3D Transformation: Introduction

3D-Transformation

Manipulation, viewing, and construction of three-dimensional graphic images require the use of three-dimensional geometric and coordinate transformations. In geometric transformation, the coordinate system is fixed, and the desired transformation of the object is done with respect to the coordinate system. In coordinate transformation, the object is fixed and the desired transformation of the object is done on the coordinate system itself. These transformations are formed by composing the basic transformations of translation, scaling, and rotation. Each of these transformations can be represented as a matrix transformation. This permits more complex transformations to be built up by use of matrix multiplication or concatenation. We can construct the complex objects/pictures, by instant transformations. In order to represent all these transformations, we need to use homogeneous coordinates.

Hence, if P(x,y,z) be any point in 3-D space, then in Homogeneous Coordinate System, we add a fourth-coordinate to a point. That is instead of (x,y,z), each point can be represented by a Quadruple (x,y,z,H) such that H≠0; with the condition that x1/H1=x2/H2; y1/H1=y2/H2; z1/H1=z2/H2. For two points (x₁, y₁, z₁, H₁) = (x₂, y₂, z₂, H₂) where H₁ ≠ 0, H₂ ≠ 0. Thus any point (x,y,z) in Cartesian system can be represented by a four-dimensional vector as (x,y,z,1) in HCS. Similarly, if (x,y,z,H) be any point in HCS then (x/H,y/H,z/H) be the corresponding point in Cartesian system. Thus, a point in three-dimensional space (x,y,z) can be represented by a four-dimensional point as: (x’,y’,z’,1)=(x,y,z,1).[T], where [T] is some transformation matrix and (x’,y’z’,1) is a new coordinate of a given point (x,y,z,1), after the transformation.