4.a) Explain any two of the 3D geometric transformations.
Answer:
Three-Dimensional Translation
A position P = (x, y, z) in three-dimensional space is translated to a location P¢=(x¢,y¢,z¢) by adding translation distances tx, ty, and tz to the Cartesian coordinates of P.
x’=x+tx
y’=y+ty
z’=z+tz
Three-dimensional translation operations can be represented in matrix form. The coordinate positions, P and P¢ , are represented in homogeneous coordinates with four-element column matrices, and the translation operator T is a 4 × 4 matrix:
Figure: Moving a coordinate position with translation vector T = (tx, ty, tz )
An inverse of a three-dimensional translation matrix is obtained by negating the translation distances tx, ty, and tz
Figure: Shifting the position of a three-dimensional object using translation vector T
Three-Dimensional Scaling
The matrix expression for the three-dimensional scaling transformation of a position P =(x, y, z) is given by
The three-dimensional scaling transformation for a point position can be represented as
P¢=S.P where scaling parameters sx, sy, and sz are assigned any positive values.
Explicit expressions for the scaling transformation relative to the origin are
x¢ = x · sx , y¢= y·sy , z¢ = z ·sz
Scaling an object with transformation changes the position of object relative to the coordinate origin. A parameter value greater than 1 move a point farther from the origin. A parameter value less than 1 move a point closer to the origin.
Uniform scaling is performed when sx=sy=sz. If the scaling parameters are not all equal, relative dimensions of a transformed object are changed.
Scaling transformation with respect to any selected fixed position (xf,yf,zf) can be constructed using the following transformation sequence:
1. Translate the fixed point to the origin.
2. Apply the scaling transformation relative to the coordinate origin.
3. Translate the fixed point back to its original position.
The matrix representation for an arbitrary fixed-point scaling can be expressed as the concatenation of translate-scale-translate transformations:
