Geometric Transforms

1
Geometric Transforms

• Changing coordinate systems

2
The plan

• Describe coordinate systems and transforms
• Introduce homogeneous coordinates
• Introduce numerical integration for animation
• Later
• camera transforms
• physics
• more on vectors and matrices

3
World Coordinates

• Objects exist in the world without need for
  coordinate systems
• But, describing their positions is easier with a
  frame of reference
• World or object-space or global coordinates

4
Screen vs World Coordinates

• Some objects only exist on the screen
• mouse pointer, radar display, score, UI
  elements
• Screen coordinates typically 2D Cartesian
• x, y
• Objects in the world also appear on the screen
  need to project world coordinates to screen
  coordinates
• camera transform

5
Coordinate Transforms

• A Cartesian coordinate system has three things
• an origin reference point measurements are
  taken from
• axes canonical directions
• scale meaning of units of distance

6
Types of transforms

• Translation
• moving in a fixed position
• Rotation
• changing orientation
• Scale
• changing size
• All can be viewed either as affecting the model
  or as affecting the coordinate system

7
Changing coordinate systems

• Translation changing the origin of the
  coordinate system

(5,13)
(2,12)
(0,0)
(0,0)
8
Changing coordinate systems

• Rotation changing the axes of the coordinate
  system

(7,1)
(5,5)
(0,0)
(0,0)
9
Changing coordinate systems

• Scaling changing the units of the coordinate
  system

(8,8)
(4,4)
(0,0)
(0,0)
10
Matrix Transforms

• For "simplicity"s sake, we want to represent our
  transforms as matrix operations
• Why is this simple?
• single type of operation for all transforms
• IMPORTANT collection of transforms can be
  expressed as a single matrix
• efficiency

11
Scaling

• If we express our 2D coordinates as (x,y),
  scaling by a factor a multiplies each coordinate
  so (ax, ay)
• Can write as a matrix multiplication

a 0
0 b
x
y
ax
by

12
Scaling Syntax

• Matrix.CreateScale( s )
• scalar argument s describes amount of scaling
• alternatively,
• Matrix.CreateScale( v )
• vector argument v describes scale amount in x y z
  

13
Rotation

• Similarly, rotating a point by an angle ?
• (x,y) ? (x cos ? - y sin ?, x sin ? y cos ?)

x cos ? y sin ?
x sin ? y cos ?
cos ? -sin ?
sin ? cos ?
x
y

14
Rotation Syntax

• Matrix.CreateRotationZ(angle)
• Note, specify the axis about which the rotation
  occurs
• also have CreateRotationX, CreateRotationY
• argument is angle of rotation, in radians

15
Translation

• Translating a point by a vector involves a vector
  addition
• (x,y) (s,t) (xs, yt)

16
Translation

• Translating a point by a vector involves a vector
  addition
• (x,y) (s,t) (xs, yt)
• Problem cannot compose multiplications and
  additions into one operation

17
Coordinate Transforms

• Scaling p Sp
• Rotation p Rp
• Translation p t p
• Problem Translation is treated differently.

18
Homogeneous Coordinates

• Add an extra coordinate w
• 2D point written (x,y,w)
• Cartesian coordinates of this point are (x/w,
  y/w)
• w0 represents points at infinity and should be
  avoided if representing location
• Aside w0 also used to represent vector, e.g.,
  normal
• in practice, usually have w1

19
Translation now

• (x,y,1) (s,t,0) (xs,yt, 1)

1 0 s
0 1 t
0 0 1
x
y
1
xs
yt
1

20
Composing Translations
1 0 s1
0 1 t1
0 0 1
1 0 s2
0 1 t2
0 0 1
1 0 s1s2
0 1 t1t2
0 0 1

21
Translation Syntax

• Matrix.CreateTranslation( v )
• vector v is the translation vector
• Also,
• Matrix.CreateTranslation( x, y, z)
• x, y, z are scalars (floats)

22
Composing Transformations

• Rotation, scale, and translate are all matrix
  multiplications
• We can compose an arbitrary sequence of
  transformations into one matrix
• C T0T1Tn-1Tn
• Remember order matters
• In XNA, later transforms are added on the right

23
Kinds of Transformations

• Rigid transformations
• composed of only rotations and translations
• preserve distances between points
• Affine transformations
• include scales
• preserve parallelism of lines
• distances and angles might change

24
More on Rotations

• R matrix we wrote allows rotation about origin
• Usually, origin and point of rotation do not
  coincide

25
More on Rotations

• Want to rotate about arbitrary point
• How to achieve this?

26
More on Rotations

• Want to rotate about arbitrary point
• How to achieve this?
• Composite transform
• first translate to origin
• next, rotate desired amount
• finally, translate back
• C T-1RT

27
More on Order of Operations

• AB ! BA in general
• But, AB BA if
• A is translation, B is translation
• A is scale, B is scale
• A is rotation, B is rotation
• A is rotation, B is scale (that preserves aspect
  ratio)
• All in two dimensions, note

Like transforms commute
28
ISROT

• ISROT a mnemonic for the order of
  transformations
• Identity gets you started
• Scale change the size of your model
• Rotation rotate in place
• Orbit rotate about an external point
• first translate
• then rotate (about origin)
• Translation move to final position

29
3D transforms

• Translation same as in 2D
• use 4D homogeneous coordinates x,y,z,w
• Scaling same as 2D
• For rotation, need to specify axis
• unique in 2D, but different possibilities
  available in 3D
• can represent any 3D rotation by a composition of
  rotations about axes

30
Transforming objects

• So far, talked about transforming points
• How about objects?
• Lines get transformed sensibly with transforms on
  endpoints
• Polygons same, mutatis mutandis

31
Hierarchical Models

• Many times, structures (and models) will be built
  as a hierarchy

Body
Arm
Other limbs...
Hand
Thumb
other fingers...
32
Transforms on Hierarchies

• Transforms applied to nodes higher in the
  hierarchy are also applied to lower nodes
• parent transforms propagate to children
• How to achieve this?
• Simple multiply your transform with your
  parent's transform
• May want a stack for hierarchy management
• push transform on descending
• pop when current node finished

33
Transforms on Hierarchies

• pseudocode for computing final transform
• applyWorld myWorld parentWorld
• applyWorld is the final transform
• myWorld is the local transform
• parentWorld is the parent node's transform
• The power of matrix transforms! Complex
  hierarchical models can be expressed simply.

34
Moving objects

• Want to get things to move
• Can adjust transform parameters
• translation amount
• rotation, orbit angles
• Just incorporate changes into Update()
• Need to do this in a disciplined way

35
Speed ? position

• If we know
• where an object started
• how fast it is moving (and the direction)
• and how much time has passed
• we can figure out where it is now
• x(t dt) x(t) v(t)dt
• Numerical integration!

36
Euler integration

• Assumes constant speed within a timestep
• still gives an answer if speed changes, but
  answer might be wrong
• x(t dt) x(t) v(t)dt
• "Explicit integration"
• Most commonly used in graphics/game applications

37
Implementation

• Time available in Update as gameTime
• object.position object.velocity
  gameTime.ElapsedGameTime.TotalSeconds
• if speed expressed in distance units per second
• assumes that position and velocity are vectors
• may have some way of updating velocity
• physics, player control, AI, ...
• Similar approach for angle updates (need angular
  velocity)

38
Recap

• How to use world transformations
• scale, rotate, translate
• Use of homogeneous coordinates for translation
• Order of operations
• ISROT
• Animation and Euler integration

39
Future lectures

• matrix and vector math
• camera transformations
• illumination and texture
• particle systems
• linear physics