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Chapter 2
Introduction to Implicit Surfaces
Implicit surfaces are defined with scalar functions. A point p ∈ R 3 is assigned a value
called a field value f(p) ∈ R by applying a continuous scalar function f : R 3 → R.
The points which store the value called the iso-value form a surface called an iso-
surface. An iso-surface can be defined as follows:
S = {p ∈ R 3 : f(p) = viso} (2.1)
where S is the iso-surface and viso is the iso-value. If an iso-surface is closed, this
scalar function can also define an implicit volume by simply changing the equality
into the inequality as follows:
V = {p ∈ R 3 : f(p) ≥ viso} (2.2)
This equation divides 3D space into inside and outside using the iso-surface as the
border. For example, a point p where f(p) ≥ viso is inside a volume V .
In practice, an iso-surface is visualized on a screen and recognized as a 3D model,
however the strength of implicit surfaces is that an underlying model is a volume
model and has several advantages of 3D modeling. Fundamental issues of implicit
surfaces are then how to define a continuous scalar function f (Section 2.1 Modeling)
and how to visualize the scalar field (Section 2.2 Rendering).
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