3Dvolume

3Dvolume is a term used in several disciplines to refer to a three-dimensional volume or to a data representation of such a volume. In mathematics and geometry, a 3D volume is the measure of the space enclosed by a solid in three-dimensional Euclidean space. In computer graphics and scientific visualization, a 3D volume commonly denotes a volumetric dataset stored as a regular grid of samples (voxels) that encode properties like density, color, or scalar fields.

Mathematically, the volume of a region can be computed by a triple integral over the region or by summing the volumes of simple shapes that partition the region. In simple shapes, formulas are well known (for example, a cube of side a has volume a^3, a sphere of radius r has volume 4/3 pi r^3). The concept extends to more complex regions via calculus and measure theory.

In volume rendering and simulation, data are stored in voxel grids. Each voxel stores values; rendering often uses ray marching or illumination models to produce images without explicit polygon surfaces. Volumetric representations enable effects such as fog, smoke, and translucent materials, and are used in medical imaging, geology, and fluid dynamics. Data can be regular grids or sparse representations such as octrees or voxel-based libraries like OpenVDB.

In practical applications, computing the 3D volume is essential for material estimation, collision detection bounding volumes, and in manufacturing and additive manufacturing to estimate material usage and print time.