ComputerGraphics 09

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Computer Graphics

Spring 2007, #9

3D graphics





Computer graphics in 3D• Define 3D objects

– geometry – translations, rotations, scalings

• Calculate their appearance

– lighting• Project onto camera plane

– perspective transformation

– in 2D: parallel projection – depth cueing: light intensities, removal of hiddenlines, shadows



3D Objects• Typically defined by specifying the

surfaces enclosing a non-zero volume• Properties of 3D objects:

– vertices – edges – polygons, plane or non-plane

– plane normals



3D Objects

V1

V2

V3

V4

V5

V6

E1 E2

E3E4

E5

E6

E7

E8

Polygon list:

P1: E

1, E

2, E

3

P 2: E 2, E 4, E 5

P 3: E 6, E 7, E 8, E 4

Edge list:

E1: V1, V2

E2: V2, V3

E3: V3, V1

E9



2D Plane in 3D• Defined by three points in a plane, not

along a line:P 1 = (x 1,y1,z 1), P 2= (x 2,y2,z 2), P 3 = (x 3,y3,z 3)

• Equation for any point in the plane:A∗x+B∗y+C∗z+D=0• Actually a three parameter equation:

A/D∗x+B/D∗y+C/D∗z=-1



2D Plane in 3D• Solve for parameters:

• Succeeds if Det | | ≠ 0. If determinant iszero then points are collinear.

z3y3x3

z2y2x2

z1y1x1

C/DB/DA/D

-1-1-1

=



2D Plane in 3D• Solutions:

A/D = [y2z3-y3z2 –(y 1z3-y3z1)+y1z2-y2z1](-1/det)

D = -(x 1y2z3- x1y3z2- x2y1z3+

x2y3z1+ x 3y1z2- x3y2z1)



Normal to Plane• We calculate a normal to the plane as follows:

• Denote the vector from the origin to V 1 by V 1

• (V2 - V1) and (V 3 - V1) lie in the plane defined bythe three vertices V 1,V2,V3

• Their cross product is perpendicular to the plane= a normal

N = (V2-V1)x(V3-V1) = Ai + Bj +Ckwhere i = (1,0,0), j = (0,1,0), k = (0,0,1)



Points In and Outisde the Plane• Denote any point in space by P = (x,y,z)• N · P = -D for points in the plane• A∗x+B∗y+C∗z+D > 0 for points above the

plane• A∗x+B∗y+C∗z+D < 0 for point below the

plane



File Formats• Files defining 3D objects should contain

– vertex information – edges, polygons

– colour information – normal, tangent, binormal

• Choose your favourite file format!



.ply File Formatplyformat ascii 1.0 { ascii/binary, format version number }comment made by Greg Turk { comments keyword specified, like all lines }

comment this file is a cubeelement vertex 8 { define "vertex" element, 8 of them in file }property float x { vertex contains float "x" coordinate }property float y { y coordinate is also a vertex property }property float z { z coordinate, too }element face 6 { there are 6 "face" elements in the file }property list uchar int vertex_index { "vertex_indices" is a list of ints }end_header { delimits the end of the header }0 0 0 { start of vertex list }0 0 10 1 10 1 01 0 01 0 11 1 1

1 1 04 0 1 2 3 { start of face list }4 7 6 5 44 0 4 5 14 1 5 6 24 2 6 7 34 3 7 4 0



Linear Algebra• Given two vectors in n-dimensional space

a = (a 1,a 2,..,a n) and b = (b 1,b 2,..,b n)• Vector addition

a + b = (a 1+b 1,a 2+b 2, ..., a n+b n)• Length |a| = (a 1∗a 1+a 2∗a 2+ ... +a n∗a n)1/2

• Dot product (a.k.a. inner product)a·b = a 1∗b1 + a 2∗b2 + ... + a n∗bn

= |a| ∗|b|∗cos( θab )



Linear Algebra: Dot Product• Angle θab between vectors a and b• a and b define a plane and a normal n

x

z

yθ

ab

a

b

n



3D Linear Algebra: Cross Product• The following applies only in 3D.• The normal to the plane formed by a and b

according to a RH coordinate system is

denoted by n• Cross product: a ×b = |a| ∗|b|∗sin( θab )∗ni j ka 1 a 2 a 3b1 b2 b3



3D Translations• A translation of a point at (x,y,z) by the

amount t = (t x,ty,tz) to the position(x+tx,y+t y,z+t z) is given by a translationmatrix in homogeneous coordinates:

1000tz100

ty010tx001

wz

yx

w’z’

y’x’

=



3D Translations• Translation from red to green by t

x

z

y

t =(tx,ty,tz)



3D Translations• The inverse translation T -1(t) is such that

T-1(t) T(t) = 1 4x4

• Explicitly T -1(t) = T(-t)

1000 -tz100

-ty010-tx001

=T-1(t)



3D Rotations: Z-Axis• Rotation w.r.t z-axis

x

z

y

1000010000cos( θ)sin( θ)

00-sin( θ)cos( θ)

wzy

x

w’z’y’

x’

=

θ



3D Rotations: X-Axis• Rotation w.r.t x-axis

x

z

y

10000cos( θ)sin( θ)00-sin( θ)cos( θ)0

0001

wzy

x

w’z’y’

x’

=θ



3D Rotations: Y-Axis• Rotation w.r.t y-axis

x

z

y

10000cos( θ)0-sin( θ)0010

0sin( θ)0cos( θ)

wzy

x

w’z’y’

x’

=θ



3D Rotations: Inverse• The inverse rotation R -1(θ) is such that

R -1(θ)R(θ) = R ( θ)R -1(θ) = 1 4x4

• Explicitly,R -1(θ) = R T(θ)

• The transpose of a matrix is such thatMT

ij = M ji

• This holds true for arbitrary pure rotations.For matrix products

[R1(α )R2(β)]T = R T2(β)RT

1(α )



General Rotation in 3D• Given a rotation axis u and a rotation

angle θ: rotate by θ around u• Basic idea:

– Translate and rotate u to align it with acoordinate axis, e.g. z-axis – Rotate around the coordinate axis by θ

– Rotate and translate u back



General Rotation in 3D• Given the rotation axis u (normalized) and

the positive sense rotation angle θ

z

y

x

u =(a,b,c)

P 1 = (x 1,y1,z 1)

P 2 = (x 2,y2,z 2)



General Rotation in 3D1: Translate u to the origin by T(-P 1)

1000-z1100-y1010-x1001

=T (-P 1)



General Rotation in 3D2: Project u onto yz-plane ≡ u’, and obtain α3: Rotate u w.r.t x-axis by α and u will be

rotated into the xz-plane with positive z-

component. Not needed if u || u x.

z

y

x

u =(a,b,c)u’ =(0,b,c)

α

u’’ = (a,0,d)



General Rotation in 3D4: Denote u x = (1,0,0) and u z = (0,0,1),

u’= (0,b,c) and let |u’| = (b 2+c 2)1/2 = d.uz·u’ = c = |u’| ∗cos( α ) = d∗cos( α )u’×u

z= d∗sin( α ) n = b ∗u

x

10000c/db/d0 0-b/dc/d0

0001

Rx(α ) =



General Rotation in 3D5: Rotate u´´ w.r.t. y-axis by β s.t. u’’||u z

– u’’ has the same x-component as u: (a, , ) – u’’ has y-component zero: (a,0, ) – u’’ has as z-component |u’| because u’ has

been rotated onto the xz-plane: (a,0,d)

z

y

x

u’’ = (a,0,d)

β



General Rotation in 3D• We now calculate cos( β) and sin( β):

|u’’| = (a 2+d 2)1/2 = (a 2+b 2+c 2)1/2 = 1uz·u’’ = d = |u’’| ∗cos( β) = cos( α )u’’×uz = sin( β) n = -a ∗ux

10000d0a0010

0-a0d

Ry(β) =



General Rotation in 3D• Now rotate by θ w.r.t the z-axis, then

rotate back first by R y-1(β) then by R x-1(α ),and finally translate back by P 1

-1:

R(θ) = T -1(-P 1)R x-1(α )Ry

-1(β)R z(θ)R y(β)R x(α )T(-P 1)



Quaternions• Another representation of rotations in

three dimensions• Other representations have advantages

and disadvantages!• Quaternions originally developed as

extensions of real numbers to complex

numbers to ???



Quaternions• Quaternion number given by

q = s+a ∗i+b∗ j+c∗k = (s,a,b,c) = (s, v)• s,a,b,c ∈ℜ

• i,j,k satisfy the algebra:i∗i = j∗ j = k∗k = -1i∗ j = -j∗i = k

j∗k = -k∗ j = ik∗i = -i∗k = j



Quaternions• Algebra:

|q| 2 = s 2 + a 2 + b 2 + c 2 = s 2 + v·v|q| -1∗|q| = (1,0)|q| -1 = (s,-v)/|q| 2

• Not commutative: q 1∗q2 generally unequalq2∗q1

• A point P = (x,y,z) in 3D space is repre-sented by the quaternion (0,P) = (0,x,y,z)



Quaternions: Rotations• For any rotation by θ along an axis u

through the origin :

P’ = (0,x’) = q ∗P∗q -1

where q = (cos( θ /2),u∗sin( θ /2))



3D Scaling• With the origin as fixed point, scale by s x in

the x-direction, s y in the y-direction and s zin the z-direction:

10000s

z00

00s y0000s x

wzyx

w’z’y’x’

=

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ComputerGraphics 09

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