3-D Transformational Geometry CS418 Computer Graphics John C. Hart - - PowerPoint PPT Presentation

3-D Transformational Geometry

CS418 Computer Graphics John C. Hart

Graphics Pipeline

Homogeneous Divide Model Coords Model Xform World Coords Viewing Xform Still Clip Coords. Clipping Window Coordinates Window to Viewport Viewport Coordinates Clip Coords. Viewing Coords Perspective Distortion

3-D Affine Transformations

• General
• Translation

3-D Coordinates

• Points represented

by 4-vectors

• Need to decide
• rientation of

coordinate axes

1 x y z            

y z x x z y

Right Handed Coord. Sys. Left Handed Coord. Sys.

x y +z (rhc) +z (lhc)

Scale

1 1 1 a x ax b y by c z cz                                     

Uniform Scale a = b = c = ¼

z y x

Squash a = b = 1, c = ¼ Stretch a = b = 1, c = 4 Project a = b = 1, c = 0 Invert a = b = 1, c = -1

3-D Rotations

• About x-axis

– rotates y  z

• About y-axis

– rotates z  x

• About z-axis

– rotates x  y

• Rotations do not commute!

1 cos sin sin cos 1                  cos sin 1 sin cos 1                  cos sin sin cos 1 1                 

Arbitrary Axis Rotation

• Rotations about x, y and z axes
• Rotation x rotation = rotation
• Can rotate about any axis direction
• Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (pv)v – Let a = p – o – Let b = v  a, (note that ||b||=||a||) – Then p’ = o + a cos  + b sin 

• Simple solution to rotate a single point
• Difficult to generate a rotation matrix

to rotate all vertices in a meshed model x y z v



p p’ a b a b



p p’

||a||cos ||a||sin

Arbitrary Rotation

• Find a rotation matrix that rotates by

an angle  about an arbitrary unit direction vector v x z v v = (xv,yv,zv), xv

2+yv 2+zv 2=1

Rotate by  about v

[ ]=[ ][ ][ ]

Rotate z to v Rotate v to z Rotate by  about z

y



z v x z



v z

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

x y z v vyz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

zv yv d

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

• 2. Then cosfx = zv/d and sinfx = yv/d

x y z v vyz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

zv yv d fx

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

• 2. Then cosfx = zv/d and sinfx = yv/d
• 3. Rotate v by fx about x into the xz plane

x y z v vyz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

zv yv d fx

1 1

z y d d y z d d

                

v v v v

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

• 2. Then cosfx = zv/d and sinfx = yv/d
• 3. Rotate v by fx about x into the xz plane

x y z v vxz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

fx

1 1

z y d d y z d d

                

v v v v

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

• 2. Then cosfx = zv/d and sinfx = yv/d
• 3. Rotate v by fx about x into the xz plane
• 4. Then cosfy = d and sinfy = xv

x y z v vxz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

fy 1 d xv

1 1

z y d d y z d d

                

v v v v

Rotate v to z

• 1. Project v onto the yz plane and

let d = sqrt(yv

2 + zv 2)

• 2. Then cosfx = zv/d and sinfx = yv/d
• 3. Rotate v by fx about x into the xz plane
• 4. Then cosfy = d and sinfy = xv
• 5. Rotate vxz by fy about y into the z axis

x y z v vxz v = (xv,yv,zv), xv

2+yv 2+zv 2=1

fy fx

1 1 1 1

z y d d y z d d

d x x d                              

v v v v

v v

Rotate  about v

• Let Rv() be the rotation matrix for

rotation by  about arbitrary axis direction v

• Recall (Rx Ry) is the matrix

(product) that rotates direction v to z axis

• Then

Rv() = (Ry Rx)-1 Rz() (Ry Rx) = Rx

• 1 Ry
• 1 Rz() Ry Rx

= Rx

T Ry T Rz() Ry Rx

(since the inverse of a rotation matrix is the transpose of the rotation matrix)

Easier Way

• Find an orthonormal vector system

x y z u v

Easier Way

• Find an orthonormal vector system

– Let r = u  v/||uv|| x y z u v r

Easier Way

• Find an orthonormal vector system

– Let r = u  v/||uv|| – Let u’ = v  r x y z u v r u’

Easier Way

• Find an orthonormal vector system

– Let r = u  v/||uv|| – Let u’ = v  r

• Find a rotation

from <r,u’,v>  <x,y,z>

' 1 ' ' 1 1 1

x x x x y y y y z z z z

r u v r r u v r r u v r                                      1 ' ' ' 1 1 1

x y z x x y z y x y z z

r r r r u u u r v v v r                                       

x y z u v r u’

Graphics Pipeline

Homogeneous Divide Model Coords Model Xform World Coords Viewing Xform Still Clip Coords. Clipping Window Coordinates Window to Viewport Viewport Coordinates Clip Coords. Viewing Coords Perspective Distortion

W2V Persp View Model                                        

Screen Vertices Model Vertices

Graphics Pipeline

W2V Persp View Model 1 1

s m s m m

x x y y z                                                                         

Graphics Pipeline

1

s s

x y                    1

m m m

x y z                  

M

Transformation Order

1

s s

x y                    1

m m m

x y z                  

M

x y z glutSolidTeapot(1); glRotate3f(-90, 0,0,1); glTranslate3f(0,1,0); glutSolidTeapot(1); glTranslate3f(0,1,0); glRotate3f(-90, 0,0,1); glutSolidTeapot(1);

1

s s

x y                    1

m m m

x y z                                          

M R T

1

s s

x y                    1

m m m

x y z                                          

M T R