## Iso Surfaces

‹ Direct Vs Indirect Methods | ● | Marching Hexahedra ›

Analogue to iso contour lines in 2D, we talk about **iso surfaces** in 3D.

An iso contour line is the set of points on a plane with the same iso value.

An iso surface is the set of points of a volume with the same iso value of the scalar function $ f(x,y,z) = const $.

- The iso surface of a C1-continuous scalar function is C1-continuous, also.
- The gradient vector of the scalar function is orthogonal to the iso surface:
**normal vector**. - The gradient vector is the vector of of partial derivatives of the scalar function
- Iso surfaces
**do not intersect**each other (onion shell model). - Iso surfaces are
**water tight**(if the scalar function is C1-continuous).

Display of the implicit definition of the iso surface with $f(x,y,z) = c_{iso}$ by extracting the surface geometry explicitly:

- Per discrete cell a single surface element is extracted.
- Sum of extracted surface elements forms the iso surface.

For hexahedra: **Marching Hexahedra Algorithm**

- [Lorensen & Cline ‘87]

For tetrahedra: **Marching Tetrahedra Algorithm**

**Iso surface extraction**:

- Decomposition of the volume into tetrahedra
- Decomposition of a single voxel into tetrahedra
- 5 asymmetrical tets
- 6 symmetrical tets

- Decomposition of a single voxel into tetrahedra
*Marching Tetrahedra Algorithm*for all decomposed tetrahedra

As the surface geometry needs to be extracted explicitly from the scalar volume before the volume can be displayed, iso surface extraction is an indirect volume visualization method.