Introduction
An implicit surface is defined as the set of points in space that satisfy
an equation 
.
Differently from parametric surfaces, implicit surfaces do not have a natural
coordinate system defined on them, and traditional parametric texture mapping cannot
be used.
The use of solid texture
[1]
leaves little room for controlling texture
placement since there is a fixed correspondence between points on the
surface and points on the texture. A technique based on projecting a bidimensional
texture onto the surface
[2]
may associate the same texture attribute to all points
that intersect a ray, which is not desirable.
We describe an effective method to apply a bidimensional texture onto implicit
surfaces with a good degree of mapping control.
The Method
We establish a correspondence between points on the implicit surface
and texture attributes with the help of a dynamic particle system.
The implicit surface's gradient vector field,
,
is interpreted as a force field and used to govern the particle system.
The motion of a particle in this system is given by the differential equation
where 
is a viscosity constant
[3,4].
Points on the implicit surface are treated as particles that
move in the direction of a support object,
where the texture is defined beforehand.
The texture attribute for each point at the surface are "read" at the
intersection of the corresponding particle trajectory with the support surface.
Texture Mapping via Simulation
We start the simulation with particles placed at the vertices of
a simplicial approximation of the implicit surface
[5].
Next, particle trajectories
are generated by numeric integration of the motion equation.
At each integration step, we test whether a particle
trajectory has crossed the support object.
If so, we obtain
the texture coordinate for the corresponding point on the implicit surface
from the intersection point on the support object.
Once the simulation is over,
linear interpolation of texture coordinates on each face
completes the texturing of the implicit surface.
Control
An important characteristic of a particle system is its dynamic behavior,
which depends on internal forces (mass, position, velocity) and on external
forces (user supplied forces and the medium).
In our method, the internal force is the gradient vector field of
the implicit function. User control of particles trajectories, and hence of
texture mapping, can be achieved with external forces, such as attractors and
repulsors introduced in the system.
References
1. G. Wyvill, C. McPheeters, B. Wyvill, "Solid texture of soft objects", IEEE CG&A 7 (1987) 20-26.
2. A. Barr, "Decals", in: State-of-the-Art in Image Synthesis, SIGGRAPH Course Notes (1983).
3. L. H. de Figueiredo, J. Gomes, D. Terzopoulos, L. Velho, "Physically Based Methods for Polygonization of Implicit Surfaces", Proceedings of Graphics Interface (1992) 250-257. [pdf]
4. L. H. de Figueiredo, J. Gomes, "Sampling implicit objects with physically-based particle systems", Computers & Graphics 20 (1996) 365-375. [pdf]
5.
L. Velho,
"Adaptive polygonization made simple",
Anais do VIII SIBGRAPI (1995) 111-118.
[pdf]