3D Affine Transforms and Point Representation in Computer Graphics, Slides of Computer Graphics

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Computer Graphics3D Affine transforms

Introduction to Transformations n Introduce 3D affine transformation: n Position (translation) n n Size (scaling)Orientation (rotation) n Previously developed 2D (x,y)^ n^ Shapes (shear) n n Now, extend to 3DExtend transform matrices to 3D or (x,y,z) case n Enable transformation of points by multiplication

Transforms in 3D n n 2D:3D: 3x3 matrix multiplication4x4 matrix multiplication: homogenous coordinates n n Recall: transform object = transform each verticeGeneral form:



 

 

 



031 032 033 134

21 22 23 24

11 12 13 14 m m m m

m m m m

m m m m M 

 

 

 





 

 

 



1 1^ z

y

x z

y

x P

P

P QQ M Xform of P Q

Recall: 3x3 2D Translation Matrix

 x y^ ' ' ^ =^  x y  +  tt yx 

y

x

y

x

t

t

y

x

= (^) *

ßPreviously, 2D :

2D Scaling ßScale: Alter object size by scaling factor (sx, sy). i.e

x’ = x. Sxy’ = y. Sy

(1,1)

(2,2) Sx = 2, Sy = 2 (2,2)

(4,4)

 yx '' = Sx 0 Sy^0  yx 

Recall: 3x3 2D Scaling Matrix

 xy '' =  Sx 0 Sy^0  yx 

y

x Sy

Sx y

x

Example: OpenGL Table Leg // define table leg//-------------------------------------------------------------------------- void tableLeg(double thick, double len){------ glPushMatrix();glTranslated(0, len/2, 0); glScaled(thick, len, thick);glutSolidCube(1.0); } glPopMatrix();

Recall: 3x3 2D Rotation Matrix

(x,y)

(x’,y’) θ r φ

 yx '' = ^ cos(sin( θθ )) −cos(sin( θθ )) yx 

sin( ) cos( ) 0

cos( ) sin( ) 0 1

y

x y

x θ θ

θ θ

Rotating in 3D n New terminology n X-roll: rotation about x-axis n n Y-roll: rotation about y-axisZ-roll: rotation about z-axis n Which way is +ve rotation n Look in –ve direction (into +ve arrow) n CCW is +ve rotation

x

y z+

Rotating in 3D

Rotating in 3D





 

 



 

 = − 0 0 0 1

0 0

0 1 0 0

0 0 s c

c s Ry β

A y-roll:





 

 



 

 −

0 0 0 1

0 0 1 0

0 0

0 0 s c

c s Rz β

A z-roll:

Rules:•Rotate row, column int. is 1•Rest of row/col is 0 •c,s in rect pattern

OpenGL: glrotated( θ, 0,1,0 )

OpenGL: glrotated( θ, 0,0,1 )

Example: Rotating in 3D





 

 



 







 

 



 

 



 

 



 

 = − 1

1. 964

1

1. 6

1

4

1

3

0 0 0 1

0 0

0 1 0 0

0 0 s c

c s Q

Q: Using y-roll equation, rotate P = (3,1,4) by 30 degrees: A: c = cos(30) = 0.866, s = sin(30) = 0.5, and

E.g. first line: 3.c + 1.0 + 4.s + 1.0 = 4.

Matrix Multiplication Code n Outline of solution: n Declare P,Q as array: n Declare transform matrix as 2-dimensional array^ • Double P[4], Q[4]; n Remember: C indexes from 0, not 1^ • Double M[4][4]; n Long way: • Write out equations line by line expression for Q[i]

• E.g. Q[0] = P[0]M[0][0] + P[1]M[0][1] + P[2]M[0][2] +P[3]M[0][3] n Cute way: • Use indexing, say i for outer loop, j for inner loop

Matrix Multiplication Code n Using loops looks like: n for(i=0;i<4;i++) {temp = 0; for(j=0;j<4;j++){ } temp += P[j]*M[i][j]; }Q[i] = temp; n n Test matrice code rigorouslyUse known results (or by hand) and plug into your code