Today’s Outline
• Transformations in matrix form
- scalar multiplication
- rotation / change of basis
- translation / displacement
- other types of transformations
Scalar Multiplication
• in geometric form:
• in matrix form:
• what is S?~v
~w
~w = s~v, s 2 R
[~w] = S [~v] , S 2 R2⇥2
S =
s 00 s
�
General Transforms
What happens if we put in generic values for the matrix M?
[~w] = M [~v] ,
M =
a bc d
�
x0 = ax+ by
y0 = cx+ dy
Affine Transforms
• Matrix multiplication gives as a linear transform:
• For translation, we need an affine transform:
x0 = x+ u
y0 = y + v
for M =
a bc d
�
for
uv
�2 R2
Homogeneous Coordinates
• A brilliantly convenient “trick” is to add an extra coordinate to our vectors and matrices:
• The result multiplication gives us an affine transform:
2
4x0
y0
1
3
5 =
2
4a b uc d v0 0 1
3
5
2
4xy1
3
5
x0 = ax+ by + u
y0 = cx+ dy + v
Hold on a second!
ax
ay
bx
by
~v ~v = v1ax + v2ay =
2
4v1v2vh
3
5
If we want [~v]B ) [~v]A ,
then what is vh?
Reference Frame Transformations
• Our final homogeneous transformation matrix to go between references frames A and B looks like this:
• Where free vectors and bound (position) vectors are encoded differently:
ATB =
ARB ~rB/A
0 1
�
h~rP/B
i=
2
4r1r21
3
5[~v] =
2
4v1v20
3
5