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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
Creator: ADOBEPS4.DRV Version 4.11
Producer: Acrobat Distiller 3.02
CreationDate: Wed Jun 16 11:55:04 1999
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