Conventions

1
Conventions

• Vector quantities denoted as v or
• We will use column format vectors
• Each vector is defined with respect to a set of
  basis vectors (which define a co-ordinate
  system).
• Basis vectors not necessarily mutually
  orthogonal.

2
Row vs. Column Formats

• Both formats, though appearing equivalent, are in
  fact fundamentally different
• be wary of different formats used in textbooks

row format
column format
transposed
3
Vectors Points

• Although vectors and points are often used
  inter-changeably in graphics texts, it is
  important to distinguish between them.
• vectors represent directions
• points represent positions
• Both are meaningless without reference to a
  coordinate system
• vectors require a set of basis vectors
• points require an origin and a vector space

both vectors equal
4
Vector Addition Subtraction

• Addition of vectors follows the parallelogram law
  in 2D and the parallelepiped law in higher
  dimensions
• Subtraction

5
Vector Multiplication by a Scalar

• Each vector has an associated length
• Multiplication by a scalar scales the vectors
  length appropriately (but does not affect
  direction)

6
Linear Combinations

• The linear combination of a set of vectors is the
  sum of scalar multiples of those vectors
• Fixing vectors vi yields an infinite number of u
  depending on the scalars ai.
• The set u is called the span of the vectors vi
• The vectors vi are term basis vectors for the
  space.
• If none of the vi can be created as a linear
  combination of the others, the vectors vi are
  said to be linearly independent.
• All linear combinations contain the zero vector.

7
Linear Combinations

• Linear combinations of 1 vector an infinite
  line

8
Linear Combinations

• Linear combinations of 2 vectors a plane

9
Linear Combinations

• The linear combination of 3 vectors a 3D
  volume.
• The 3D Cartesian coordinate system employs the
  well-known 3D co-ordinate basis x, y and z

The vector v here is a linear combination of the
basis vectors x, y and z
10
Vector Magnitude

• The magnitude or norm of a vector of dimension n
  is given by the standard Euclidean distance
  metric
• For example
• Vectors of length 1 (unit vectors) are often
  termed normal vectors.

11
Normal Vectors

• When we wish to describe direction we use
  normalised vectors.
• We often need to normalise a vector

12
Dot Product

• Dot product (inner product) is defined as
• Note
• Therefore we can redefine magnitude in terms of
  the dot-product operator
• Dot product operator is commutative and
  associative.

13
Dot Product

• If one of the vectors is normal, the dot product
  defines the projection of the other onto it
  (perpendicularly)
• In this example, a is positive and b is negative.
• Note that if both vectors are pointing in same
  direction, the dot-product is positive.

q
14
Dot Product

• If both vectors are normal, the dot product
  defines the cosine of the angle between the
  vectors

In general
15
Dot Product

• Note that is q 90 then the dot product 0,
  i.e. the projection of one onto the other has
  zero length ? vectors are orthogonal.
• Also, if q gt 90 then the dot product is negative.
• Example

16
Cross Product

• Used for defining orientation and constructing
  co-ordinate axes.
• Cross product defined as
• The result is a vector, perpendicular to the
  plane defined by u and v

17
Cross Product
Right Handed Coordinate System
18
Cross Product

• Cross product is anti-commutative
• It is not associative
• Direction of resulting vector defined by operand
  order

R.H.S.
19
Normals Polygons

• Polygons are (usually) planar regions bounded by
  n edges connecting n1 points or vertices.
• For lighting and viewing calculations we need to
  define the normal to a polygon
• The normal distinguishes the front-face from the
  back-face of the polygon.

v1
e3
e1
v3
e2
v2
20
Normals Polygons

• First determine the 2 edge vectors from the
  vertices
• The polygon normal is given by

u2
v1
u1
v3
v2
21
Homogeneous Co-ordinates

• Basis of the homogeneous co-ordinate system is
  the set of n basis vectors and the origin
  position
• All points and vectors are therefore compactly
  represented using their ordinates

22
Homogeneous Co-ordinates

• Vectors have no positional information and are
  represented using ao 0 whereas points are
  represented with ao 1
• Examples

Points
Associated vectors
23
Hierarchical Transformations

• For geometries with an implicit hierarchy we wish
  to associate local frames with sub-objects in the
  assembly.
• Parent-child frames are related via a
  transformation.
• Transformation linkage is described by a tree
• Each node has its own local co-ordinate system.

24
Hierarchical Transformations
R
R
R
T
Hierarchical transformation allow independent
control over sub-parts of an assembly
25
translate base
rotate joint1
rotate joint2
complex hierarchical transformation
26
OpenGL Implementation
glMatrixMode(GL_MODELVIEW) glLoadIdentity() glTr
anslatef(bx, by, bz) create_base() glTranslat
ef(0, j1y, 0) glRotatef(joint1_orientation)
create_joint1() glTranslatef(0, uay, 0)
create_upperarm() glTranslatef(0,
j2y) glRotatef(joint2_orientation)
create_joint2() glTranslatef(0, lay, 0)
create_lowerarm() glTranslatef(0, py,
0) glRotatef(pointer_orientation)
create_pointer()
27
Hierarchical Transformations

• The previous example had simple one-to-one
  parent-child linkages.
• In general there may be many child frames derived
  from a single parent frame.
• we need some mechanism to remember the parent
  frame and return to it when creating new
  children.
• OpenGL provide a matrix stack for just this
  purpose
• glPushMatrix() saves the CTM
• glPopMatrix() returns to the last saved CTM

28
Hierarchical Transformations
Each finger is a child of the parent (wrist) ?
independent control over the orientation of the
fingers relative to the wrist
29
Hierarchical Transformations
30
glMatrixMode(GL_MODELVIEW) glLoadIdentity() glTr
anslatef(bx, by, bz) create_base() glTranslat
ef(0, jy, 0) glRotatef(joint1_orientation)
create_joint1() glTranslatef(0, ay, 0)
create_upperarm() glTranslatef(0,
wy) glRotatef(wrist_orientation)
create_wrist() glPushMatrix() // save frame
glTranslatef(-xf, fy0, 0) glRotatef(lowerfinge
r1_orientation) glTranslatef(0, fy1, 0)
create_lowerfinger1() glTranslatef(0, fy2,
0) glRotatef(upperfinger1_orientation)
create_fingerjoint1() glTranslatef(0, fy3,
0) create_upperfinger1() glPopMatrix() //
restore frame glPushMatrix() // do finger
2... glPopMatrix() glPushMatrix() // do
finger 3... glPopMatrix()
Finger1
31
Project2 part 1

• Hierarchical Transformations
• Demonstrate the use of hierarchical coordinate
  transforms in computer graphics. Create a small
  OpenGL program which implements a hierarchical
  transformation. Examples are
• (1) A small planetary system
• Sun at the center rotates about origin
• planet1 revolves about the sun in an orbit of
  radius d1 and rotates about it's centre. Orbit
  lies on a plane tilted at an angle theta1
• planet2 revolves about the sun in an orbit of
  radius d2 and rotates about it's centre. Orbit
  lies on a plane tilted at an angle theta2
• satelite revolves about planet 2 at an orbit of
  radius d3 and rotates about it's centre.

32

• (2) A robotic claw
• wrist is a fixed vertically cylinder connected to
  origin
• palm (or knuckle) is a fixed horizontal cylinder
  connected to wrist
• finger_bone1a is a free cylinder and is connected
  to palm and swings about axis of the palm
• finger_bone1b is connected to finger_bone1a and
  swings about axis at end of the finger_bone1a
• finger_bone2a is a free cylinder and is connected
  to palm and swings about axis of the palm
• finger_bone2b is connected to finger_bone2a and
  swings about axis at end of the finger_bone2a
• or any decent example which demonstrates
  hierarchical transforms with a hierarchy of at
  least two branches and a 3 levels deep