1. Introduction to Computer Graphics Geometric Transformations
EEL
Introduction to Computer Graphics
Geometric Transformations By
Kiranmayi Karri U

2. Outline Transformation Geometric Transformation 2D Transformation
References

3. Transformation What is a transformation?
An operation that changes one configuration into another. A geometric transformation maps positions that define the object to other positions Linear transformation means the transformation is defined by a linear function.
Why use them?
Modeling
Moving the objects to the desired location in the environment
Multiple instances of a prototype shape

4. Geometric Transformation
In computer graphics many applications need to alter or manipulate a picture, for example, by changing its size, position or orientation. This can be done by applying a geometric transformation to the coordinate points defining the picture.
Once the models are prepared, we need to place them in the environment .Objects are defined in their own local coordinate system .We need to translate, rotate and scale them to put them into the world coordinate system .

5. Types of Transformations
Geometric Transformations
Translation
Rotation
Scaling
Linear Transformation
Non-uniform scales, shears or skews
Non-linear Transformation
• Twists, bends, warps, morphs

6. 2-D Transformations
Translation
A common requirement is to move a picture to a new position. Translation transform simply moves every point by a certain amount horizontally and a certain amount vertically. The original object and its translation have the same shape and size, and they face in the same direction.

7. 2D Translation-Matrix Notation
A translation moves all points in an object along the same straight-line path to new positions.
The path is represented by a vector, called the translation or shift vector.
We can write the components:
p'x = px + tx
p'y = py + ty
or in matrix form:
P' = P + T

8. Rotation 2-D Transformations
A rotation is a transformation that turns a figure about a fixed point called the center of rotation.  An object and its rotation are the same shape and size, but the figures may be turned in different directions.
Another common type of transformation is rotation. This is used to orientate objects. A rotation transform, rotates each point about the origin, (0,0). Every point is rotated through the same angle, called the angle of rotation. For this purpose, angles can be measured either in degrees or in radians. A rotation with a positive angle rotates points in the direction from the positive x-axis to the positive y-axis.

9. 2D Rotation – Matrix Notation
A rotation repositions all points in an object along a circular path in the plane centered at the pivot point.

10. Scaling 2-D Transformations
Changing the size of an object is called a scale. We scale an object by scaling the x and y coordinates of each vertex in the object.
We have two types of scaling, Uniform and Non-Uniform Scaling
Uniform Scaling
Uniform scaling is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar  to the original. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.

11. Scaling 2-D Transformations
Non Uniform Scaling
Is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching. Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it.

12. 2D Scaling-Matrix Notation

13. Homogeneous Coordinates
Homogeneous coordinates are a way to handle projective geometric spaces easily. These are non-Euclidean geometries that represent non-linear projections, like a perspective projection. Now, they also use an extra coordinate. The transformation between a Cartesian coordinate system and a Homogeneous one is done by dividing all of the components of the Homogeneous coordinates by the last coordinate.

14. Homogenous Coordinates
In Euclidean space (geometry), two parallel lines on the same plane cannot intersect, or cannot meet each other forever. It is a common sense that everyone is familiar with.  However, it is not true any more in projective space. Euclidean space (or Cartesian space) describe our 2D/3D geometry so well, but they are not sufficient to handle the projective space (Actually, Euclidean geometry is a subset of projective geometry). The Cartesian coordinates of a 2D point can be expressed as (x, y).  What if this point goes far away to infinity? The point at infinity would be (∞,∞), and it becomes meaningless in Euclidean space. The parallel lines should meet at infinity in projective space, but cannot do in Euclidean space.
Solution to the problem of Euclidean Space is Homogeneous Coordinates
Homogeneous coordinates, introduced by August Ferdinand Möbius, make calculations of graphics and geometry possible in projective space. Homogeneous coordinates are a way of representing N-dimensional coordinates with N+1 numbers.  To make 2D Homogeneous coordinates, we simply add an additional variable, w, into existing coordinates. Therefore, a point in Cartesian coordinates, (X, Y) becomes (x, y, w) in Homogeneous coordinates. And X and Yin Cartesian are re-expressed with x, y and w in Homogeneous as;  X = x/w  Y = y/w  For instance, a point in Cartesian (1, 2) becomes (1, 2, 1) in Homogeneous. If a point, (1, 2), moves toward infinity, it becomes (∞,∞) in Cartesian coordinates. And it becomes (1, 2, 0) in Homogeneous coordinates, because of (1/0, 2/0) = (∞,∞). Notice that we can express the point at infinity without using "∞".

15. Inverse Transformations

16. Composite 2-D Transformations
We can represent any sequence of transformations as a single matrix.
No special cases when transforming a point – matrix * vector.
Composite transformations – matrix * matrix.
Composite transformations:
Rotate about an arbitrary point – translate, rotate, translate
Scale about an arbitrary point – translate, scale, translate
Change coordinate systems – translate, rotate, scale

17. Composite Transformation
Transformation T followed by
Transformation Q followed by
Transformation R

18. Order of Composite transformation
In composite transformations, the order of transformations is very important.
Rotation followed by Translation:
Translation followed by Rotation:

19. Rotation wrt x-axis and y-axis
„To rotate about an arbitrary point P (px,py) by θ: „ Translate the object so that P will coincide with the origin: T(-px, -py) „ Rotate the object: R(θ) „ Translate the object back: T(px,py) .
„ Translate the object so that P will coincide with the origin: T(-px, -py) „ Rotate the object: R(θ) „ Translate the object back: T(px,py) „

20. Rotation wrt. Arbitrary point
To rotate about P1 (x1,y1), the following sequence of the fundamental transformations are needed: 1.Translate the object by (-x1, -y1)  2.Rotate (Ø) 3.Translate the object by (x1, y1)
The matrix representation of these steps is

21. Scaling wrt. Arbitrary point
To scale about P1 (x1,y1), the following sequence of the fundamental transformations is needed 1.Translate the object by (-x1, -y1) 2.Scale by (Sx, Sy) 3.Translate the object by (x1, y1)
This series of steps can be performed as following: 

22. Composite transformation order
In a composite transformation, the order of the individual transformations is very important. Matrix operations are not cumulative. For example, the result of a Graphics → Rotate → Translate → Scale → Graphics operation will be different from the result of a Graphics → Scale → Rotate → Translate → Graphics operation. The main reason that order is significant is that transformations like rotation and scaling are done with respect to the origin of the coordinate system. The result of scaling an object that is centered at the origin is different from the result of scaling an object that has been moved away from the origin. Similarly, the result of rotating an object that is centered at the origin is different from the result of rotating an object that has been moved away from the origin.
Scale → Rotate → Translate composite transformation

23. Composite transformation order
Translate → Rotate → Scale composite transformation with Append
Translate → Rotate → Scale composite transformation with Prepend

24. Rigid Motion
A rigid motion is the action of taking an object and moving it to a different location without altering its shape or size. Reflections, rotations, translations, and glide reflections are all examples of rigid motions. In fact, every rigid motion is one of these four kinds.

25. Rigid Motion Reflection
Every reflection is completely determined by its axis of reflection. 2. Every reflection is completely determined by a single point-image pair P and P 0 (provided that P 6= P 0 ). 3. The fixed points of a reflection are exactly the points on its axis of reflection. 4. Reflections are improper rigid motions (they reverse clockwise and counterclockwise orientations). 5. Applying the same reflection twice gives the identity motion.

26. Rigid Motion
Rotation
Every rotation is completely determined by its rotocenter and the angle of rotation. 2. A 360◦ rotation is equivalent to the identity motion. (So are720◦ , 1040◦ , ) 3. A rotation that is not the identity has exactly one fixed point: the rotocenter. 4. Rotations are proper rigid motions (they preserve clockwise/counterclockwise orientations). 5. A rotation is determined by any two point-image pairs P, P 0 and Q, Q 0

27. Rigid Motion Translation
1. Every translation is completely determined by a vector of translation, v. 2. In a translation other than the identity, every point gets moved, so there are no fixed points. 3. Translations are proper rigid motions (they preserve clockwise/counterclockwise orientations). 4. A translation is determined by any one point-image pair P, P The effect of a translation can be reversed by a translation in the opposite direction (with vector −v).

28. Reflections
Reflection is nothing more than a rotation of the object by 180o. In case of reflection the image formed is on the opposite side of the reflective medium with the same size. Therefore we use the identity matrix with positive and negative signs according to the situation respectively.

29. Reflections

30. Reflections

31. Reflections wrt to an arbitrary axis

32. Shear in x and y -direction
A shear along the x direction in 2D changes a rectangle (with lower right corner at the origin) into a parallelogram as follows
The transformation has the properties: a) The y coordinate of any point (x,y) is unchanged b) the x coordinate is stretched in a
linear way, based on the height of the point above the x axis - i.e. on y. Thus the coordinate change has the form:
x' = x + ay y' = y
where a is a constant that measures the degree of shearing. If a is negative then the shearing is in the opposite direction.

33. Matrix Notation
A suitable multiplicative matrix description of the transform is given by:
SHx(a) = | 1 a 0 |
| |
| |
We can similarly introduce a shearing along the y axis, which would have the form:
x' = x y' = y + bx
and in this case a suitable matrix description is given by:
SHy(b) = | |
| b 1 0 |

34. Raster Methods Raster Graphics
Image is represented as a rectangular grid of colored pixels
PIXEL = Pictu(X)re ELement
Translation
If you want to move a point (x0 , y0) to a new position (xt , yt)
The transformation would be
xt = x0 + dx and yt = y0 + dy
For motion dx and dy are the components of the velocity vector
dx = cos v and dy = - sin v

35. Raster Methods Scaling Rotation
Multiply the coordinates of each vertex by the scale factor
Everything will expand from the center
The transformation would be
xt = x0 * scale and yt = y0 * scale
Rotation
Spin and object around it centered around the z-axis
Rotate each point the same angle
Positive angles are clockwise
Negative angles are counterclockwise
C++ uses radians (not degrees)
xt = x0 * cos() – y0 * sin()
yt = y0 * cos() + x0 * sin()

36. 3D Transformation A 3D point (x,y,z) – x,y, and z coordinates
We will still use column vectors to represent points
Homogeneous coordinates of a 3D point
(x,y,z,1)
Transformation will be performed using 4x4 matrix

37. Homogenous coordinates
3D Translation
Very similar to 2D transformation
Translation transformation
Homogenous coordinates

38. Homogenous coordinates
3D Scaling
Very similar to 2D transformation
Scaling transformation
Homogenous coordinates

39. 3D Rotation 3D rotation is done around a rotation axis
Fundamental rotations – rotate about x, y, or z axes
Counter-clockwise rotation is referred to as positive rotation (when you look down negative axis)

40. 3D Rotation wrt z-axis Rotation about z – similar to 2D rotation y + x

41. 3D Rotation wrt y-axis Rotation about y: z -> y, y -> x, x->z

42. 3D Rotation wrt x-axis
Rotation about x (z -> x, y -> z, x->y) x z y
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, y x z

43. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, 43

44. Inverse of 3D Transformations
Invert the transformation
In general, X= AX’->x’=A-1X
T(-tx,-ty,-tz)
T(tx,ty,tz)

45. 3D Rotation about Arbitrary Axes
Rotate p about the by the angle
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

46. 3D Rotation about Arbitrary Axes
Translate so that rotation axis passes through the origin
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

47. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

48. 3D Rotation about Arbitrary Axes
Rotation by about y-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

49. 3D Rotation about Arbitrary Axes
Rotation by about z-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

50. 3D Rotation about Arbitrary Axes
Rotation by about y-axis
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters,

51. 3D Rotation about Arbitrary Axes
Translate the object back to original point

52. 3D Shearing

53. 3D Reflection

54. 3D Reflection Transformation

55. Affine Transformation

56. 3D Coordinate Transformation
Transform object description from to p

57. 3D Coordinate Transformation
Transform object description from to p

58. 3D Coordinate Transformation
Transform object description from to p

59. 3D Coordinate Transformation
Transform object description from to y x z

60. Composite 3D Transformation
Similarly, we can easily extend composite transformation from 2D to 3D

61. Composite 3D Transformation

62. References

63. References