There are many different representations of geometries. We can divide them into two large categories: explicit and implicit, and another way to categorize geometry representation is on whether they represent surfaces or volumes.

Implicit Representations Vs Explicit Representation

Implicit representations means that points aren’t known directly, but satisfy some relationship. implicit surfaces make some tasks hard (like finding points on a surface), but some other tasks easy (for example: finding if a point is inside )

On the other hand, all points are given directly in an explicit representation. For example. we can say all points on sphere are

Surface Vs Volume Representations

Surface representation is currently more prevalent in computer graphics Polygon mesh, in particular triangle mesh, is the most widely used 3D format because of its properties and efficiency in rendering.

However, using a volume representation also has its advantages. Volumes is a better source of truth since all volumes can be convert to surfaces, but only watertight surfaces can be converted to volumes. 1

Surface and volume representation can be converted between each other using various algorithms. These conversions work in both directions:

  1. volume to surface: isosurface extraction methods, such as marching cubes or dual contouring, can generate a surface representation from volumetric data
  2. surface to volume: techniques like voxelization can transform surface models into volumetric representations

Representations

Explicit

Surfaces

Volumes

Implicit

Surfaces

Volumes

Others

Footnotes

  1. Claire Blackshaw: Surface vs Volume Formats in Tools