Download CSCI480/582 Lecture 8 Chap.2.1 Principles of Key

CSCI480/582 Lecture 11 Chap.2.4
Orientation Along Path and Deformation
Feb, 16, 2009
Outline


Orientation along a path
Free-form deformation
Representing orientations by Euler Angles
Euler angles
 ,  ,
Gimbal Lock example
using the z-x-z convention:
Representing orientations by Quarternion
A general Quaternion
q  [s, x, y, z ]
T
A Quaternion that can represent a rotation in 3D space
q  [s, x, y, z ] with s  x  y  z  1
T
2
2
2
2
The 4 elements can be further decomposed to
q  [cos( / 2), sin(  / 2) x0 , sin(  / 2) y0 , sin(  / 2) z0 ]T
where
[ x0 , y0 , z0 ]T is a unit vecto r which th e rotation is around,
and  is the rotation angle
[x0,y0,z0]

Rotation Interpolation using Qurternion
Given the begin rotation q1 and ending rotation q2, the
interpolation function that gives the orientation in between
is called spherical linear interpolation (slerp)
q1 sin( 1  u )  q2 sin u
slerp (q1 , q2 , u ) 
sin(  )
with
cos( )  q1  q2
3D Object Deformation

Interpolation problems in 3-D parameter space:



In contrast to


Temporal: Given the beginning and ending shapes of
an object, get the deformation function that produce
the shapes in between the two boundary shapes
Spatial: Given the control points, interpolate a 3D
volume inside the control points.
curve interpolation in 1-D parameter space,
 or surface interpolation in 2-D parameter space
The same basis functions still apply to object
deformation with increased amount of
integrations.
Free-form Deformation using Bezier Basis


Free-form refers to a grid of control points that
are free to move in 3D space.
If Bezier basis function is chosen, then

Geometry interpolation from control points to
object volume
n
Bezier Basis : Bi , n ( s )   (1  s ) n i s i
i
Given a grid of control points {Pi , j , k } with i  [0, l ], j  [0, m], and k  [0, n], any points
inside the volume of the object can be interpolat ed in the 3 - D parameter bounding box
of [0, l, 0, m, 0, n] as
l
m
n
i 0
j 0
k 0
p ( s, t , u )   Bi ,l ( s ) B j , m (t ) Bk , n (u )Pi , j , k
with
cos( )  q1  q2

Temporal change is then defined by changing
the control points directly