The Planar/Hyper-Planar Rotated Polar Coordinate System and Its Mathematical Solid Vector Addition, Multiplication, Division, Dot Product and Cross Product Operations

Abstract

In geometry, the curvilinear coordinate systems have curved coordinate lines. They are used to localize physical or virtual quantities in image processing computer science, space science, earth science, cartography, quantum mechanics, relativity, and other engineering sciences. Their Mathematical operations are useful for defining the motion of these quantities and objects under the influence of central forces, and solving equation of curved boundary conditions. The typical ones are the correct 2D Polar mathematical operations; and the wrong 3D Spherical mathematical operations. The problem statement is that there is no available definition of curvilinear one that is 3D and correct, except the Solid Vector Subtraction operation which was recently proposed and used to develop 3D spatial filters of field of vectors for geometrical edge detection in 3D image processing science. In this research, this Solid Vector Subtraction is extended by proposing its 3D and correct definitions of: (1) Rotated Polar coordinate system; (2) 2D, 3D, and hyper dimensional space; and (3) Other whole complementary set of Solid Vector Addition, Multiplication, Division, Dot Product, and Cross Product operations. Because their justification has a long derivation, it is for future work. These Solid Vector operations are 3D and correct.