Geometric Objects and Transformation

1
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 gt 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)Qav
• aR (1-a)Q
• For 0ltalt1 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? Cant 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
• va1v1 a2v2 .anvn
• They appear to have the similar representations
• Pb1 b2 b3 va1 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 0P 0 and 1P P then we can write
  
• va1v1 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
• px 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 w0, 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 w0 for
  vectors and w1 for points
• For perspective we need a perspective division

13
Representing a Mesh
e2
v5

• 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
• PPd

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-bodytransformations
26
Scaling

• Expand or contract along each axis (fixed point
  of origin)

xsxx ysyx zszx
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 1T
• px y z 1T
• ddx dy dz 0T
• Hence p p d or
• xxdx
• yydy
• zzdz

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
• pTp 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
• pRz(q)p

xx cos q y sin q y x sin q y cos q zz
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

• xsxx
• ysyx
• zszx
• pSp

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
  MABCD 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
• pT pTCTBTAT

38
Rotation about a Fixed Point and about the Z axis
f
?
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 y0, and
the line segment is foreshortened to
?
?
?
?
?
?
?
Rotate clockwise about the y-axis, so
?
?
Final transformation matrix