Introduction to
Computer Graphics
Farhana Bandukwala, PhD
Lecture 9: Objects and
Transformations in 3D
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Lecture 9: Objects and Transformations in 3D Computer Graphics, Study notes of Introduction to Sociology
A lecture note from dr. Farhana bandukwala's introduction to computer graphics course, focusing on geometric objects and transformations in 3d. Topics include curves, surfaces, volumetric objects, coordinate systems, and matrix representations of transformations such as translation, scaling, and rotations. The document also covers composite transformations and changing coordinate systems.
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Introduction toComputer GraphicsFarhana Bandukwala, PhDLecture 9: Objects andTransformations in 3D
Outline
• Geometric objects & operators
(Angel, chapter 4.1)
• Coordinate systems and transformations^ (Angel, chapter 4.2) • Matrix representations
(Angel, chapter 4.3)
• Composite transformations
Coordinate systems
-^ Points in 3-d w/respect to 3 linearly independent vectors(basis or coordinate system)•^ Frame: coordinate system and defined origin
∆^ defines
all 3-d points uniquely• All 3-d points (or vectors) can be represented as a three-tuple in terms of basis vectors:• For example: a
1 = ae 1
2 + ae 2
3 + ae 3
a (^2) e (^1) e
(^3) e
Transformations in 3D
Use homogeneous coordinates, and represent 3Dpoints as four-tuples (x,y,z,W)
dx 0 1 0
d^ y 0 0 1
d^ z 0 0 0
Translation
s^0 0 x^
0 s^0 y^
0 0 sz
Scaling
Composite Transformations • To rotate about a fixed point: 1.^ Translate point to origin,2.^ Rotate object3.^ Reverse previous translationz x
z P x^
y P
z x P y^ x
z P y^ x
z
y P
-px 0 1 0
-p^ y 0 0 1
-p^ z 0 0 0
Translation T(-P) =
0 cosT^
-sinT^0 0 sinT^
cosT^0 0 0
Rotation around xR(T) =
px 0 1 0
p^ y 0 0 1
p^ z 0 0 0
TranslationT(P)=
M = T(P)R(
T)T(-P)
Composite TransformationTo rotate about an arbitrary axis (
A) by^ α:
-^ rotate such that rotation vector (
A) aligned with z-axis
-^ then rotate by desired angle–^ Reverse first rotation^ M = R(-
T) R(-T) R(x^ y
D) R(T) R(zy
T)x
x^
y z
z x Tx y
z x A P y
z x
y Ty
x^
y z
Rotate by
Tx around x axis
Rotate by
Ty around y axis
Rotate by
D
around z axis
Reverse rotationsaround x & y axes Remember:Dot products give anglesbetween 2 vectors
Changing coordinate systemsConsider three linearly independent vectors:^1 –^ v= v
1 e+ v 1112
2 e+ ve^13
3
2 – v= v
1 e+ v 2122
2 e+ ve^23
3
3 – v= v
1 e+ v 3132
2 e+ ve^33
3
- coordinate system as a transformation matrix
(^2) e (^1) e
(^3) e (^2) v^1 v
(^3) v
Suppose b
is the vector coincident with a
but with respect to the 2
a^ same as nd^ basis
b
(^1) v^ = (^2) v (^3) v -1^ a = Vb
v^ v^11
v^13 vv^21
v^23 vv^31
e v 33 (^12) e (^3) e