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# 3D Geometric Transformation - TAU - geometric transformation rotation calculator 3D Geometric
Transformation
(Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker)
3D Coordinate Systems
? Right-handed coordinate system:
z
y
x
? Left-handed coordinate system:
y
z
x
Reminder: Vector Product
UxV
V

U
VxU
U ?V = n^ U V sin
x^ y^ z^ u yvz - uzv y
U ?V = ux uy uz =

uzvx - uxvz
vx v y vz uxv y -u yvx
3D Point
? A 3D point P is represented in
homogeneous coordinates by a
4-dim. vector:
P=

x

y

z
1

? Note, that
p=

x x

y
y
z

1

z

3D Transformations
? In homogeneous coordinates, 3D
transformations are represented by
4x4 matrices:

a b c tx
d e f ty
g h i tz

0 0 0 1
? A point transformation is
performed:

x' a b c tx x
y'
d e f ty
z' =

1
g h i tz
y

0 0 0 1

z
1

3D Translation
? P in translated to P' by:

1 0 0 tx x x' x + tx
0 1 0 ty
0 0 1 tz
y

0 0 0 1

y'
z =

1
y+ty
z' =

1

z + tz
1

Or T P = P'
z
x
y
? Inverse translation: T -1P' = P

#### Pdf INFO

Title: Microsoft PowerPoint - geom3d
Author: Sagit