Basic Transformation methods
- Aim
- To implement set of basic transformations on polygon i.e. Translation, Rotation, Scaling, Reflection and Shearing
- Pre-requisite:
- Knowledge of matrix fundamentals and basic transformations on polygon i.e. translation, rotation, scaling, reflection and shearing.
- Description
- Transformations allows us to uniformly alter the entire picture. The geometric transformations considered here - translation, scaling and rotation are expressed in terms of matrix multiplication. Homogeneous coordinates are considered to uniformly treat the translations
Scaling Transformation
A 2D point can be scaled by multiplication of the coordinate values (x,y) by scaling factors Sx and Sy to produce the transformed coordinates (x',y').
Matrix format: [X'] = [S]x[X] Where [X']- Transformed Matrix; [X]- Point or Object Matrix; and [S]- Scaling Matrix [x1' x2' x3'] [sx 0 0] [x1 x2 x3] [y1' y2' y3'] = [0 sy 0] x [y1 y2 y3] [1 1 1 ] [0 0 1] [1 1 1]
Translation Transformation
A 2D point can be translated by adding the coordinate values (x,y) by Translation distances tx and ty to produce the transformed coordinates (x',y').
Matrix format: [X'] = [T]x[X] Where [X']- Transformed Matrix; [X]- Point or Object Matrix; and [T]- Translation Matrix [x1' x2' x3'] [1 0 tx] [x1 x2 x3] [y1' y2' y3'] = [0 1 ty] x [y1 y2 y3] [1 1 1 ] [0 0 1] [1 1 1]
Rotaion Transformation
A 2D point can be rotated by re-positioning it along a circular path in the xy plane. We specify the rotation angle and the position of the rotation point about which the object is to be rotated. Multiplication of the coordinate values (x,y) by rotation matrix produce the transformed coordinates (x',y').
Matrix format: [X'] = [R]x[X] Where [X']- Transformed Matrix; [X]- Point or Object Matrix; and [R]- Rotation Matrix [x1' x2' x3'] [cos(Θ) -sin(Θ) 0] [x1 x2 x3] [y1' y2' y3'] = [sin(Θ) cos(Θ) 0] x [y1 y2 y3] [1 1 1 ] [0 0 1] [1 1 1]
