Plane Equation - Curves and Surfaces
Plane is a polygonal surface that bisects its environment in two halves. One is termed to as forward and another as backward half of some plane. Currently the question is that half is forward and that backward, since both are relative terms. Thus to get rid of this dilemma, we utilize the mathematical representation of planes, that is, concepts as equations of planes, normal to a plane and so on, that we have already studied. Now we know that both forward and backward halves are associative terms but with respect to what? Yes, it's the plane itself in respect of that we can say any point in the surrounding environment is in back or front of the plane. Hence we see any point on the plane must satisfy the equation of a plane to be zero, that is: Ax + By + Cz + D=0. This equation implies any point (x,y,z) will only lie upon the plane whether it satisfies the equation to be zero any point (x,y,z) will lie upon the front of the plane whether it satisfies the equation to be greater than zero and one of point (x,y,z) will lie upon the back of the plane whether it satisfies the equation to be less than zero.
Here (x, y, z) is any point on the plane, and the coefficients A, B, C and D are constants explaining the spatial properties of the plane? This idea of space partitioning is used commonly in method of BSP that is Binary Space Partitioning, trees generation a polygon representation scheme, fairly similar to Octrees.
The significance of plane equations is that they assist in producing display of any 3-D object, but for such we have to to process the input data representation for the object by several procedures that may include the follow steps of processing.
- To transform the modeling and world-coordinate explanations to viewing coordinates,
- To devise coordinates,
- To know visible surfaces,
- To apply surface-rendering processes