CSE 681 Brief Review: Matrices, Vectors, and Transformations

1
CSE 681Brief ReviewMatrices, Vectors, and
Transformations
2
Overview

• Mathematical toolbox
• Review basic matrix and vector algebra
• Line representation
• Affine transformations
• Composing transformations
• Axial rotations Euler angles
• Rotation about an arbitrary axis

3
What is a Matrix?

• A matrix is a set of elements, organized into
  rows and columns

rows
columns
4
Basic Operations

• Addition, Subtraction, Multiplication

Just add elements
Just subtract elements
Multiply each row by each column
5
Multiplication

• Is AB BA? Maybe, but maybe not!
• Multiplication is NOT commutative!

6
Inverse of a Matrix

• Identity matrix AI A
• Inverse MatrixAA-1 I
• Useful equivalence(ABC)-1 C-1B-1A-1

7
Determinant of a 2 x 2 Matrix

• Useful for inversion
• If det(A) 0
• Then A has no inverse
• Easy for a 2 x 2 matrix

8
Determinant of a 3 x 3 Matrix
Sum from left to right Subtract from right to
left Note n! terms
9
Inverse of a 3 x 3 Matrix

• Append the identity matrix to A
• Apply Gaussian Elimination so that A turns into
  the identity
• Transform the identity matrix as you go
• The identity has become the inverse!

10
Vector Operations

• Vector 1 x N matrix
• Looks like a point
• Vector has its tail at the origin head at the
  point
• Magnitude (length) and direction

y
v
x
11
Vectors Dot Product
Think of the dot product as a matrix
multiplication
The magnitude is the dot product of a vector with
itself
The dot product is also related to the angle
between the two vectors
12
Dot Product

• A . B B . A
• If A and B are unit vectors, A . B cos(q)
• Remember, A . B A B cos(q)
• If A is a unit vector, A . B is length of B
  projected onto A

13
Vectors Cross Product

• A x B C, where vector C is perpendicular to A
  and B
• The magnitude of C is proportional to the sine of
  the angle between A and B
• The direction of C depends on the handedness of
  the coordinate system.

14
Cross Product

• A x B -(B x A)
• A x B is orthogonal to plane defined by A and B
• If A and B are unit vectors, A x B sin(?)

15
Line Equation

• We need a way to represent a ray, i.e. an
  infinite line
• How would you define a line mathematically?

16
Line Equation

• Two points determine a line, say between
  endpoints P0(x, y, z) and P1(x, y, z)
• A line equation of the form f (x, y, z) is
  cumbersome since it depends upon on all three
  variables simultaneously
• More convenient to define the line with a single
  variable
• Will become clearer when we implement our ray
  tracer
• How do we do this?

17
Parametric Line Equation

• Define a single variable t that parameterizes the
  line simultaneously for all dimensions x, y, and
  z
• P(t) P0(x, y, z) (1 t) P1(x, y, z) t
• where 0 t 1.0

t 1
t 0
P1
P0
18
Parametric Line Equation

• Re-write P(t)
• P(t) P0(x, y, z) t D(x, y, z)
• where D(x, y, z) P1(x, y, z) P0(x, y, z)
• Furthermore,
• x(t) P0(x) t Dx
• y(t) P0(y) t Dy
• z(t) P0(z) t Dz

19
ReviewTransformations
20
Transformations - Modeling
21
Modeling
How do I place the spheres ?
22
Viewing
Orthographic
Perspective
23
Modeling Transformations

• Transform objects/points

Transform coordinate system
24
Affine Transformations

• A transformation that preserves
• Collinearity
• All points lying on a line initially still lie on
  a line after the transformation
• Parallel lines stay parallel
• Ratios of distances
• The midpoint of a line segment remains the
  midpoint after transformation
• Does not necessarily preserve lengths or angles

25
Affine Transformations

• Lets talk about transforming points
• Transform P (x, y, z) to Q (x, y, z)
• General form of an Affine transformation
• x m11x m12 y m13z m14
• y m21 x m22 y m23z m24
• z m31 x m32 y m33z m34

26
Translation

• Translation by (tx, ty, tz)
• x x tx
• y y ty
• z z tz

(tx, ty, tz)
27
Scaling

• Scaling by (sx, sy, sz)
• x sx x
• y sy y
• z sz z

28
Rotation

• Rotation counter-clockwise by angle ? around the
  z-axis
• x x cos(?) y sin(?)
• y x sin(?) y cos(?)
• z z

Derivation x r cos(?) y r sin(?) x r
cos(? ?) r cos(?) cos(?) r sin(?) sin(?)
x cos(?) y sin(?) y r sin(? ?) r cos(?)
sin(?) r sin(?) cos(?) x sin(?) y cos(?)
29
Rotation Around the x-axis

• Rotation counter-clockwise by angle ? around the
  x-axis
• x x
• y y cos(?) z sin(?)
• z y sin(?) z cos(?)

z
?
y
x
30
Rotation Around the y-axis

• Rotation counter-clockwise by angle ? around the
  y-axis
• y y
• z z cos(?) x sin(?)
• x z sin(?) x cos(?)
• Or
• x x cos(?) z sin(?)
• y y
• z x sin(?) z cos(?)

x
?
z
y
31
Matrix Multiplication

• Scaling by (sx, sy, sz)
• x sx x
• y sy y
• z sz z
• Rotation counter-clockwise by angle ? around the
  z-axis
• x x cos(?) y sin(?)
• y x sin(?) y cos(?)
• z z
• Translation by (tx, ty, tz)
• x x tx
• y y ty
• z z tz

32
Homogeneous Coordinates

• Represent P by (x, y, z, 1) and Q by (x, y, z,
  1)
• Translation by (tx, ty, tz)
• x x tx
• y y ty
• z z tz
• Scaling by (sx, sy, sz)
• x sx x
• y sy y
• z sz z

33
Rotation Matrices

• Rotation counter-clockwise by angle ? around the
  z-axis
• x x cos(?) y sin(?)
• y x sin(?) y cos(?)
• z z
• Rotation counter-clockwise by angle ? around the
  x-axis
• x x
• y y cos(?) z sin(?)
• z y sin(?) z cos(?)
• Rotation counter-clockwise by angle ? around the
  y-axis
• x x cos(?) z sin(?)
• y y
• z x sin(?) z cos(?)

34
Affine Transformation Matrix

• Affine transformation
• x m11 x m12 y m13 z m14
• y m21 x m22 y m23 z m24
• z m31 x m32 y m33 z m34
• Transformation Matrix

35
Shearing

• Shear along the x-axis
• x x hy
• y y
• z z

36
Reflection

• Reflection across the x-axis
• x x
• y (-1) y
• z z

Reflection is a special case of scaling!
37
Compose Transformations
38
Compose Transformation Matrices

• Scale by (2, 1, 1)

• Rotate by 30 counter-clockwise around the
  z-axis

• Translate by (0, -50, 0).

Translate
Rotate
Scale
39
Compose Transformation Matrices

• Scale by (2, 1, 1)

• Translate by (20, 5, 0)

• Rotate by 30 counter-clockwise around the
  z-axis

• Translate by (0, -50, 0).

40
Order Matters!

• Transformations are not necessarily commutative!

• Translate by (20, 0, 0).
• Rotate by 30?.

• Rotate by 30?.
• Translate by (20, 0, 0).

41
Order of Transformation Matrices

• Apply transformation matrices from right to left.

• Translate by (20, 0, 0).
• Rotate by 30?.

• Rotate by 30?.
• Translate by (20, 0, 0).

42
Elementary Transformations

• Theorem Every affine transformation can be
  decomposed into elementary operations
• (For example a rotation, then a scaling and
  finally a translation)
• Elementary transformations
• Translation
• Rotation
• Scaling
• Shear

43
Eulers Theorem

• Every rotation around the origin can be
  decomposed into a rotation around the x-axis
  followed by a rotation around the y-axis followed
  by a rotation around the z-axis

44
Fixed Point Transformations

• Sometimes it is convenient to transform to the
  origin first, then perform your operations
• Take the hatchet in this example

45
Fixed Point Transforms
46
Fixed Point Scaling

• Assuming that (xp, yp) is our fixed point (2D
  example)
• 1. Translate (xp, yp) to the origin
• 2. Scale by Sx and Sy
• 3. Translate the origin back to (xp, yp)
• Expressing (x, y) in terms of (x, y) we
  get

47
Rotation About An Arbitrary Axis
y
u
Q
Rotate P to Q by angle b about an arbitrary axis
(vector) u?
P
x
z
48
Rotation About An Arbitrary Axis

• Make axis u coincident with axis x
• Two rotations Rz(-f) Ry(q)
• Apply transformation to P as well
• Rotate P around x-axis Rx(b)
• Undo earlier rotations Ry(-q) Rz(f)
• R Ry(-q) Rz(f) Rx(b) Rz(-f) Ry(q) Eulers
  Theorem!

What OpenGL command can construct this matrix?
glRotate