1. Geometric Objects and Transformation
Chapter 4
Geometric Objects and Transformation

2. Lines Consider all points of the form P(a)=P0 + a d
Set of all points that pass through P0 in the direction of the vector d

3. Parametric Form This form is known as the parametric form of the line
More robust and general than other forms
Extends to curves and surfaces
Two-dimensional forms
Explicit: y = mx +h
Implicit: ax + by +c =0
Parametric:
x(a) = ax0 + (1-a)x1
y(a) = ay0 + (1-a)y1

4. Rays and Line Segments
If a >= 0, then P(a) is the ray leaving P0 in the direction d
If we use two points to define v, then
P( a) = Q + a (R-Q)=Q+av
=aR + (1-a)Q
For 0<=a<=1 we get all the
points on the line segment
joining R and Q

5. Dot and Cross: Products

6. Three-Dimensional Primitives
Hollow objects
Objects can be specified by vertices
Simple and flat polygons (triangles)
Constructive Solid Geometry (CSG)
3D curves
3D surfaces
Volumetric Objects

7. Constructive Solid Geometry

8. Representation
Until now we have been able to work with geometric entities without using any frame of reference, such a coordinate system
Need a frame of reference to relate points and objects to our physical world.
For example, where is a point? Can’t answer without a reference system
World coordinates
Camera coordinates

9. Confusing Points and Vectors
Consider the point and the vector
P = P0 + b1v1+ b2v2 +….+bnvn
v=a1v1+ a2v2 +….+anvn
They appear to have the similar representations
P=[b1 b2 b3] v=[a1 a2 a3]
which confuse the point with the vector
A vector has no position v p v
can place anywhere
fixed

10. A Single Representation
If we define 0•P = 0 and 1•P =P then we can write
v=a1v1+ a2v2 +a3v3 = [a1 a2 a3 0 ] [v1 v2 v3 P0] T
P = P0 + b1v1+ b2v2 +b3v3= [b1 b2 b3 1 ] [v1 v2 v3 P0] T
Thus we obtain the four-dimensional homogeneous coordinate representation
v = [a1 a2 a3 0 ] T
P = [b1 b2 b3 1 ] T

11. Homogeneous Coordinates
The general form of four dimensional homogeneous coordinates is
p=[x y x w] T
We return to a three dimensional point (for w0) by
xx/w
yy/w
zz/w
If w=0, the representation is that of a vector
Note that homogeneous coordinates replaces points in three dimensions by lines through the origin in four dimensions

12. Homogeneous Coordinates and Computer Graphics
Homogeneous coordinates are key to all computer graphics systems
All standard transformations (rotation, translation, scaling) can be implemented by matrix multiplications with 4 x 4 matrices
Hardware pipeline works with 4 dimensional representations
For orthographic viewing, we can maintain w=0 for vectors and w=1 for points
For perspective we need a perspective division

13. Representing a Mesh Consider a mesh There are 8 nodes and 12 edgesv5
Consider a mesh
There are 8 nodes and 12 edges
5 interior polygons
6 interior (shared) edges
Each vertex has a location vi = (xi yi zi) v6 e3 e9 e8 v8 v4 e1
e11
e10 v7 e4 e7 v1
e12 v2 v3 e6 e5

14. Inward and Outward Facing Polygons
The order {v0, v3, v2, v1} and {v1, v0, v3, v2} are equivalent in that the same polygon will be rendered by OpenGL but the order {{v0, v1, v2, v3} is different
The first two describe outwardly
facing polygons
Use the right-hand rule =
counter-clockwise encirclement
of outward-pointing normal
OpenGL treats inward and
outward facing polygons differently

15. Geometry versus Topology
Generally it is a good idea to look for data structures that separate the geometry from the topology
Geometry: locations of the vertices
Topology: organization of the vertices and edges
Example: a polygon is an ordered list of vertices with an edge connecting successive pairs of vertices and the last to the first
Topology holds even if geometry changes

16. Bilinear Interpolation
Assuming a linear variation, then we can make use of the
same interpolation coefficients in coordinates for the
interpolation of other attributes.

17. Scan-line Interpolation
A polygon is filled only when it is displayed
It is filled scan line by scan line
Can be used for other associated attributes with each vertex

18. General Transformations
A transformation maps points to other points and/or vectors to other vectors

19. Linear Function (Transformation)
Transformation matrix for homogeneous coordinate system:

20. Affine Transformations – 1/2
Line preserving
Characteristic of many physically important transformations
Rigid body transformations: rotation, translation
Scaling, shear
Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

21. Affine Transformations – 2/2
Every linear transformation (if the corresponding matrix is nonsingular) is equivalent to a change in frames
However, an affine transformation has only 12 degrees of freedom because 4 of the elements in the matrix are fixed and are a subset of all possible 4 x 4 linear transformations

22. Translation Move (translate, displace) a point to a new location
Displacement determined by a vector d
Three degrees of freedom
P’=P+d P´ d P

23. Rotation (2D) – 1/2 Consider rotation about the origin by q degrees
radius stays the same, angle increases by q
x´ = r cos (f + q) = r cosf cosq - r sinf sinq
y´ = r sin (f + q) = r cosf sinq + r sinf cosq
x´ =x cos q –y sin q
y´ = x sin q + y cos q
x = r cos f
y = r sin f

24. Rotation (2D) – 2/2 Using the matrix form: There is a fixed point
Could be extended to 3D
Positive direction of rotation is counterclockwise
2D rotation is equivalent to 3D rotation about the z-axis

25. (Non-)Rigid-Body Transformation
Translation and rotation are rigid-body transformation
Non-rigid-body transformations

26. Scaling x´=sxx y´=syx z´=szx Uniform and non-uniform scaling
Expand or contract along each axis (fixed point of origin)
x´=sxx
y´=syx
z´=szx
Uniform and non-uniform
scaling

27. Reflection corresponds to negative scale factors sx = -1 sy = 1
original
sx = -1 sy = -1
sx = 1 sy = -1

28. Transformation in Homogeneous Coordinates
With a frame, each affine transformation is represented by a 44 matrix of the form:

29. Translation
Using the homogeneous coordinate representation in some frame
p=[ x y z 1]T
p´=[x´ y´ z´ 1]T
d=[dx dy dz 0]T
Hence p´= p + d or
x´=x+dx
y´=y+dy
z´=z+dz
note that this expression is in
four dimensions and expresses
that point = vector + point

30. Translation Matrix
We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p´=Tp where This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

31. Rotation about the Z axis
Rotation about z axis in three dimensions leaves all points with the same z
Equivalent to rotation in two dimensions in planes of constant z
or in homogeneous coordinates
p’=Rz(q)p
x´=x cos q –y sin q
y´= x sin q + y cos q
z´=z

32. Rotation Matrix

33. Rotation about X and Y axes
Same argument as for rotation about z axis
For rotation about x axis, x is unchanged
For rotation about y axis, y is unchanged

34. Scaling Matrix
x´=sxx
y´=syx
z´=szx
p´=Sp

35. Inverses
Although we could compute inverse matrices by general formulas, we can use simple geometric observations
Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)
Rotation: R -1(q) = R(-q)
Holds for any rotation matrix
Note that since cos(-q) = cos(q) and sin(-q)=-sin(q)
R -1(q) = R T(q)
Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

36. Concatenation
We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices
Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p
The difficult part is how to form a desired transformation from the specifications in the application

37. Order of Transformations
Note that matrix on the right is the first applied
Mathematically, the following are equivalent
p´ = ABCp = A(B(Cp))
Note many references use column matrices to present points. In terms of column matrices
p´T = pTCTBTAT

38. Rotation about a Fixed Point and about the Z axis

39. General Rotation about the Origin
A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes
R(q) = Rx() Ry() Rz() y
, ,  are called the Euler angles v q
Note that rotations do not commute
We can use rotations in another order but
with different angles x z

40. Rotation about an Arbitrary Axis – 1/2
Final rotation matrix:    
Normalize u:  

41. Rotation about an Arbitrary Axis – 2/2
Rotate the line segment to the plane of y=0, and the line segment is foreshortened to:       
Rotate clockwise about the y-axis, so:  
Final transformation matrix: