Lect9 transformation3d

Graphics

3D Geometric
Transformation

Sudipta Mondal

sudipta.hit@gmail.com

Contents
BCET

n Translation
n Scaling
n Rotation
n Rotations about an arbitrary axis in space
n Other Transformations
n Coordinate Transformations


Transformation in 3D
BCET

n Transformation Matrix
éA D G Jù é ù
êB E H Kú ê 3´ 3 3 ´ 1ú
ê ú Þ ê ú
êC F I Lú ê ú
ê ú ê ú
ë0 0 0 Sû ë 1´ 3 1 ´ 1û

3´3 : Scaling, Reflection, Shearing, Rotation

3´1 : Translation

1´1 : Uniform global Scaling

1´3 : Homogeneous representation


3D Translation
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n Translation of a Point

x' = x + t x , y' = y + t y , z' = z + t z

y

é x 'ù é1 0 0 tx ùé xù
ê y 'ú ê 0 1 0 ty úê yú
ê ú=ê úê ú
ê z ' ú ê0 0 1 tz úê z ú
ê ú ê úê ú
x ë 1 û ë0 0 0 1 ûë1 û
z


3D Scaling
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n Uniform Scaling

x' = x × s x , y' = y × s y , z' = z × sz

y

é x 'ù é s x 0 0 0ù é x ù
ê y 'ú ê 0 sy 0 0ú ê y ú
ê ú=ê úê ú
ê z 'ú ê 0 0 sz 0ú ê z ú
ê ú ê úê ú
x
ë1û ë 0 0 0 1û ë1 û
z


Relative Scaling
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n Scaling with a Selected Fixed Position

y y y y

z x z x z x z x

Original position Translate Scaling Inverse Translate

éx'ù é1 0 0 x f ùésx 0 0 0ùé1 0 0 - x f ùéxù
ê y'ú ê0 1 0 y f úê 0 sy 0 0úê0 1 0 - y f úê yú
T (x f , y f , z f ) × S(sx , sy , sz ) ×T(-x f ,-y f ,-z f ) = ê ú = ê úê úê úê ú
ê z'ú ê0 0 1 z f úê 0 0 sz 0úê0 0 1 - z f úê z ú
ê ú ê úê úê úê ú
ë 1 û ë0 0 0 1 ûë 0 0 0 1ûë0 0 0 1 ûë1û


3D Rotation
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n Coordinate-Axes Rotations
n X-axis rotation
n Y-axis rotation
n Z-axis rotation

n General 3D Rotations
n Rotation about an axis that is parallel to one of the
coordinate axes
n Rotation about an arbitrary axis


Coordinate-Axes Rotations
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n Z-Axis Rotation n X-Axis Rotation n Y-Axis Rotation

é x'ù écos q - sin q 0 0ù é x ù é x'ù é1 0 0 0ù é x ù é x'ù é cos q 0 sin q 0ù é x ù
ê y'ú ê sin q cos q 0 0ú ê y ú ê y'ú ê0 cos q - sin q 0ú ê y ú ê y'ú ê 0 1 0 0ú ê y ú
ê ú=ê úê ú ê ú=ê úê ú ê ú=ê úê ú
ê z 'ú ê 0 0 1 0ú ê z ú ê z 'ú ê0 sin q cos q 0ú ê z ú ê z 'ú ê- sin q 0 cos q 0ú ê z ú
ê ú ê úê ú ê ú ê úê ú ê ú ê úê ú
ë1û ë 0 0 0 1û ë 1 û ë 1 û ë0 0 0 1û ë 1 û ë1û ë 0 0 0 1û ë 1 û

y y y

x x x
z z z


Order of Rotations
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n Order of Rotation Affects Final Position
n X-axis è Z-axis

n Z-axis è X-axis


General 3D Rotations
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n Rotation about an Axis that is Parallel to One of
the Coordinate Axes
n Translate the object so that the rotation axis
coincides with the parallel coordinate axis
n Perform the specified rotation about that axis
n Translate the object so that the rotation axis is
moved back to its original position


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n Rotation about an Arbitrary Axis
y Basic Idea
T 1. Translate (x1, y1, z1) to the origin
(x2,y2,z2) 2. Rotate (x’2, y’2, z’2) on to the z
R
axis
(x1,y1,z1) 3. Rotate the object around the z-axis
4. Rotate the axis to the original
x R-1
orientation
z 5. Translate the rotation axis to the
T-1
original position

R (q ) = T -1R -1 (a )R -1 (b )R z (q )R y (b )R x (a )T
x y


BCET

n Step 1. Translation
y

(x2,y2,z2) é1 0 0 - x1 ù
ê0 1 0 - y1 ú
T=ê ú
(x1,y1,z1) ê0 0 1 - z1 ú
ê ú
x
ë0 0 0 1 û

z


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n Step 2. Establish [ TR ]ax x axis
y b b
sin a = =
b +c
2 2 d
(0,b,c) c c
(a,b,c) cos a = =
b +c
2 2 d
Projected
Point a a é1 0 0 0ù é1 0 0 0ù
ê0 cos a - sin a 0 ú ê0 c / d -b/d 0ú
x R x (a ) = ê ú=ê ú
ê0 sin a cos a 0 ú ê0 b / d c/d 0ú
z ê ú ê ú
ë0 0 0 1 û ë0 0 0 1û
Rotated
Point


Arbitrary Axis Rotation
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n Step 3. Rotate about y axis by b
y a d
sin b = , cos b =
l l
l 2 = a 2 + b2 + c2 = a2 + d 2
(a,b,c)
d = b2 + c2
Projected l
Point écos b 0 - sin b 0ù éd / l 0 - a / l 0ù
d ê 0 1 0 0ú ê 0 1 0 0ú
b x R y (b ) = ê ú=ê ú
ê sin b 0 cos b 0ú ê a / l 0 d / l 0ú
(a,0,d) ê ú ê ú
ë 0 0 0 1û ë 0 0 0 1û
Rotated
z Point


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n Step 4. Rotate about z axis by the desired
angle q
y

écosq - sin q 0 0ù
ê sin q cosq 0 0ú
R z (q ) = ê ú
ê 0 0 1 0ú
l
x ê ú
q ë 0 0 0 1û

z


BCET

n Step 5. Apply the reverse transformation to
place the axis back in its initial position
y
é1 0 0 - x1 ù é1 0 0 0ù
ê0 1 0 - y1 ú ê0 cos a sin a 0ú
T -1R -1 (a )R -1 ( )= ê
y b
úê ú
l
x
ê0 0 1 - z1 ú ê0 - sin a cos a 0ú
ê úê ú
ë0 0 0 1 û ë0 0 0 1û
é cos b 0 sin b 0ù
l ê 0 1 0 0ú
x ê ú
ê- sin b 0 cos b 0ú
ê ú
ë 0 0 0 1û
z

R (q ) = T -1R -1 (a )R -1 (b )R z (q )R y (b )R x (a )T
x y


Example
BCET

Find the new coordinates of a unit cube 90º-rotated
about an axis defined by its endpoints A(2,1,0) and
B(3,3,1).

A Unit Cube


Example
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n Step1. Translate point A to the origin

y
é1 0 0 - 2ù
B’(1,2,1) ê0 1 0 - 1ú
T=ê ú
ê0 0 1 0ú
A’(0,0,0)
x ê ú
ë0 0 0 1û
z


Example
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n Step 2. Rotate axis A’B’ about the x axis by and
angle a, until it lies on the xz plane.
2 2 2 5
sin a = = =
2 2 + 12 5 5
1 5
cos a = =
y 5 5
Projected point
(0,2,1) B’(1,2,1)
l = 12 + 2 2 + 12 = 6

a l
é1 0 0 0ù
x ê 5 2 5 ú
z ê0 - 0ú
B”(1,0,Ö5) R x (a ) = ê 5 5 ú
ê0 2 5 5
0ú
ê 5 5 ú
ê0
ë 0 0 1ú
û

Example
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n Step 3. Rotate axis A’B’’ about the y axis by and
angle b , until it coincides with the z axis.
1 6
sin b = =
y 6 6
5 30
cos b = =
6 6

l é 30 6 ù
b x ê 0 - 0ú
(0,0,Ö6)
ê 6 6 ú
B”(1,0, Ö 5)
R y (b ) = ê 0 1 0 0ú
ê 6 30
z 0 0ú
ê 6 6 ú
ê
ë 0 0 0 1ú
û


Example
BCET

n Step 4. Rotate the cube 90° about the z axis
é0 - 1 0 0 ù
ê1 0 0 0ú
R z (90°) = ê ú
ê0 0 1 0 ú
ê ú
ë0 0 0 1 û
n Finally, the concatenated rotation matrix about the
arbitrary axis AB becomes,

R (q ) = T -1R -1 (a )R -1 (b )R z (90°)R y (b )R x (a )T
x y


Example
BCET

é1 0 0 0ù é 30 6 ù
é1 0 0 2ù ê 5 2 5 úê 0 0ú é 0 - 1 0 0ù
ê0 ú ê0 0ú ê 6 6 úê
1 0 1ú 5 5 0ú ê 1 0 0 0ú
R (q ) = ê ê úê 0 1 0 ú
ê0 0 1 0ú ê 2 5 5 6 30 ê0 0 1 0ú
ê ú 0 - 0ú ê - 0 0ú ê ú
ë0 0 0 1û ê 5 5 úê 6 6 ú 0 0
ë 0 1û
ê0
ë 0 0 1ú ê 0
ûë 0 0 1ú
û
é 30 6 ù é1 0 0 0ù
ê 0 - 0ú ê 5 2 5 ú é1 0 0 - 2ù
6 6 - 0 ú ê0
ê
0 1 0
ú ê0
0ú ê 5 5 1 0 - 1ú
ê úê ú
ê 6 30 2 5 5 ê0 0 1 0ú
0 0 ú ê0 0ú ê ú
ê 6 6 úê 5 5 ú 0 0 0 1û
ê ë
ë 0 0 0 1 ú ê0
ûë 0 0 1úû
é 0.166 - 0.075 0.983 1.742 ù
ê 0.742 0.667 0.075 - 1.151ú
=ê ú
ê - 0.650 0.741 0.167 0.560 ú
ê ú
ë 0 0 0 1 û


Example
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n Multiplying R(θ) by the point matrix of the original cube

[P¢] =R(q)×[P]
é 0.166 - 0.075 0.983 1.742 ù é0 0 1 1 0 0 1 1ù
ê 0.742 0.667 0.075 - 1.151ú ê1 1 1 1 0 0 0 0ú
[P¢] = ê úê ú
ê- 0.650 0.741 0.167 0.560 ú ê1 0 0 1 1 0 0 1ú
ê úê ú
ë 0 0 0 1 û ë1 1 1 1 1 1 1 1û
é 2.650 1.667 1.834 2.816 2.725 1.742 1.909 2.891 ù
ê- 0.558 - 0.484 0.258 0.184 - 1.225 - 1.151 - 0.409 - 0.483ú
=ê ú
ê 1.467 1.301 0.650 0.817 0.726 0.560 - 0.091 0.076 ú
ê ú
ë 1 1 1 1 1 1 1 1 û


Other Transformations
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n Reflection Relative to the xy Plane
é x' ù é1 0 0 0ù é x ù
y y ê y 'ú ê 0 1 0 0ú ê y ú
ê ú=ê úê ú
z ê z' ú ê0 0 - 1 0ú ê z ú
ê ú ê úê ú
z x x ë 1 û ë0 0 0 1û ë1 û

n Z-axis Shear
é x 'ù é1 0 a 0ù é x ù
ê y 'ú ê 0 1 b 0ú ê y ú
ê ú=ê úê ú
ê z ' ú ê0 0 1 0ú ê z ú
ê ú ê úê ú
ë 1 û ë0 0 0 1ûë1 û


Other Transformations
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n Z-axis Shear
The effect of this transformation matrix is to alter x and y-coordinate
values by an amount that is proportional to the z value, while leaving the
z coordinate unchanged.
q Boundaries of planes that are perpendicular to the z axis are thus shifted
by an amount proportional to z.
q An example of the effect of this shearing matrix on a unit cube is
shown in Fig., for shearing values a = b = 1. Shearing matrices for
the x axis and y axis are defined similarly.

é x 'ù é1 0 a 0ù é x ù
ê y 'ú ê 0 1 b 0ú ê y ú
ê ú=ê úê ú
ê z ' ú ê0 0 1 0ú ê z ú
ê ú ê úê ú
ë 1 û ë0 0 0 1ûë1 û


Coordinate Transformations
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n Multiple Coordinate System
n Local (modeling) coordinate system
n World coordinate scene

Local Coordinate System

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n Multiple Coordinate System
n Local (modeling) coordinate system
n World coordinate scene

Word Coordinate System

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n Example – Simulation of Tractor movement
n As tractor moves, tractor coordinate system and
front-wheel coordinate system move in world
coordinate system
n Front wheels rotate in wheel coordinate system


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n Transformation of an Object Description from
One Coordinate System to Another
n Transformation Matrix
n Bring the two coordinates systems into alignment
1. Translation
y’
y
u'y x’
y u'x u'y
(x0,y0,z0) u'x
x
z’ u'z z (0,0,0)
u'z
(0,0,0) x
z
T(- x0 , - y0 , - z0 )

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2. Rotation & Scaling

y’ y y y
u'y x’
u'x
x x (0,0,0) x
z (0,0,0) z (0,0,0)
z
z’ u'z

é u ¢1
x u¢2x u¢3x 0ù
êu ¢ u ¢y 2 u ¢y 3 0ú
R = ê y1 ú
ê u ¢1
z u ¢2
z u ¢3
z 0ú
ê ú
ë 0 0 0 1û


Lect9 transformation3d

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