# Modelling with Implicit Surfaces that Interpolate

- "Modelling with Implicit Surfaces that Interpolate
- Greg Turk and James F. O'Brien
- ACM Transactions on Graphics
- Vol. 21, No. 4, October 2002

Full Paper (PDF, 1.4 Mbytes).

## Abstract

We introduce new techniques for modelling with interpolating implicit
surfaces. This form of implicit surface was first used for problems of
surface reconstruction and shape transformation, but the emphasis of our
work is on model creation. These implicit surfaces are described by
specifying locations in 3D through which the surface should pass, and also
identifying locations that are interior or exterior to the surface. A 3D
implicit function is created from these constraints using a variational
scattered data interpolation approach, and the iso-surface of this function
describes a surface. Like other implicit surface descriptions, these
surfaces can be used for CSG and interference detection, may be
interactively manipulated, are readily approximated by polygonal tilings,
and are easy to ray trace. A key strength for model creation is that
interpolating implicit surfaces allow the direct specification of both the
location of points on the surface and the surface normals. These are two
important manipulation techniques that are difficult to achieve using other
implicit surface representations such as sums of spherical or ellipsoidal
Gaussian functions ("blobbies"). We show that these properties make this
form of implicit surface particularly attractive for interactive sculpting
using the particle sampling technique introduced by Witkin and Heckbert.
Our formulation also yields a simple method for converting a polygonal model
to a smooth implicit model, as well as a new way to form blends between
objects.

## Click on images below for larger pictures.

Surfaces from constraints based on tetrahedron (left) and branching
structure (right).

Polygonal mesh (left) and resulting implicit bunny (right).

Polygonal fist (left) and resulting implicit surface (right).

Particle-based surface (left) and similar surface after normal
constraint changed (right).

Blended tori.

Ray traced implicits using sphere tracing.

Boolean implicits, also rendered with ray tracing.

Go to
Greg Turk's Home Page.