Introduction to Computer Graphics Modeling (1) April 16, 2020 - - PowerPoint PPT Presentation

Introduction to Computer Graphics – Modeling (1) –

April 16, 2020 Kenshi Takayama

Some additional notes on quaternions

2

Another explanation for quaternions (overview)

• 1. Any rotation can be decomposed

into even number of reflections

• 2. Quaternions can concisely describe reflections in 3D

𝑆 ⃗

" ⃗

𝑦 = − ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔#$

• 3. Combining two reflections equivalent to the rotation leads to the formula

𝑆! 𝑆 ⃗

# ⃗

𝑦 = cos $

% + 𝜕 sin $ %

⃗ 𝑦 cos $

% − 𝜕 sin $ %

3

Any rotation can be decomposed into even number of reflections

• Mathematically proven
• Valid for any dimensions
• Not unique (of course!)

4

R

! " ! #

𝜄

R R

! " $! #

𝜄

R

! "

𝜄

R R

One way Another way

Quaternions recap

• Complex number: real + imaginary

𝑏 + 𝑐 𝐣

• Quaternion: scalar + vector

𝑏 + ⃗ 𝑤

• Definition of quaternion multiplication:

𝑏! + 𝑤! 𝑏" + 𝑤" ≔ 𝑏!𝑏" − 𝑤! ⋅ 𝑤" + 𝑏!𝑤" + 𝑏"𝑤! + 𝑤!×𝑤"

• Pure vectors can take multiplication by interpreting them as quaternions:

𝑤! 𝑤" = −𝑤! ⋅ 𝑤" + 𝑤!×𝑤"

• Notable properties:
• (Relevant later)

5

⃗ 𝑤 ⃗ 𝑤 = − ⃗ 𝑤 " ⃗ 𝑤#! = − ⃗ 𝑤 ⃗ 𝑤 "

If ⃗ 𝑤 ⋅ 𝑥 = 0 , then ⃗ 𝑤 𝑥 = −𝑥 ⃗ 𝑤

⃗ 𝑤× ⃗ 𝑤 is always zero Scalar part Vector part ⃗ 𝑤 𝑥 = ⃗ 𝑤×𝑥 = −𝑥× ⃗ 𝑤 = −𝑥 ⃗ 𝑤 Multiplying ⃗ 𝑤 to rhs produces 1

Describing reflections using quaternions

• Reflection of a point ⃗

𝑦 across a plane

• rthogonal to ⃗

𝑔: 𝑆 ⃗

" ⃗

𝑦 ≔ − ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔#$

• Holds essential properties of reflections:
• Linearity:

𝑆 ⃗

' 𝑏 ⃗

𝑦 + 𝑐 ⃗ 𝑧 = 𝑏 𝑆 ⃗

' ⃗

𝑦 + 𝑐 𝑆 ⃗

' ⃗

𝑧

• ⃗

𝑔 gets mapped to − ⃗ 𝑔 :

𝑆 ⃗

' ⃗

𝑔 = − ⃗ 𝑔 ⃗ 𝑔 ⃗ 𝑔() = − ⃗ 𝑔

• If a point ⃗

𝑦 satisfies ⃗ 𝑦 ⋅ ⃗ 𝑔 = 0 (i.e. on the plane), ⃗ 𝑦 doesn’t move:

𝑆 ⃗

' ⃗

𝑦 = − ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔() = − − ⃗ 𝑦 ⃗ 𝑔 ⃗ 𝑔() = ⃗ 𝑦

6

⃗ 𝑦 𝑆 ⃗

# ⃗

𝑦 ⃗ 𝑔 𝑃

https://math.stackexchange.com/a/7263

Because if ⃗ 𝑦 ⋅ ⃗ 𝑔 = 0 , then ⃗ 𝑔 ⃗ 𝑦 = − ⃗ 𝑦 ⃗ 𝑔

Setup for rotation around arbitrary axis

• Rotation axis (unit vector)： 𝜕
• Rotation angle： 𝜄
• Point before rotation： ⃗

𝑦

• Point after rotation： ⃗

𝑧 ≔ 𝑆%, ' ⃗ 𝑦

• Think of local 2D coordinate system：
• “Right” vector ： 𝑣 ≔ ⃗

𝑦 − 𝜕 ⋅ ⃗ 𝑦 𝜕

• “Up” vector： ⃗

𝑤 ≔ 𝜕× ⃗ 𝑦

• Note that 𝑣

= ⃗ 𝑤

• (Let’s call it 𝑀)

7

𝜕 𝜄 ⃗ 𝑦 ⃗ 𝑧 𝑣 ⃗ 𝑤

https://legacygl-js.glitch.me/demo/quaternion-schematic.html

𝜕 ⋅ ⃗ 𝑦 𝑀

𝑃

Decompose rotation into two reflections

8

1st reflection：

⃗ 𝑔 ≔ ⃗ 𝑤

2nd reflection：

⃗ 𝑕 ≔ − sin $

% 𝑣 + cos $ % ⃗

𝑤

Top view

⃗ 𝑔 ⃗ 𝑕 𝜄

https://legacygl-js.glitch.me/demo/quaternion-schematic.html

' ,

Combining two reflections

• Formula： 𝑆! 𝑆 ⃗

# ⃗

𝑦 = 𝑆! − ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔23 = − ⃗ 𝑕 − ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔23 ⃗ 𝑕23 = ⃗ 𝑕 ⃗ 𝑔 ⃗ 𝑦 ⃗ 𝑔23 ⃗ 𝑕23

• Substitute

⃗ 𝑔 ≔ ⃗ 𝑤, 𝑕 ≔ − sin !

" 𝑣 + cos ! " ⃗

𝑤 to the above

• For the left part ⃗

𝑕 ⃗ 𝑔 ：

⃗ 𝑕 ⋅ ⃗ 𝑔 = − sin !

" 𝑣 + cos ! " ⃗

𝑤 ⋅ ⃗ 𝑤 = 𝑀# cos !

"

⃗ 𝑕× ⃗ 𝑔 = − sin !

" 𝑣 + cos ! " ⃗

𝑤 × ⃗ 𝑤 = −𝑀# sin !

" 𝜕

Therefore,

⃗ 𝑕 ⃗ 𝑔 = − ⃗ 𝑕 ⋅ ⃗ 𝑔 + ⃗ 𝑕× ⃗ 𝑔 = −𝑀% cos .

/ + 𝜕 sin . /

• The right part ⃗

𝑔23 ⃗ 𝑕23 = 0 1

23 is analogous： ⃗

𝑔23 ⃗ 𝑕23 = −𝑀2% cos .

/ − 𝜕 sin . /

• Finally, we get the formula：

𝑆!, # ⃗ 𝑦 = 𝑆$ 𝑆 ⃗

& ⃗

𝑦 = −𝑀' cos !

" + 𝜕 sin ! "

⃗ 𝑦 −𝑀(' cos !

" − 𝜕 sin ! "

= cos !

" + 𝜕 sin ! "

⃗ 𝑦 cos !

" − 𝜕 sin ! " 9

(because 𝑣 ⋅ ⃗ 𝑤 = 0 ) (because 𝑣× ⃗ 𝑤 = 𝑀"𝜕 )

• Any rotations (poses) can be

represented as unit quaternions

• Points on hypersphere of 4D space
• Fix 𝜕 and vary 𝜄

è unit circle in 4D space

• A pose after rotating 360°about a certain axis

is represented as another quaternion

• One pose corresponds to two quaternions

(double cover)

• A geodesic between two points 𝑞, 𝑟 on the

hypersphere represents interpolation of these poses

• Should pick either 𝑟 or −𝑟 which is closer to 𝑞 (i.e. 4D dot product is positive)

Representing and blending poses using quaternions

10

ℝ# 𝑞 = cos !

" + 𝜕 sin ! "

𝑟 −𝑟 (1, 0,0,0) (-1, 0,0,0) (0, 𝜕%,𝜕&,𝜕') (0, -𝜕%,-𝜕&,-𝜕') 𝜄=0 𝜄=𝜌 𝜄=2𝜌 𝜄=3𝜌

Normalize quaternions or not?

• Any quaternions can be written as scaling of unit quaternions

𝑟 = 𝑠 cos !

" + 𝜕 sin ! " , 𝑟() = 𝑠() cos ! " − 𝜕 sin ! "

• In the rotation formula, the scaling part is cancelled

𝑟 ⃗ 𝑦 𝑟() = 𝑠 cos !

" + 𝜕 sin ! "

⃗ 𝑦 𝑠() cos !

" − 𝜕 sin ! "

= cos !

" + 𝜕 sin ! "

⃗ 𝑦 cos !

" − 𝜕 sin ! "

è so, normalization isn’t needed?

• In practice, don’t use quaternion mults for computing

coordinate transformation (because inefficient)

• Just do explicit vector calc using axis & angle

⃗ 𝑦 − 𝜕 ⋅ ⃗ 𝑦 𝜕 cos 𝜄 + 𝜕× ⃗ 𝑦 sin 𝜄 + 𝜕 ⋅ ⃗ 𝑦 𝜕

• Can get axis & angle only after normalization
• Un-normalized can cause artifact when interpolated

11

𝑠 = 1

𝑟( 𝑟"

)!*)" "

Modeling curves

12

Parametric curves

• X & Y coordinates defined by parameter t (≅ time)
• Example: Cycloid

𝑦 𝑢 = 𝑢 − sin 𝑢 𝑧 𝑢 = 1 − cos 𝑢

• Tangent (aka. derivative, gradient) vector: 𝑦< 𝑢 , 𝑧< 𝑢
• Polynomial curve: 𝑦 𝑢 = ∑= 𝑏=𝑢=

13

Cubic Hermite curves

• Cubic polynomial curve interpolating

derivative constraints at both ends (Hermite interpolation)

• 4 constraints è 4 DoF needed

è 4 coefficients è cubic

• 𝑦 𝑢 = 𝑏@ + 𝑏3𝑢 + 𝑏%𝑢% + 𝑏A𝑢A
• 𝑦′ 𝑢 = 𝑏3 + 2𝑏%𝑢 + 3𝑏A𝑢%
• Coeffs determined by substituting

constrained values & derivatives

14

𝑦 0 = 𝑦@ 𝑦 1 = 𝑦3 𝑦′(0) = 𝑦@

B

𝑦B 1 = 𝑦3

B

𝑦 0 = 𝑏@ = 𝑦@ 𝑦 1 = 𝑏@ + 𝑏3 + 𝑏% + 𝑏A = 𝑦3 𝑦B 0 = 𝑏3 = 𝑦@

B

𝑦B 1 = 𝑏3 + 2 𝑏% + 3 𝑏A = 𝑦3

B

è 𝑏@ = 𝑦@ 𝑏3 = 𝑦@

B

𝑏% = −3 𝑦@ + 3 𝑦3 − 2 𝑦@

B − 𝑦3 B

𝑏A = 2 𝑦@ − 2 𝑦3 + 𝑦@

B + 𝑦3 B t x 1

Bezier curves

• Input: 3 control points (CPs) 𝑄>, 𝑄

$, 𝑄,

• Coordinates of points in

arbitrary domain (2D, 3D, ...)

• Output: Curve 𝑄 𝑢 satisfying

𝑄 0 = 𝑄> 𝑄 1 = 𝑄, while being “pulled” by 𝑄

$

15

𝑄 𝑢 = 1 − 𝑢 𝑄@ + 𝑢 𝑄% 𝑄 𝑢 = ? 𝑄

3

𝑄% 𝑄@

t=0 t=1

• Eq. of line segment
• Eq. of Bezier curve

Bezier curves

• 𝑄@3 𝑢 = 1 − 𝑢 𝑄@ + 𝑢 𝑄

3

• 𝑄

3% 𝑢 = 1 − 𝑢 𝑄 3 + 𝑢 𝑄%

• 𝑄*) 0 = 𝑄*
• 𝑄

)' 1 = 𝑄'

• Idea: ”Interpolate the interpolation”

As 𝑢 changes 0 → 1 , smoothly transition from 𝑄@3 to 𝑄

3%

• 𝑄@3% 𝑢 = 1 − 𝑢 𝑄@3 𝑢 + 𝑢 𝑄

3% 𝑢

= 1 − 𝑢 1 − 𝑢 𝑄@ + 𝑢 𝑄

3 + 𝑢

1 − 𝑢 𝑄

3 + 𝑢 𝑄%

= 1 − 𝑢 %𝑄@ + 2𝑢 1 − 𝑢 𝑄

3 + 𝑢%𝑄%

16

𝑄

3

𝑄% 𝑄@ 𝑄

3% 𝑢 𝑢 𝑢 𝑢 1 − 𝑢 1 − 𝑢

𝑄@3 𝑢 𝑄@3% 𝑢

1 − 𝑢 Quadratic Bezier curve

• Eq. of line
• Eq. of line

Bezier curves

• 𝑄@3 𝑢 = 1 − 𝑢 𝑄@ + 𝑢 𝑄

3

• 𝑄

3% 𝑢 = 1 − 𝑢 𝑄 3 + 𝑢 𝑄%

• 𝑄*) 0 = 𝑄*
• 𝑄

)' 1 = 𝑄'

• Idea: ”Interpolate the interpolation”

As 𝑢 changes 0 → 1 , smoothly transition from 𝑄@3 to 𝑄

3%

• 𝑄@3% 𝑢 = 1 − 𝑢 𝑄@3 𝑢 + 𝑢 𝑄

3% 𝑢

= 1 − 𝑢 1 − 𝑢 𝑄@ + 𝑢 𝑄

3 + 𝑢

1 − 𝑢 𝑄

3 + 𝑢 𝑄%

= 1 − 𝑢 %𝑄@ + 2𝑢 1 − 𝑢 𝑄

3 + 𝑢%𝑄%

17

𝑄

3

𝑄% 𝑄@

Quadratic Bezier curve

• Exact same idea applied to 4 points 𝑄@, 𝑄

3, 𝑄% 𝑄A:

• As 𝑢 changes 0 → 1, transition from 𝑄*)' to 𝑄

)'+

• 𝑄@3%A 𝑢 = 1 − 𝑢 𝑄@3% 𝑢 + 𝑢 𝑄

3%A 𝑢

= 1 − 𝑢

1 − 𝑢 '𝑄* + 2𝑢 1 − 𝑢 𝑄

) + 𝑢'𝑄' + 𝑢

1 − 𝑢 '𝑄

) + 2𝑢 1 − 𝑢 𝑄' + 𝑢'𝑄+

= 1 − 𝑢 A𝑄@ + 3𝑢 1 − 𝑢 %𝑄

3 + 3𝑢% 1 − 𝑢 𝑄% + 𝑢A𝑄A

Cubic Bezier curve

18

𝑄

3%A 𝑢

𝑄@3% 𝑢 𝑄@3%A 𝑢

𝑢 1 − 𝑢

𝑄

3

𝑄A 𝑄@ 𝑄%

Cubic Bezier curve

• Quad. Bezier
• Quad. Bezier

• Exact same idea applied to 4 points 𝑄@, 𝑄

3, 𝑄% 𝑄A:

• As 𝑢 changes 0 → 1, transition from 𝑄*)' to 𝑄

)'+

• 𝑄@3%A 𝑢 = 1 − 𝑢 𝑄@3% 𝑢 + 𝑢 𝑄

3%A 𝑢

= 1 − 𝑢

1 − 𝑢 '𝑄* + 2𝑢 1 − 𝑢 𝑄

) + 𝑢'𝑄' + 𝑢

1 − 𝑢 '𝑄

) + 2𝑢 1 − 𝑢 𝑄' + 𝑢'𝑄+

= 1 − 𝑢 A𝑄@ + 3𝑢 1 − 𝑢 %𝑄

3 + 3𝑢% 1 − 𝑢 𝑄% + 𝑢A𝑄A

• Can easily control tangent at endpoints è ubiquitously used in CG

Cubic Bezier curve

19

𝑄

3

𝑄A 𝑄@ 𝑄% 𝑄@3%A 𝑢

Cubic Bezier curve

n-th order Bezier curve

• Input: n+1 control points 𝑄>, ⋯ , 𝑄

E

𝑄 𝑢 = 8

=F> E EC= 𝑢= 1 − 𝑢 E#= 𝑄=

20

𝑐1

2(𝑢)

Bernstein basis function

1 − 𝑢 ,𝑄* + 4𝑢 1 − 𝑢 +𝑄

)

+ 6𝑢' 1 − 𝑢 '𝑄' + 4𝑢+ 1 − 𝑢 𝑄+ + 𝑢,𝑄

,

1 − 𝑢 -𝑄* + 5𝑢 1 − 𝑢 ,𝑄

)

+ 10𝑢' 1 − 𝑢 +𝑄' + 10𝑢+ 1 − 𝑢 '𝑄+ + 5𝑢, 1 − 𝑢 𝑄

,

+ 𝑢-𝑄-

Quartic (4th) Quintic (5th)

Cubic Bezier curves & cubic Hermite curves

• Cubic Bezier curve & its derivative:
• 𝑄 𝑢 = 1 − 𝑢 A𝑄@ + 3𝑢 1 − 𝑢 %𝑄

3 + 3𝑢% 1 − 𝑢 𝑄% + 𝑢A𝑄A

• 𝑄B(𝑢) = −3 1 − 𝑢 %𝑄@ + 3

1 − 𝑢 % − 2𝑢(1 − 𝑢) 𝑄

3 + 3 2𝑢 1 − 𝑢 − 𝑢% 𝑄% + 3𝑢%𝑄A

• Derivatives at endpoints:
• 𝑄B 0 = −3𝑄@ + 3𝑄

3

è 𝑄

3 = 𝑄@ + 3 A 𝑄B 0

• 𝑄B 1 = −3𝑄% + 3𝑄A

è 𝑄% = 𝑄A − 3

A 𝑄B 1

• Different ways of looking at cubic curves,

essentially the same

21

𝑄

3

𝑄A 𝑄@ 𝑄%

𝑄′(0) 𝑄′(1)

Evaluating Bezier curves

• Method 1: Direct evaluation of polynomials
• Simple & fastJ, could be numerically unstableL
• Method 2: de Casteljau’s algorithm
• Directly after the recursive definition of Bezier curves
• More computation stepsL, numerically stableJ
• Also useful for splitting Bezier curves

22

t 1-t t 1-t t 1

• t

t 1-t t 1-t t 1-t

Drawing Bezier curves

• In the end, everything is drawn as polyline
• Main question: How to sample parameter t?
• Method 1: Uniform sampling
• Simple
• Potentially insufficient sampling density
• Method 2: Adaptive sampling
• If control points deviate too much from straight line,

split by de Casteljau’s algorithm

23

Further control: Rational Bezier curve

• Another view on Bezier curve:

“Weighted average” of control points

• 𝑄*)' 𝑢 = 1 − 𝑢 '𝑄* + 2𝑢 1 − 𝑢 𝑄

) + 𝑢'

𝑄' = 𝜇* 𝑢 𝑄*+ 𝜇) 𝑢 𝑄

)+ 𝜇' 𝑢 𝑄'

• Important property: partition of unity

𝜇* 𝑢 + 𝜇) 𝑢 + 𝜇' 𝑢 = 1 ∀𝑢

• Multiply each 𝜇I 𝑢 by arbitrary coeff 𝑥I:

𝜊I 𝑢 = 𝑥I 𝜇I(𝑢)

• Normalize to obtain new weights:

𝜇I

B 𝑢 = J6 K ∑7 J7 K

24

w* = w' = 1

Non-polynomial curve è can represent arcs etc.

Cubic splines

• Series of connected cubic curves
• Piecewise polynomial
• Share value & derivative at every transition
• f intervals (C1 continuity)
• Parameter range can be other than [0, 1]
• Assumption: 𝑢M < 𝑢MN3
• Given values as only input,

we want to automatically set derivatives

25

t x t0 t1 t2 t3 t4 t5 x0(t) x1(t) x2(t) x3(t) x4(t) Curve tool in PowerPoint

Cubic Catmull-Rom spline

• Cubic function 𝑦M(𝑢) for range 𝑢M ≤ 𝑢 ≤ 𝑢MN$ is defined by

adjacent constrained values 𝑦M#$, 𝑦M, 𝑦MN$, 𝑦MN,

26

𝑢.() 𝑢. 𝑢./) 𝑢./' 𝑢 𝑦 𝑦. 𝑦.() 𝑦./) 𝑦./' 𝑦.(𝑢)

Cubic Catmull-Rom spline: Step 1

• As 𝑢M → 𝑢MN$, interpolate such that 𝑦M → 𝑦MN$ è Line

𝑚M(𝑢) = 1 − 𝑢 − 𝑢M 𝑢MN$ − 𝑢M 𝑦M + 𝑢 − 𝑢M 𝑢MN$ − 𝑢M 𝑦MN$

27

𝑢.() 𝑢. 𝑢./) 𝑢./' 𝑢 𝑦 𝑚.()(𝑢) 𝑚.(𝑢) 𝑚./)(𝑢)

Cubic Catmull-Rom spline: Step 2

• As 𝑢M#$ → 𝑢MN$, interpolate such that 𝑚M#$ → 𝑚M è Quadratic curve

𝑟M(𝑢) = 1 − 𝑢 − 𝑢M#$ 𝑢MN$ − 𝑢M#$ 𝑚M#$(𝑢) + 𝑢 − 𝑢M#$ 𝑢MN$ − 𝑢M#$ 𝑚M(𝑢)

• Passes through 3 points 𝑢M23, 𝑦M23 , 𝑢M, 𝑦M , 𝑢MN3, 𝑦MN3

28

𝑢.() 𝑢. 𝑢./) 𝑢./' 𝑢 𝑦 𝑟.(𝑢) 𝑟./)(𝑢)

Cubic Catmull-Rom spline: Step 3

• As 𝑢M → 𝑢MN$, interpolate such that 𝑟M → 𝑟MN$ è Cubic curve

𝑦M 𝑢 = 1 − 𝑢 − 𝑢M 𝑢MN$ − 𝑢M 𝑟M 𝑢 + 𝑢 − 𝑢M 𝑢MN$ − 𝑢M 𝑟MN$(𝑢)

29

𝑢.() 𝑢. 𝑢./) 𝑢./' 𝑢 𝑦 𝑦.(𝑢)

Summary:

• Derivative at each CP is defined by a quadratic

curve passing through its adjacent CPs

• Each interval is a cubic curve satisfying derivative

constraints at both ends

Evaluating cubic Catmull-Rom spline

30

A recursive evaluation algorithm for a class of Catmull-Rom splines [Barry,Colgman,SIGGRAPH88]

𝑄 𝑢@ = 𝑄@ 𝑄 𝑢3 = 𝑄

3

𝑄 𝑢% = 𝑄% 𝑄 𝑢A = 𝑄A 𝑄 𝑢 𝑢3 ≤ 𝑢 ≤ 𝑢%

Ways of setting parameter values 𝑢! (aka. knot sequence)

• Assume: 𝑢@ = 0
• Uniform

𝑢M = 𝑢M23 + 1

• Chordal

𝑢M = 𝑢M23 + 𝑄M23 − 𝑄M

• Centripetal

𝑢M = 𝑢M23 + 𝑄M23 − 𝑄M

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Parameterization of Catmull-Rom Curves [Yuksel,Schaefer,Keyser,CAD11]

Application of cubic Catmull-Rom spline: Hair modeling

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Parameterization of Catmull-Rom Curves [Yuksel,Schaefer,Keyser,CAD11]

Recent exciting development: 𝜆-Curves

• Collaboration between

university & company (Adobe)

• Features:
• C2 continuous (smoother)
• Curvature maxima always on control points

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𝜆-Curves: Interpolation at Local Maximum Curvature [Yan, Schiller, Wilensky, Carr, Schaefer, SIGGRAPH 2017] Control points Catmull-Rom 𝜆-Curves

https://www.youtube.com/watch?v=MLg8xQgEIfk

Key ideas of 𝜆-Curves

• Cubic Bezier is difficult to control
• Actually, quadratic Bezier is easier to use!
• At most one curvature maximum can exist
• User specifies curvature maxima

è reverse compute control points of quadratic Bezier

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Curvature maxima

i-th quad Bezier (i+1)-th quad Bezier

Possible configurations with cubic Bezier

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Global/nonlinear formulation = iterative computation Change of one CP = change of entire shape ”Buckling” always occurs on CPs Curvature discontinuity at convex/concave boundary

B-spline

• Another way of defining polynomial spline
• Represent curve as sum of basis functions
• Cubic basis is the most commonly used
• Deeply related to subdivision surfaces

è Next lecture

• Non-Uniform Rational B-Spline
• Non-Uniform = varying spacing of knots (𝑢$)
• Rational = arbitrary weights for CPs
• (Complex stuff, not covered)
• Cool Flash demo:

http://geometrie.foretnik.net/files/NURBS-en.swf

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Parametric surfaces

• One parameter è Curve 𝑄(𝑢)
• Two parameters è Surface 𝑄 𝑡, 𝑢
• Cubic Bezier surface:
• Input: 4×4=16 control points 𝑄IP

𝑄 𝑡, 𝑢 = J

IQ@ A

J

PQ@ A

𝑐I

A 𝑡 𝑐 P A 𝑢 𝑄IP

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Bernstein basis functions 𝑐+

$ 𝑢 = 1 − 𝑢 $

𝑐(

$ 𝑢 = 3𝑢 1 − 𝑢 "

𝑐"

$ 𝑢 = 3𝑢"(1 − 𝑢)

𝑐$

$ 𝑢 = 𝑢$

Coons patch

• Given four curves joining at endpoints,

evaluate pairs of opposing curves, then interpolate them è a surface patch

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https://en.wikipedia.org/wiki/Coons_patch Patch function: where is the bilinear interpolation of four corners c0 c1 d0 d1 s t

3D modeling using parametric surface patches

• Pros
• Can compactly represent smooth surfaces
• Can accurately represent spheres, cones, etc
• Cons
• Hard to design nice layout of patches
• Hard to maintain continuity across patches
• Often used for designing man-made objects

consisting of simple parts

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Pointers

• http://en.wikipedia.org/wiki/Bezier_curve
• http://antigrain.com/research/adaptive_bezier/
• https://groups.google.com/forum/#!topic/comp.graphics.algorithms/2FypAv

29dG4

• http://en.wikipedia.org/wiki/Cubic_Hermite_spline
• http://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline

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