A point CGAL transformation between two coordinate systems

Question

Using CGAL, how to determine the coordinates of a point from one system to another? Suppose we have:

Point p1(1.0, 1.0, 1.0);  

which is expressed in the typical coordinate system determined by the vectors:

Vector vx1(1.0, 0.0, 0.0);  
Vector vy1(0.0, 1.0, 0.0);  
Vector vz1(0.0, 0.0, 1.0);  

representing the coordinate axis. Now taking the vectors of the coordinate axis of another system, how do I determine the coordinates of p1 in that system?

Vector vx2(1.0, -1.0, -1.0);  
Vector vy2(-1.0, 1.0, -1.0);  
Vector vz2(1.0, 1.0, 0.0);  

I think I must determine a matrix for pass it to an object of CGAL::Aff_transformation_3, but I don't know how.

Vector p2 = p1.transform(??);  

Any tips?

Answer 1

Well, after researching a bit, I found here the theory for the solution to my problem. https://en.wikipedia.org/wiki/Change_of_basis

To clarify a bit, these definitions have built with the purpose of making the code more understandable.

typedef CGAL::Cartesian<long double>   KC;

typedef KC::Point_3                    Point;
typedef KC::Vector_3                   Vector;
typedef CGAL::Aff_transformation_3<KC> Transform3;

After considering the above, built the affine transformation as follows:

Transform3 tr3(
    vx2.x(), vx2.y(), vx2.z(),
    vy2.x(), vy2.y(), vy2.z(),
    vz2.x(), vz2.y(), vz2.z());

Then, with this transform object, I can get the coordinates of a point in the desired system:

Point p1_out = p1.transform (tr3);

Thanks!