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)
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