# ppt - Reza Ali

Software and s/w Development

Dec 14, 2013 (4 years and 4 months ago)

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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 &

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

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 &

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

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)