Isosurface

An isosurface of a scalar field is the set of points in three-dimensional space where the field takes a constant value. Formally, for a scalar function f: R^3 → R and a constant c, the isosurface is the level set { x ∈ R^3 | f(x) = c }. It is the three-dimensional analog of a contour line in two dimensions and represents a surface that separates regions with f < c from those with f > c.

In practice, many data sets come on a discrete grid. The isosurface is extracted by interpolation between samples, producing a polygonal mesh. Common algorithms include marching cubes and marching tetrahedra, which traverse grid cells and generate triangles that approximate the surface. The quality of the isosurface depends on data resolution and the interpolation method; the surface can be closed, have holes, or experience topological changes with varying c.

Normals to the surface can be obtained from the gradient ∇f, and the surface is typically oriented accordingly. Isosurfaces can be smooth or piecewise-linear depending on representation. They are examples of level-set surfaces and can be related to implicit surfaces defined by F(x) = 0, where F is typically f(x) − c.

Applications include visualization of volumetric data in medicine (CT and MRI), computational fluid dynamics, geology, and meteorology, where isosurfaces represent constant density, velocity magnitude, pressure, or energy. In computer graphics, isosurfaces are used for surface rendering and shape analysis; advances also include time-varying isosurfaces and level-set methods for evolving surfaces.