Length, Area and Volume Computation in Homogeneous Coordinates

Many problems solved in computer graphics, computer vision, visualization etc. require fast and robust computation of an area of a triangle or volume of a tetrahedron. These very often used algorithms are well known and robust if vertices coordinates of triangles or tetrahedrons are given in Euclidean coordinates. The homogeneous coordinates are often used for the representation of geometric transformations. They enable us to represent translation, rotation, scaling and projection operations in a unique way and handle them properly. Today’s graphics hardware based on GPU offers very high computational power using pixel shaders and fragment shaders not only for graphical elements processing, but also for general computation using GPU as well. This paper presents simple methods for the area of a triangle and the volume of a tetrahedron computation if vertices are given in homogeneous coordinates without the need to use the division operation for vertices coordinates transformation from the homogeneous coordinates to the Euclidian coordinates. Area or volume computation is transferred to the cross product computation that is fast, simple, robust and can be supported in hardware or implemented on GPU that uses vector operations with homogeneous coordinates natively. The presented formula can be used directly for Euclidean representation just setting w equal to 1.