SM3107
CG Fundamentals for Animators
From object to world coordinate
SM3107 CG Fundamentals for Animators
Last week…
Vertices of a triangle
Object-to-world transform
World-to-camera transform
Projection
Viewport transform
2D image
SM3107 CG Fundamentals for Animators
This week…
Vertices of a triangle
Object-to-world transform
World-to-camera transform
Projection
Viewport transform
2D image
SM3107 CG Fundamentals for Animators
From object to world coordinate
Each object has its own coordinate system.
Each object has its own center and 3 axes.
We call it “object coordinate system”.
Object coordinate is useful for defining the relation
between an object and its components.
For example, a robot’s head is always 5 units from its pelvis,
no matter where the robot is in the virtual world.
The virtual world has a “world coordinate system”.
There is a shared origin and 3 axes.
We put objects in the virtual world by using object-toworld transformation.
Basically translate, rotate and scale.
SM3107 CG Fundamentals for Animators
From object to world
Example
translateX = 1
translateY = 0.5
translateZ = -4
SM3107 CG Fundamentals for Animators
Math calculation
Translate: how to calculate (x’,y’) from (x,y)?
Y
Y
(x’,y’)
(x,y)
X
X
SM3107 CG Fundamentals for Animators
x’ = x + translateX
y’ = y + translateY
Math calculation
Scale
(x’,y’)
(x,y)
X
X
SM3107 CG Fundamentals for Animators
x’ = x * scaleX
y’ = y * scaleY
Math calculation
Rotate (about z-axis)
NOTE: in these equations,
rotation follows the “left-hand”
rule.
(x’,y’)
(x,y)
X
X
x’ = x*cosθ - y*sinθ
y’ = x*sinθ + y*cosθ
NOTE: in these equations, scale and rotate are
always referring to the “world” origin.
SM3107 CG Fundamentals for Animators
A sequence of transformation
Note that the order of a sequence of transformation is important.
Different order gives you different result.
Rotate Y first, then rotate X
Rotate X first, then rotate Y
A sequence of transformation:
another example
Translate and then rotate
Y
Y
X
Y
X
X
Rotate and then translate
Y
Y
X
Y
X
X
Some remarks:
In most 3D software (e.g. Maya), there is some fundamental
differences from our formula:
In our formula, all rotation and scaling are about the “world”
origin.
In most 3D software, usually all rotation and scale are
relative to the object’s local axes. Therefore, whether the
object is translated-before-rotated or rotated-beforetranslated doesn’t matter.
In most 3D software, the order of rotations (about the 3 axes)
can be specified.
The default is rotate about X first, then rotates about Y,
and then rotates about Z.
SM3107 CG Fundamentals for Animators
Transformation in Maya
SM3107 CG Fundamentals for Animators
Transformation in Maya
SM3107 CG Fundamentals for Animators
A sequence of transformation:
math calculation
Example: first translate (translateX=2, translateY=0),
and then rotate 45 degree.
First translate:
x’ = x + 2
y’ = y + 0
Then rotate:
x’’ = x’ * cos45 – y’ * sin45
y’’ = x’ * sin45 + y’ * cos45
Combine together:
x’’ = (x+2)* cos45 – (y+0) * sin45
y’’ = (x+2)* sin45 + (y+0) * cos45
SM3107 CG Fundamentals for Animators
A sequence of transformation:
math calculation
Example: first rotate 45 degree, and then translate
(translateX=2, translateY=0).
First rotate:
x’ = x * cos45 – y * sin45
y’ = x * sin45 + y * cos45
Then translate
x’’ = x’ + 2
y’’ = y’ + 0
Combine together:
x’’ = (x * cos45 – y * sin45) + 2
y’’ = (x * sin45 + y * cos45) + 0
SM3107 CG Fundamentals for Animators
Math calculation
Extend the mathematics to 3D, we have:
Translate:
x’ = x + translateX
y’ = y + translateY
z’ = z + translateZ
Scale:
x’ = x * scaleX
y’ = y * scaleY
z’ = z * scaleZ
Rotate about x-axis:
x’ = x
y’ = y*cosθ - z*sinθ
z’ = y*sinθ + z*cosθ
Rotate about y-axis:
x’ = x*cosθ + z*sinθ
y’ = y
z’ = -x*sinθ + z*cosθ
SM3107 CG Fundamentals for Animators
Rotate about z-axis:
x’ = x*cosθ - y*sinθ
y’ = x*sinθ + y*cosθ
z’ = z