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Transformation Matrix

Transformation Matrix

Jul 02 2018, 5:27 PM PST 2 min

Martrix transformation is an advanced math concept. This article is meant primarily as a reference, rather than a teaching guide. To teach yourself how to perform matrix transformations, we recommend Khan Academy's lessons on Linear Algebra.

Simple multiplication

A transformation matrix can be thought of as containing the three new coordinate axes in its columns, since transforming the coordinate axes results in a vector matching the contents of a column:

a d g
b e h
c f i
×
1
0
0
=
a
b
c
a d g
b e h
c f i
×
0
1
0
=
d
e
f


a d g
b e h
c f i
×
0
0
1
=
g
h
i



Identity

The identity matrix is a transformation matrix that maps every point onto itself (i.e. transforming by it has no effect)

I =

1 0 0
0 1 0
0 0 1



Scaling

S(x, y, z) =

x 0 0
0 y 0
0 0 z



Rotation

X-axis rotation

Rx(θ) =

1 0 0
0 cos θ -sin θ
0 sin θ cos θ

Y-axis rotation

Ry(θ) =

cos θ 0 sin θ
0 1 0
-sin θ 0 cos θ

Z-axis rotation

Rz(θ) =

cos θ -sin θ 0
sin θ cos θ 0
0 0 1
Tags:
  • movement
  • reference