1. GEOMETRIC TRANSFORMATIONS Yingcai Xiao

2. Roadmap to Geometric Transformations
Start with 2D GT
Extend to 3D GT
Preview mathematics
Express GT in four type of notations
Algebraic
Matrix
Logical
Visual

3. Review of Mathematical Preliminaries Mathematical Notations Points:
Vectors: directional lines
P1(x1,y1)
P0(x0,y0)

4. 1. A vector has a direction and a length: Length: |V|=(x2 + y2)1/2
A unit vector: |V| = 1
e.g.:
Normalize a vector:
2. Add two vectors
3. Scalar Multiplication

5. 4. Dot product of two vectors scalar, V W 
V  W if  = 90 => cos = 0 => V  W = 0.
if  < 90 => cos > 0 => V  W > 0.
if  > 90 => cos < 0 => V  W < 0.
5.Normal: a unit vector perpendicular to a surface

6. A point p(xp,yp) is on the line if f(xp,yp) = 0. When b < 0:
Lines
y = mx+b
e.g.: y = x;
ax+by+c=0
Let f(x,y) = ax + by + c
A point p(xp,yp) is on the line if f(xp,yp) = 0.
When b < 0:
p(xp,yp) is above the line if f(xp,yp) < 0.
p(xp,yp) is below the line if f(xp,yp) > 0.
When b > 0:
p(xp,yp) is above the line if f(xp,yp) > 0.
p(xp,yp) is below the line if f(xp,yp) < 0.

7. Parametric Form: P(t)=P0+t(P1-P0); 0 <= t <= 1
P1(x1,y1)
P0(x0,y0)
P(t)=P0+t(P1-P0)

8. Where,
x(t) = x0 + t * (x1 – x0)
y(t) = y0 + t * (y1 – y0)
0 <= t <= 1
x(t) = (1-t) * x0 + t * x1
y(t) = (1-t) * y0 + t * y1

9. 2D Transformations Translate a point
The algebraic representation of translation of point P(x,y) by D(dx,dy) is
x’= x + dx
y’= y + dy
Its “matrix” representation is
Its logical representation is
P’ = P + D
Its visual representation is
P’ (x’,y’)
P(x,y)
D(dx,dy)

10. To move a shape: translate every vertex of the shape5 10 X Y
Before Translation
(4,5)
(7,5)
After Translation 5 10 X Y
(7,1)
(10,1)

11. Scaling (relative to the origin) Scale a point P(x,y) by S(sx, sy)
Algebraic:
x’=sx* x
y’=sy* y
Matrix:
Logic:
P’ = SP or
P’ = S(sx, sy)P
Scale a line P0P1 (scale each point)
P0’=SP0
P1’=SP1

12. Scale a shape: scale every vertex of the shape. Visual representation5 10 X Y
After Scaling
(2,5/4)
(7/2,5/4) 5 10 X Y
Before Scaling
(4,5)
(7,5)
Uniform Scaling: sx=sy

13. Rotate (around the origin)
Positive angles are measured counterclockwise from x axis to y axis.
Rotate point (x,y) around the origin
Algebraic representation:
x’ = x * cos - y * sin
y’ = x * sin + y * cos
Matrix representation:
Logic representation:
P’ = R P

14. Visual representation5 10 X Y 5 10 X Y 
Before
After
Rotate a shape: rotate every vertex of the shape 5 10 X Y
(2.1, 4.9)
(4.9, 7.8)
(5,2)
(9,2)
Before Rotation
After Rotation

15. Summary Translation: P’ = P + D Scaling: P’ = S P Rotation: P’ = R P
Homogeneous Coordinates: P(x,y,w)
(x,y) to (x,y,w) :
(x,y,w) to (x,y)

16. y’= y + dy w’=1 x’= x + dx P’’ = T(dx2 , dy2)P’
= T(dx2 , dy2) T(dx1 , dy1) P
= T(dx2+dx1, dy2+dy1) P

17. Scaling P’=S(Sx, Sy)P P’’ = S(Sx2 , Sy2)P’
= S(Sx2 , Sy2)S(Sx1 , Sy1) P
= S(Sx2Sx1, Sy2Sy1)P

18. P’’=R(2)R(1)PP”=R(2+1)P
Rotation
  P’=R()P
P’’=R(2)R(1)PP”=R(2+1)P

19. Shear Transformation: SHx(a) and SHy(b) P’=SHx(a)P
Shear in x against y by a (or an angle).
x’ = x+ay
y’ = y
w’=1

20. P’=SHy(b)P Shear in y against x by b (or an angle). x’=x y’=y + bx
w’=1.

21. 5 10 X Y 5 10 X Y
Sheared in x 5 10 X Y
Before Shear
Sheared in y

22. Rigid-body Transformation: T and R.
change: location, orientation;
not change: size, angle between elements.
Affine Transformation: S, SH
change: size, location, angle;
not change: line (parallelism).
Note: uniform scaling is between the two. It changes size but not angle.
So far, our S, R, SH are all around the origin.

23. Composition of 2D Transformation
Rotate the house around P1
For every vertex (P) on the object: P’ = T(P1)R()T(-P1)P y y x
After translation
of P1 to origin y x
After rotation  y x P1
After translation
To P1

24. Scale and rotate the house around P1 and move it to P2y x P1
Original
house
Translate P1
to origin
Scale
Rotate P2
Translate to
final position P2

25. For every vertex (P) on the object:
P’ = T(P2) R()S(Sx,Sy)T(-P1)P.
Swap the order of operations: not permitted,
except uniform scaling (sx=sy) can be swapped with rotation.

26. Window-to-Viewport Transformation
world-coordinate: inches, feet etc & screen-coordinate: pixels y
World coordinates
Window x
Screen coordinates
Viewport 1
Viewport 2
Window in
world coordinates
Window translated
to origin
Window scale to
size of viewport
Translated by (u,v)
to final position
Maximum range of
screen coordinates

27. Matrix Representation of 3D Transformationy z
Translation:
Scaling:

28. Rotation, around Z-axis:
Rotation, around X-axis:
Rotation, around Y-axis:

29. Projection: Project 3D objects on to a 2D surface
(0,0,0) d z
P( x, y, z ) x
P’(xp, yp, zp) xp
Projection Plane (View Plane)

30. xp= x’ / w’ = x / (z/d + 1)
yp= y’ / w’ = y / (z/d + 1)

31. Foreshortening: The size of the projected object becomes smaller when the object moves away from the eye. z
P’1
P’2 P1 P2
Perspective Projection: the projection that has the foreshortening effect.

32. For computer-aided design (CAD), we can not have foreshortening
For computer-aided design (CAD), we can not have foreshortening. A meter long object should always measures to 1 meter regardless where it is. Parallel Projection: d  , lines of sight become parallel P1 z P2
View Plan
(0,0,0)

33. Summary 2D GT Homogeneous coordinates 3D GT Rigid body transformation
Affine transformation
Perspective Projection & Foreshortening
Parallel Projection