ppt - Reza Ali

Constructing Bezier Curves on
the Surface of a Sphere

By Reza Ali

Fundamentals of Spatial Computing

UCSB MAT 594CM

Spring 2009

Presentation Outline

•
Purpose/Goal

•
Spherical Coordinates & Properties

•
SLERP (
s
pherical
l
inear int
erp
olation)

•
OpenGL Bezier Curves

•
Particle Systems

•
Voronoi &
Shaders

Purpose/Goal

•
Partition the surface of
a sphere using the
voronoi algorithm

•
Allow the points that
define the voronoi to be
an interactive magnetic
particle system

•
Real Time manipulate of
particles


Spherical Voronoi

Spherical Coordinates


•
Sphere Equations:





•
Cartesian Equivalents:



Sphere Properties

•
Equation of a sphere

–
Cartesian Coordinates:


–
Spherical Coordinates:


•
2 values define a sphere

–
Center & Radius

•
Geodesic: curve that is
the shortest distance
between two points




Sphere Properties

•
Antipodal: points are
located directly
opposite of each other
on a sphere (no
geodesic)

•
Great Circle: the
intersection of a plane
containing the origin
and the unit sphere

Spherical Linear Interpolation

•
A method of
interpolating between
two points on a sphere

•
Estimation:



•
Not good enough this
will traverse the
geodesic at non
-
constant rate

Spherical Linear Interpolation

•
Z=
slerp(x,y,α
) (constant
rate)





•
Watch for the case
Ω
=180
°

(antipodal case)

•
Related to Quaternion



Bezier Curves

•
Develop a set of
parametric cubic
equations to represent
curves and surfaces
using only a small set of
control points (4)


Bezier Curves & OpenGL

•
OpenGL evaluator
functions allow you to
use polynomial
equations to produce
vertices, normals,
textures coordinates,
and colors

•
Evaluator functions
define a Bezier Curve
(also the basis for
NURBS)

Bezier Curves & OpenGL

•
Function: glMap1f()

•
Data
Glfloat

ctlpts[4][3]

•
glMap1f(

•
target type

•
Lower
t

range

•
Higher
t

range

•
Stride

•
Number of points

•
Reference to points)

•
glEnable(GL_MAP1_VERTEX_3)

•
glMapGrid1d(20,0,1)

•
glEvaMesh1(GL_LINE,0,20)

•
t
=(0,1/20,2/20,…1)

•
20 = number of points to evaluate




Particle Systems


•
Developing 3D Particle
System

•
The particles will
distribute themselves
along the surface of a
sphere

•
Electromagnetic
repulsion

•
Voronoi Pattern
Creation based of
particle system

Voronoi &
Shaders

•
Create a voronoi curves
that will define a sphere
and use these curves as
points where light
escapes like
-
>

•
Allow user to interactive
with system via GUI
(GLV)

•
Real Time, maybe?

Inspiration

References

•
Principles of Computer Graphics (
Shalini

Govil
-
Pai
)

•
3D Computer Graphics (Samuel R. Buss)

•
Wikipedia
:
Spherical Coordinates

•
Wikipedia
:
Sphere

•
Google Image Search

•
Efficient Reconstruction of Functions on the Sphere from Scattered Data
(
Keiner
,
Kunis
, Potts)

•
Vimeo
:
Mass_Ins

•
Spherical Centroid Voronoi Tesselation

•
Distributing Points on a Sphere


•
Voronoi Diagram on the sphere

•
Voronoi Diagram of Curves Objects

•
Voronoi diagrams on the sphere (Na, Lee, Cheong)