Determine affine transformation that transform one plane into a parallel plane to another

Question

How i can determine the CGAL affine transformation (Aff_transformation_3) that transform one plane (plane1) into a parallel plane to another (plane2)?

Suppose that i have two object planes:

Plane_3  pl1;
Plane_3  pl2;

and they are not parallels, how determine this kind of affine transformation?

Aff_transformation_3 t3 = ??? (pl1, pl2);

I consulted this question and your answer: CGAL: Transformation Matrix for Rotation given two lines/vectors/directions, but i don't know how it may helpme. I have two planes, but in 3d dimensions.

Thanks.

Answer 1

I don't know how a 2d affine transformation (Aff_transformation_2) may helpme to apply a 3d affine transformation (Aff_transformation_3).

However, i found the solution to my question. This is may bit code that i hope to help someone.

typedef CGAL::Cartesian<double>         KC;

typedef KC::Line_3                      Line3;
typedef KC::Vector_3                    Vector3;
typedef KC::Plane_3                     Plane3;
typedef CGAL::Aff_transformation_3<KC>  Transform3;

// forwards
struct axis_angle;

typedef boost::shared_ptr<axis_angle>   RAxisAngle;

struct axis_angle
{
    axis_angle()
    {
        angle = 0;
        axis = Vector3(0.0, 0.0, 0.0);
    }

    double  angle;
    Vector3 axis;
};

Vector3 normalize(const Vector3 &v)
{
    ldouble len = ::sqrt(v.squared_length());

    if (len == 0.0)
        return v;

    return v / len;
}

// return the angle and axis from two planes that there are not parallels
RAxisAngle axis_angle_from_planes(const Plane3 &pln1, const Plane3 &pln2)
{
    RAxisAngle result = RAxisAngle(new axis_angle());

    Vector3 norm1 = pln1.orthogonal_vector();
    Vector3 norm2 = pln2.orthogonal_vector();

    double dot_r = norm1 * norm2;
    double len_r = ::sqrt(norm1.squared_length() * norm2.squared_length());

    if (len_r)
        result->angle = ::acos(dot_r / len_r);
    else
        result->angle = 0.0;

    Line3 l1;
    CGAL::Object obj_cgal = CGAL::intersection(pln1, pln2);
    if (CGAL::assign(l1, obj_cgal))
    {
        result->axis = normalize(l1.to_vector());
    }
    else
    {
        // when planes are parallels, then use some basic axis
        result->axis = Vector3(1.0, 0.0, 0.0);
    }

    return result;
}

// return a CGAL affine transformation that is builded from a 3x3 matrix
// this transformation is for rotate an object from axis and angle
// http://en.wikipedia.org/wiki/Transformation_matrix
// http://en.wikipedia.org/wiki/Rotation_matrix
// http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToMatrix/index.htm
Transform3 axis_angle_to_matrix(const RAxisAngle &aa)
{
    double tmp1, tmp2;

    double c = ::cos(aa->angle);
    double s = ::sin(aa->angle);
    double t = 1.0 - c;

    double m00 = c + aa->axis.x() * aa->axis.x() * t;
    double m11 = c + aa->axis.y() * aa->axis.y() * t;
    double m22 = c + aa->axis.z() * aa->axis.z() * t;

    tmp1 = aa->axis.x() * aa->axis.y() * t;
    tmp2 = aa->axis.z() * s;
    double m10 = tmp1 + tmp2;
    double m01 = tmp1 - tmp2;

    tmp1 = aa->axis.x() * aa->axis.z() * t;
    tmp2 = aa->axis.y() * s;
    double m20 = tmp1 - tmp2;
    double m02 = tmp1 + tmp2;

    tmp1 = aa->axis.y() * aa->axis.z() * t;
    tmp2 = aa->axis.x() * s;
    double m21 = tmp1 + tmp2;
    double m12 = tmp1 - tmp2;

    return Transform3(m00, m01, m02, m10, m11, m12, m20, m21, m22);
}

Then, i can use there as this:

RAxisAngle aa = axis_angle_from_planes(plane1, plane2);
Transform3 t3 = axis_angle_to_matrix(aa);

Plane2 new_transform_plane = plane1.transform(t3);

or maybe a point of this plane:

Point3 new_transform_point = point_of_plane1.transform(t3);

Thanks for giveme the posibility to post my little solution.