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