Non-linear co-ordinates and their application in Computer Graphics

There are many applications that are not naturally based on the orthogonal co-ordinate systems, like ultrasound imaging, astronomy, mechanical computations etc. In those cases it is necessary to transform the problem formulation to the orthogonal co-ordinate system, where the problem is solved, transform the solution back and visualize the solution. The polar, spherical or cylindrical co-ordinate systems are natural to many problems and sometime also used for computations. Some properties of those systems are discussed including geometric transformation in the polar, cylindrical and spherical co-ordinate systems. The geometric transformations can be expressed by matrix operations in considered co-ordinate systems. As the memory capacity and hardware graphics accelerator grow fast it can be reasonable to reconsider fundamental functionality of it, as it is limited generally to the vector dot multiplication. Also use of non-linear co-ordinate systems and the influence on design of algorithms with lower algorithm complexity is presented, too.