vertex

• the edge odd controlpoint adjacent to anextraordinary vertex

67


Doo-Sabin• Vertex Split• Quad. Meshes• interpolating or approximating??

68


Mid-Edge• Midedge, the simpler version of Doo-Sabin

scheme.

69


Doo-Sabin & Mid-Edge• The interior rules can be decomposed into a

sequence of averaging steps.

Doo-Sabin scheme

Midedge scheme

70


Subdivision Schemes

Loop Butterfly

Catmull-Clark Doo-Sabin

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Modeling with Subdivision• Subdivision produces smooth continuous

surfaces. • How can “sharpness” and creases be controlled

in a modeling environment? • ANSWER: Define new subdivision rules for

“creased” edges and vertices.– Tag Edges sharp edges.– If an edge is sharp, apply

new sharp subdivision rules.– Otherwise subdivide with

normal rules. 72

361: 17 curves and surfaces - simon fraser universitytorsten/teaching/cmpt361/lecture... · •...

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