MakeHomogenousMatrix

 MakeHomogenousMatrix[m, v]creates a homogeneous transformation matrix from a 3 x 3 rotation matrix m and a 3 x 1 translation vector v. MakeHomogenousMatrix[m]creates a homogeneous transformation matrix from a 3 x 3 rotation matrix m with a {0,0,0} translation vector. MakeHomogenousMatrix[v]creates a homogeneous transformation matrix from a 3 x 1 translation vector v and with a 3 x 3 identity matrix. MakeHomogenousMatrix[o, z]creates a homogeneous transformation matrix of a coordinate frame defined by its vector of origin o and the direction vector of the z-axis z. MakeHomogenousMatrix[]creates a 4x4 identity matrix
• The homogeneous matrix is a 4 x 4 matrix with the last row {0, 0, 0, 1} .The homogeneous matrix can represent a transformation matrix or describe the spatial placement of a coordinate frame.
• If the homogeneous matrix represent the transformation of A frame to B frame then the 3 x 3 upper left minor matrix represents the rotation, which takes A frame into B and the last column represents the translational vector, which takes the origin of A frame into B.
• If the spatial placement of a coordinate frame is A, than the column vector of 3 x 3 upper left minor matrix represent the unit direction vector of x, y, z axis of A frame and the last column of A matrix is the origin of the frame
This creates the default homogenous matrix
 Out[2]//MatrixForm=
This creates a homogeneous matrix from a rotation matrix
 Out[3]//MatrixForm=
This creates a homogeneous matrix from a translational vector
 Out[4]//MatrixForm=