two-dimensional and three-dimensional cartesian …...figure 7.9 coordinate axes after translation....

Two-dimensional and Three-dimensional Cartesian Coordinate Transformation

Translation

x’ = x - tx

y’ = y - ty

Scaling

Rotation

x’ = x cosθ - y sinθ

y’ = x sinθ + y cosθ

clockwise:

Example for a given angle

Combined

Matrix Notation

R X

RXor

Matrix notation for calculations “stacks” the equations

Multiplication is from rows on the left, down columns on the right, and add

4 8 3 6 9 2

x y z[ ][ ].

is the same as

4x + 8y +3z 6x + 9y + 2z

Number of columns in first must equal rows in second

Multiplication an entire matrix by a constant (scalar) is the same as multiplying each element

a b c m n p[ ] 6a 6b 6c

6m 6n 6p[ ]6 =

Adding to matrices is just adding the corresponding rows (need equal number of rows)

4 8 3 6 9 2[ ] 2 1 2

0 1 0[ ] 6 9 5 6 10 2[ ]+ =

Matrix Notation

R X

RXor

Matrix Notation

becomes:

In 3D we have three axes, hence three rotations, one around each axis

Three Dimensional Transformation, X, Y, Z

-

X’ = RzX

-

-

X’ = Rx Ry Rz X

x’ y’ z’

=[ ]Combining all three rotations