1. Chapter 5 Geometric Transformations
Computer Graphics
Chapter 5
Geometric Transformations
Andreas Savva

2. 2D Translation
Repositioning an object along a straight line path from one co-ordinate location to another
(x,y) (x’,y’)
To translate a 2D position, we add translation distances tx and ty to the original coordinates (x,y) to obtain the new coordinate position (x’,y’)
x’= x + tx , y’= y + ty
Matrix form T x y

3. 2D Translation
Moving a polygon from position (a) to position (b) with the translation vector (-5, 10), i.e. x y 5 10 15 20 x y 5 10 15 20
(a)
(b)

4. Translating a Polygon class Point2D { public: GLfloat x, y; };
void translatePoly(Point2D P[], GLint n,
GLfloat tx, GLfloat ty) {
GLint i;
for (i=0; i<n; i++) {
P[i].x = P[i].x + tx;
P[i].y = P[i].y + ty; }
glBegin (GL_POLYGON);
for (i=0; i<n; i++)
glVertex2f(P[i].x, P[i].y);
glEnd();

5. 2D Rotation
Repositioning an object along a circular path in the xy-plane x y
(x,y)
(x’,y’) φ θ r
The original coordinates are:

6. 2D Rotation
Substituting x y
(x,y)
(x’,y’) φ θ r
Matrix form

7. 2D Rotation about a Pivot position
Rotating about pivot position (xr, yr) x y
(x,y)
(x’,y’) φ θ r xr yr

8. Translating a Polygon class Point2D { public: GLfloat x, y; };
void rotatePoly(Point2D P[], GLint n,
Point2D pivot, GLdouble theta) {
Point2D *V;
V = new Point2D[n];
GLint i;
for (i=0; i<n; i++) {
V[i].x = pivot.x + (P[i].x – pivot.x) * cos(theta)
- (P[i].y – pivot.y) * sin(theta);
V[i].y = pivot.y + (P[i].x – pivot.x) * sin(theta)
- (P[i].y – pivot.y) * cos(theta); }
glBegin (GL_POLYGON);
for (i=0; i<n; i++
glVertex2f(V[i].x, V[i].y);
glEnd();
delete[] V;

9. Reduced in size and moved
2D Scaling
Altering the size of an object. Sx and Sy are the scaling factors. If Sx = Sy then uniform scaling. x y
Sx = Sy = ½
Matrix form
Sx = Sy = ½ x’ x
Reduced in size and moved
closer to the origin

10. 2D Scaling relative to Fixed point
Scaling relative to fixed point (xf, yf) x y
Sx = ¼ , Sy = ½ P1 P2 P3
P1’
P2’
P3’
(xf , yf) OR
where the additive terms xf(1-Sx) and yf(1-Sy) are
constants for all points in the object.

11. Translating a Polygon class Point2D { public: GLfloat x, y; };
void scalePoly(Point2D P[], GLint n, Point2D fixedPt,
GLfloat Sx, GLfloat Sy) {
Point2D *V;
V = new Point2D[n];
GLfloat addx = fixedPt.x * (1 – Sx);
GLfloat addy = fixedPt.y * (1 – Sy);
GLint i;
for (i=0; i<n; i++) {
V[i].x = P[i].x * Sx + addx;
V[i].y = P[i].y * Sy + addy; }
glBegin (GL_POLYGON);
for (i=0; i<n; i++
glVertex2f(V[i].x, V[i].y);
glEnd();
delete[] V;

12. Matrix Representation
Use 3×3 matrices to combine transformations
Translation
Rotation
Scaling

13. Inverse Transformations
Translation
Rotation
Scaling

14. Example
Consider the line with endpoints (10, 10) and (30, 25). Translate it by tx = -20, ty = -10 and then rotate it by θ = 90º. x y
(10, 10)
(30, 25)
Right-to-left

15. Solution x y
(10, 10)
(30, 25)

16. Solution (continue) Point (10, 10) Point (30, 25) y x (30, 25)
(0, -10)
(-15, 10)
Point (10, 10)
Point (30, 25)

17. Result Step-by-step x y x y x y T(-20, -10) R(90º) (30, 25) (-15, 10)
(10, 10)
(30, 25)
(0, -10)
(-15, 10)
Step-by-step x y
(10, 15)
(-10, 0) x y
(0, -10)
(-15, 10)
T(-20, -10)
R(90º)

18. Exercises Consider the following object:
Apply a rotation by 145º then scale it by Sx=2 and Sy=1.5 and then translate it by tx=20 and ty=-30.
Scale it by Sx=½ and Sy=2 and then rotate it by 30º.
Apply a rotation by 90º and then another rotation by 45º.
Apply a rotation by 135º. x y 10 25 45

19. Exercises Composite 2D Transformations Translation: Show that:
Rotation: Show that:
Scaling: Show that:

20. General 2D Pivot-Point Rotationx y
(xr , yr) x y
Original position
and Pivot Point
Translate Object so that
Pivot Point is at origin x y
(xr , yr) x y
Rotation about origin
Translate object so that Pivot Point
is return to position (xr , yr)

21. General Pivot-point Rotation
Using Matrices

22. Exercises Consider the following object:
Apply a rotation by 60° on the Pivot Point (-10, 10) and display it.
Apply a rotation by 30° on the Pivot Point (45, 10) and display it.
Apply a rotation by 270° on the Pivot Point (10, 0) and then translate it by tx = -20 and ty = 5. Display the final result. x y 10 25 45

23. General 2D Fixed-Point Scalingy
(xf , yf) x y
Original position
and Fixed Point
Translate Object so that
Fixed Point is at origin
(xf , yf) x y x y
Scale Object with
respect to origin
Translate Object so that Fixed Point
is return to position (xf , yf)

24. General 2D Fixed-Point Scaling
Using Matrices

25. Exercises Consider the following object:
Scale it by sx = 2 and sy = ½ relative to the fixed point (140, 125) and display it.
Apply a rotation by 90° on the Pivot Point (50, 60) and then scale it by sx = sy = 2 relative to the Fixed Point (0, 200). Display the result.
Scale it sx = sy = ½ relative to the Fixed Point (50, 60) and then rotate it by 180° on the Pivot Point (50, 60). Display the final result. x y 60 50
125
220

26. Order of Transformations
Object is first translated in the x direction
and then rotated by 90º x y
Object is first rotated by 90º and then
translated in the x direction x y

27. Reflection About the x axis About the y axis y x y x
Reflection of an object about the x axis x y
About the y axis x y
Reflection of an object about the y axis

28. Same as a rotation with 180º
Reflection
Relative to the coordinate origin
Same as a rotation with 180º x y
With respect to the line y = x x y
y = x

29. 2D Shear x-direction shear Matrix form y x Initial object y x shx = 21
Matrix form x y
shx = 2 1 2 3

30. 2D Shear x-direction relative to other reference line Matrix form y x1
yref = -1
Matrix form y x
shx = ½, yref = -1 1 2 3
yref = -1

31. 2D Shear y-direction shear Matrix form y x Initial object y x shy = 21
Matrix form x y
shy = 2 1 2 3

32. 2D Shear y-direction relative to other reference line Matrix form y x1
xref = -1
Matrix form y x 1 2
xref = -1
shy = ½, xref = -1

33. Transformations between 2D Coordinate Systemsx y x0 x’ y’ y0 
To translate object descriptions from xy coordinates to x’y’ coordinates, we set up a transformation that superimposes the x’y’ axes onto the xy axes. This is done in two steps:
Translate so that the origin (x0, y0) of the x’y’ system is moved to the origin (0, 0) of the xy system.
Rotate the x’ axis onto the x axis.

34. Transformations between 2D Coordinate Systems
i.e. 1) 2)
Concatenating:

35. Example
Find the x’y’-coordinates of the xy points (10, 20) and (35, 20), as shown in the figure below: x y 30 x’ y’ 10
30º
(10, 20)
(35, 20)

36. y y’ y’ x’ x’ x (-12.38, 18.66) (35, 20) (10, 20) (9.31, 6.16) 30º 10

37. Exercise
Find the x’y’-coordinates of the rectangle shown in the figure below: x y 10 x’ y’
60º 20

38. x’= x + tx , y’= y + ty , z’= z + tz
3D Translation
Repositioning an object along a straight line path from one co-ordinate location to another
(x,y,z) (x’,y’,z’)
To translate a 3D position, we add translation distances tx ty and tz to the original coordinates (x,y,z) to obtain the new coordinate position (x’,y’)
x’= x + tx , y’= y + ty , z’= z + tz
Matrix form (4 × 4)
T(tx, ty, tz) x y z

39. 3D Rotation x y z
z-axis
The 2D z-axis rotation equations are extended to 3D.
Matrix form x y z

40. 3D Rotation
x-axis x y z
Matrix form

41. 3D Rotation
y-axis x y z
Matrix form

42. 3D Scaling x y z
Matrix form

43. Other 3D Transformations
Reflection z-axis x y z
Shears z-axis