Subodh Kumar Dept of Computer Sc. & Engg. IIT Delhisubodh/courses/CSL781/pdfslides/transformation.… · The original space is a projection of this n ... Orthographic ! Oblique !

Computer Graphics

Subodh Kumar

Dept of Computer Sc. & Engg. IIT Delhi

Model Representation


! List of triangles ! eg, Begin(TRI ); Vertex(v0);.. End(); ! Or even, Begin(TRISTRIP ); Vertex(v0);.. End();



! Mesh ! Vertices & Connectivity ! eg, InterleavedArrays(); DrawElements(); ! Displaylists, VBO



! Mesh ! Vertices & Connectivity ! eg, InterleavedArrays(); DrawElements(); ! Displaylists, VBO

! Instanced ! aka Scene graph

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

Sphere

Face Mesh

Cylinder

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

Cylinder

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

Cylinder

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

Cylinder

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

Cylinder

T’

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

Cylinder

T’

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

T

Sphere

Face Mesh

BotArmy

Cylinder

T’

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

World

T

Sphere

Face Mesh

BotArmy

Cylinder

T’

Scene Graph

Robot

BodyHead

ArmTrunk Mesh

Leg Mesh

EyeMouth Mesh

World

T

Sphere

Face Mesh

BotArmy

Cylinder

T0

T1

T2

T3

T4

Scene Graph Draw

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw

! DFS Traversal

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw

! DFS Traversal! Multiply World->Army matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw

! DFS Traversal! Multiply World->Army matrix! For each robot:

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw


! Multiply Army->Robot matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw


! Multiply Army->Robot matrix! Multiply Robot->Head matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw



! Multiply Head->Face Matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw



! Multiply Head->Face Matrix! Draw Face

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw



! Multiply Head->Face Matrix! Draw Face! …

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix! Un-multiply Robot->Head matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix! Un-multiply Robot->Head matrix! Multiply Robot->Body matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix! Un-multiply Robot->Head matrix! Multiply Robot->Body matrix! … Robot

BodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix! Un-multiply Robot->Head matrix! Multiply Robot->Body matrix! …! Un-multiply Robot->Body matrix

RobotBodyHead

Arm

World

BotArmy

Scene Graph Draw




! Un-multiply Head->Face Matrix! Un-multiply Robot->Head matrix! Multiply Robot->Body matrix! …! Un-multiply Robot->Body matrix! …

RobotBodyHead

Arm

World

BotArmy

OpenGL Matrix Stack

! glPushMatrix() ! glMultMatrix(M1)

! glPopMatrix() ! glLoadMatrix()

OpenGL Matrix Stack



M0

OpenGL Matrix Stack



M0

M0 M0

OpenGL Matrix Stack



M0

M0 M0

M0 M0*M1

OpenGL Matrix Stack



M0

M0

M0 M0

M0 M0*M1

Transformation

! Change an object to another ! Transform all points of an object ! E.g.,

! Move it, Translate ! Re-orient it, Rotate ! Re-size it, Scale ! Skew, etc.

! Want to compose a sequence of transformations

Interior of Triangle

V0

V1

V2


V0

V1

V2


V0

V1

V2


V0

V1

V2


V0

V1

V2

V


V0

V1

V2

(1-t ) V1 + t V2

t =(V-V1)/(V2-V1)V


V1

V2

aV0+bV1+cV2

V0

V

a =Area(V2V V1)/ Area(V2V0V1)

. .

Simple Xform Types

! Rigid body ! No length changes

! Translation, Rotation

! Linear ! aT(P) + bT(Q) = T(aP+bQ)

! Rotation, Scale, nxn Matrix-multiplication

! Affine ! aT(P) + bT(Q) = T(aP+bQ) + K ! Preserves parallel lines ! Translation, Rotation, Scale ! Linear in projective space, n+1xn+1 matrix

multiplication ! Projective

Translation

Translation

Translation

Translation

Translation

tx, ty, tz

Rotation

Rotation

θ

Rotation

θ

Vector, θ

Rotation

θ

Vector, θ

Rotation

r

rθ

Vector, θ

Rotation

r

rθ

α

Vector, θ

Rotation

r

r

x = r cos α

θ

α

Vector, θ

Rotation

r

r

x = r cos αr cos(α+θ)

θ

α

Vector, θ

Rotation

r

r


θ

α

Vector, θ

x´= r cosαcosθ - r sinαsinθ

Rotation

r

r


θ

α

Vector, θ

x´= r cosαcosθ - r sinαsinθ = xcosθ - ysinθ

Rotation

r

r


θ

cos θ -sin θ sin θ cos θ

α

Vector, θ


x y

Rotation

r

r


θ

cos θ -sin θ sin θ cos θ

cos θ -sin θ 0 sin θ cos θ 0 0 0 1

3D

α

Vector, θ


x y

Rotation

Rotation

Rotation

Rotation

Rotation

Rotation

Rotation

Rotation

1.Translate to origin 2.Rotate 3.Translate back

Rotation


Rotation


Rotation


Rotation


Scale

Scale

Scale

Scale

Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1

Scale

Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1

Scale

Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1

Scale

Sx 0 0 0 0 Sy 0 0 0 0 Sz 0 0 0 0 1

Point, Sx, Sy, Sz

Translation

1 0 0 tx

0 1 0 ty

0 0 1 tz

0 0 0 1

x y z w

Homogeneous Coordinates

Projective Space

! n-dimensional projective space is constructed from an n+1-dimensional vector space ! Where, aP = P ! (0,0,0,…) is undefined

! The original space is a projection of this n+1-dimensional vector space ! x, y, z, w = x/w, y/w, z/w, 1 = affine space ! Except when w=0: x, y, z, 0 = x/z, y/z, 1, 0

! Except when z = 0 … ! w is called the homogeneous coordinate

Projective Plane

x

yw

w=1

x = uw, y = vw, ie, uw u vw = v w 1

Projective Plane

x

yw

w=1

x = uw, y = vw, ie, uw u vw = v w 1

2

664

1 0 0 dx

0 1 0 dy

0 0 1 dz

0 0 0 1

3

775

2

664

x

y

z

1

3

775=

2

664

x+ dx

x+ dy

z + dz

1

3

775

2

664

sx 0 0 0

0 sy 0 0

0 0 sz 0

0 0 0 1

3

775

2

664

x

y

z

1

3

775=

2

664

x.sx

x.sy

z.sz

1

3

775

2

664

cos↵ sin↵ 0 0

� sin↵ cos↵ 0 0

0 0 1 0

0 0 0 1

3

775

2

664

x

y

z

1

3

775=

2

664

x cos↵+ y sin↵

�x sin↵+ y cos↵

z

1

3

775

Examples of Xform Matrices

Translation by dx, dy, dz Scaling by sx, sy, sz

Rotation by α (about Z)

Change of Basis

! New basis: ! Orthonormal axes:

V1, V2, V3 ! Origin: O

2

664

— V1 — 0— V2 — 0— V3 — 0

0 0 0 1

3

775

2

664

1 0 0 �Ox

0 1 0 �Oy

0 0 1 �Oz

0 0 0 1

3

775

Rotation about Axis

19

u,v,w

ɵ

2

4x

0

y

0

z

0

3

5=

2

4u.u(1� cos ✓) + cos ✓ w.u(1� cos ✓)� w sin ✓ u.w(1� cos ✓)� v sin ✓

v.u(1� cos ✓) + w sin ✓ w.u(1� cos ✓) + cos ✓ v.w(1� cos ✓)� u sin ✓

w.u(1� cos ✓)� v sin ✓ w.u(1� cos ✓)� u sin ✓ w.w(1� cos ✓) + cos ✓

3

5

2

4x

y

z

3

5

Transformation Recap

! General transformation = ! Composition of known special transformations

! Transformation is equivalent to an inverse change of basis

! Be mindful of space in which ! the transformation is specified ! the vertex/vector is specified

! [M1 M2 M3 … Mk] V = ! Mk then Mk-1 until M1 in world-coordinates

! M1, then M2, until Mk in object-coordinates

Composition: Fixed frame

! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p

Composition: Local frame

! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p


! T(1,0) ! R(90) ! R T p

Projection

! Projection = dimension reduction ! 3D " 2D in our case ! 3D lines map to 2D points ! Parallel

! Orthographic ! Oblique

! Projective

Perspective

! Transform plane of projection to Z = -1

Z = -1

Y

Z O

yy'

z

y'/-1 = y/z, or y' = -y/z and x' = -x/z

2

664

1 0 0 00 1 0 00 0 1 00 0 �1 1

3

7750

Perspective II

! Also normalize the image space ! Image X, Y lie in [-1:1]

Z = zv

y

ZO

zv tan(θ)

z

y' = zv y/z. Normalized y', call it y'' = y'/(zv tanθ) = y cotθ/z

θ FOV/2

Image Plane

y'

Perspective III

! What if we also want Z to lie in [0:1] ! Znear = 0

! Zfar = 1

Viewport

! What if we want ! X to lie in [0:Screen-Width] ! Y to lie in [0:Screen-Height]

! Xscreen = (Ximage + 1)/2 * Width

! Yscreen = (Yimage + 1)/2 * Height

Transformation of Normals

n

d


n.d = 0

n

d


n.d = 0nTd = 0

n

d


n.d = 0nTd = 0=> n’T Md must be 0

n

d


n.d = 0nTd = 0=> n’T Md must be 0=> (M’n)T Md = 0

n

d


n.d = 0nTd = 0=> n’T Md must be 0=> (M’n)T Md = 0=> nTM’T Md = 0

n

d


n.d = 0nTd = 0=> n’T Md must be 0=> (M’n)T Md = 0=> nTM’T Md = 0=> M’T M = k I

n

d


n.d = 0nTd = 0=> n’T Md must be 0=> (M’n)T Md = 0=> nTM’T Md = 0=> M’T M = k I=> M’T = k M-1

n

d


n.d = 0nTd = 0=> n’T Md must be 0=> (M’n)T Md = 0=> nTM’T Md = 0=> M’T M = k I=> M’T = k M-1

=> M’ = (M-1)T k = 1

n

d

3D Interaction

! Specify point (3 DOF) ! Specify orientation (3 DOF) ! Navigate (6 DOF)

! Manipulate object ! Manipulate eye/walk ! Specify path

! Select an object or region in space ! Find out its location and orientation

30

3D Interface! Mouse provide a 2D input (+ buttons)

! 2D location ! Up-down, Left-right motion ! 2D rotation

! 3D navigation has 6 DOF ! Up / Down ! Forward / Backward ! Left / Right ! Pitch (transverse axis) ! Yaw (normal axis) ! Roll (longitudinal axis)

31

cf Hoffman

Yaw

Pitch

Roll

Trackball Interface

! Inspect an object ! Imagine rotating a sphere by a handle

! Map planar 2D motion onto unit sphere ! Raise old position <x0, z0> to <x0, y0, z0>

! Raise new position <x1, z1> to <x1, y1, z1> ! Find rotation that moves handle from old to

new

Trackball

Trackball

! p1: x12+y1

2+z12 = 1

Trackball

! p1: x12+y1

2+z12 = 1

! p2: x22+y2

2+z22 = 1

Trackball

! p1: x12+y1

2+z12 = 1

! p2: x22+y2

2+z22 = 1

! Ball rotates about the axis formed by p1 and p2

! p1 X p2

Trackball

! p1: x12+y1

2+z12 = 1

! p2: x22+y2

2+z22 = 1


! p1 X p2

! Rotation angle proportional to the distance p1p2

! Or simply ||p1 X p2||

Trackball

! p1: x12+y1

2+z12 = 1

! p2: x22+y2

2+z22 = 1


! p1 X p2

! Rotation angle proportional to the distance p1p2

! Or simply ||p1 X p2||

! What happens if the object is not at the origin?

Rotation by Euler angles

34

Yaw

Pitch

Roll


34

Yaw

Pitch

Roll

Gimbal Lock


34

Yaw

Pitch

Roll

Gimbal Lock

! Interpolation, Composition is non-intuitive

Quaternions

! Relates two vectors ! Like a vector relates two points

! q = <s, x, y, z> = <s,v> = s + xi + yj + zk ! Scalar, Vector: relates length and orientation ! cos(θ/2) + U sin (θ/2)

Quaternion multiplication: q1q2 = ( s1s2 – v1.v2, s1v1 + s2v2 + v1 x v2) Quaternion inverse: If q = (s,v), then q-1 = (s,-v)/||q||2 Apply quaternion to p: P' = q P q-1

Quaternions



Quaternion multiplication: q1q2 = ( s1s2 – v1.v2, s1v1 + s2v2 + v1 x v2) Quaternion inverse: If q = (s,v), then q-1 = (s,-v)/||q||2 Apply quaternion to p: P' = q P q-1 , P = (0, p)

Quaternions



Quaternion multiplication: q1q2 = ( s1s2 – v1.v2, s1v1 + s2v2 + v1 x v2) Quaternion inverse: If q = (s,v), then q-1 = (s,-v)/||q||2 Apply quaternion to p: P' = q P q-1 (0, p') = , P = (0, p)

Quaternion Rotation

p' = s2p + v(p.v) + 2s(v x p) + v x (v x p)

36

Linear Interpolation

37

Lerp(t,a, b) = (1� t)a+ tb, 0 t 1

Linear

(Point a and b)

! Euclidean space interpolation:


37

Lerp(t,a, b) = (1� t)a+ tb, 0 t 1

Linear

(Point a and b)

! Euclidean space interpolation:

! Spherical interpolation


37

Lerp(t,a, b) = (1� t)a+ tb, 0 t 1

Slerp(t,a, b) =sin((1� t)✓)

sin ✓a+

sin(t✓)

sin ✓b, 0 t 1

LinearSpherical

(Point a and b)

(Vector a and b) θ = arccos(a.b)

Documents

Subodh Kumar Dept of Computer Sc. & Engg. IIT Delhisubodh/courses/CSL781/pdfslides/transformation.… · The original space is a projection of this n ... Orthographic ! Oblique !