Download SI460 Graphics RST Transform Cheat-Sheet v1.0

SI460 Graphics
RST Transform Cheat-Sheet
v1.0
1 0 0
� 0 1 0�
0 0 1
Diagonal matrix. All non-zero elements occur along the
diagonal. The “diagonal” are those elements whose row
number equals their column number. Must be square.
1 0
�0 1
0 0
Identity matrix. A diagonal matrix whose diagonal elements all
equal one. Often written as ‘I’.
Transpose. The transpose AT of matrix A has the same
numbers, but the rows are switched with the columns.
For a diagonal matrix, AT=A.
𝑎
A=�
𝑑
Determinant. A value associated with a square matrix.
The determinant of matrix A is written as det(A) or |A|.
det(I) = 1
10 0
�
�
0 5
0
0�
1
�
1 0
�
0 1
𝑎
𝑐
� , AT= �𝑏
𝑓
𝑐
𝑏
𝑒
𝑎 𝑏
A=�
�
𝑐 𝑑
det(A)=ad-bc
𝑎 𝑏 𝑐
B=�𝑑 𝑒 𝑓 �
𝑔 ℎ 𝑖
det(B) = aei+bfg+cdh-ceg-bdi-afh
Using matrix A to the right, det(A)= the area of a parallelogram
described by vectors (a,b) and (c,d).
1
A=�
3
Inverse. The inverse A-1 of A is the matrix that satisfies AA-1=I.
You can calculate the inverse of A if you know the determinant
and cofactor of A. (Note – cofactors are complicated to
calculate. You are not required to memorize their equation.)
2
−2 1
� , A-1= �
�
4
1.5 −.5
A-1=
1
* Acofactor
|𝐴|
QT=Q-1, det(Q)=-1 or 1
QTQ = QQT = I
Orthogonal matrix. A square matrix made up with real entries.
Each row contains an orthonormal vector. (Vector length=1.
x2+y2=1 or x2+y2+z2=1.) Its determinant = -1 or 1. Its transpose
always equals its inverse.
Rotational Matrix. Must be orthogonal.
A matrix times a constant (n):
A matrix times a matrix:
A matrix plus a matrix:
Matrix multiplication is associate:
Matrix multiplication is distributive:
Matrix multiplication is not commutative:
�
𝑎
𝑐
�
�
𝑎
𝑐
𝑛∗�
𝑎
𝑐
𝑒
𝑏
�∗�
𝑔
𝑑
𝑑
𝑒�
𝑓
𝑎
𝑐
𝑏
𝑛𝑎
�= �
𝑑
𝑛𝑐
𝑛𝑏
�
𝑛𝑑
𝑥
𝑎𝑥 + 𝑏𝑦
𝑏
� ∗ �𝑦 � = �
�
𝑐𝑥 + 𝑑𝑦
𝑑
𝑓
𝑎𝑒 + 𝑏𝑔
�= �
ℎ
𝑐𝑒 + 𝑑𝑔
𝑒
𝑏
�+�
𝑔
𝑑
𝑓
𝑎+𝑒
�= �
ℎ
𝑐+𝑔
(AB)C = A(BC)
A(B+C) = AB+AC
AB != BA
𝑎𝑓 + 𝑏ℎ
�
𝑐𝑓 + 𝑑ℎ
𝑏+𝑓
�
𝑑+ℎ
SI460 Graphics
RST Transform Cheat-Sheet
v1.0
Type of Vector
Rotate
2D,
non-homogeneous
(x, y)
Rz(φ) = �
3D,
non-homogeneous
(x, y, z)
Scale
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ
�
𝑠𝑖𝑛φ 𝑐𝑜𝑠φ
1
Rx(φ) = �0
0
S=�
0
0
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ�
𝑠𝑖𝑛φ 𝑐𝑜𝑠φ
𝑐𝑜𝑠φ 0 𝑠𝑖𝑛φ
1
0 �
Ry(φ) = � 0
−𝑠𝑖𝑛φ 0 𝑐𝑜𝑠φ
2D,
homogeneous
(x, y, 1) or (x, y, 0)
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ 0
Rz(φ) = � 𝑠𝑖𝑛φ 𝑐𝑜𝑠φ 0�
0
0
1
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ 0
Rz(φ) = � 𝑠𝑖𝑛φ 𝑐𝑜𝑠φ 0�
0
0
1
1
0
Rx(φ)=�
0
0
3D,
homogeneous
(x, y, z, 1) or
(x, y, z, 0)
0
0
0
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ 0
�
𝑠𝑖𝑛φ 𝑐𝑜𝑠φ 0
0
0
1
𝑐𝑜𝑠φ
0
Ry(φ)=�
−𝑠𝑖𝑛φ
0
𝑆𝑥
0
Angle
(deg)
0
30
45
60
90
cos
0
0�
𝑆𝑧
𝑆𝑥
S=� 0
0
0
𝑆𝑦
0
0
0�
1
sin
1
√3⁄2 ≈ .866
√2⁄2 ≈ .707
.5
0
n/a
0
𝑆𝑦
0
𝑐𝑜𝑠φ −𝑠𝑖𝑛φ 0 0
𝑠𝑖𝑛φ 𝑐𝑜𝑠φ 0 0
Rz(φ)=�
�
0
0
1 0
0
0
0 1
Angle
(rad)
0
π/6
π/4
π/3
π/2
0
�
𝑆𝑦
𝑆𝑥
S=� 0
0
𝑆𝑥
0
S=�
0
0
0 𝑠𝑖𝑛φ 0
1
0
0
�
0 𝑐𝑜𝑠φ 0
0
0
1
Translate
0
.5
√2⁄2 ≈ .707
√3⁄2 ≈ .866
1
0
𝑆𝑦
0
0
0
0
𝑆𝑧
0
n/a
0
0
�
0
1
1 0
T=�0 1
0 0
1
0
T=�
0
0
𝑇𝑥
𝑇𝑦�
1
0 0 𝑇𝑥
1 0 𝑇𝑦
�
0 1 𝑇𝑧
0 0 1