// This program is free software; you can redistribute it and/or
// modify it under the terms of the GNU General Public License

// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.

// You should have received a copy of the GNU General Public License
// along with this program; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA.

// Copyright 2007, Daniel Fontijne, University of Amsterdam -- fontijne@science.uva.nl

#include <string>

#include "c3ga.h"
#include "c3ga_util.h"
#include "e3ga_util.h"
#include "gabits.h"

namespace c3ga {

mv exp(const mv &x, int order /*= 9*/) {
// First try special cases:
// Check if (x * x == scalar) is scalar
mv x2 = x * x;
mv::Float s_x2 = _Float(x2);
if ((_Float(norm_e2(x2) - s_x2 * s_x2)) < 1e-7f) {
// OK (x * x == scalar), so use special cases:
if (s_x2 < 0.0) {
mv::Float a = sqrt(-s_x2);
return (mv::Float)cos(a) + (mv::Float)sin(a) * x * (1.0f / a);
}
else if (s_x2 > 0.0) {
mv::Float a = sqrt(s_x2);
return (mv::Float)cosh(a) + (mv::Float)sinh(a) * x * (1.0f / a);
}
else {
return 1 + x;
}
}

// now do general series eval:

int i;
mv result;

result = 1.0;
if (order == 0) {
return result;
}

// scale by power of 2 so that its norm is < 1
unsigned long max = (unsigned long)x.largestCoordinate();
unsigned long scale=1;
if (max > 1) scale <<= 1;
while (max)
{
max >>= 1;
scale <<= 1;
}

mv scaled = x * scalar(1.0f / (mv::Float)scale);

// taylor approximation
mv tmp;

tmp = 1.0;
for (i = 1; i < order; i++) {
tmp = tmp*scaled * scalar(1.0f / (mv::Float)i);
result += tmp;
}

// undo scaling
while (scale > 1)
{
result *= result;
scale >>= 1;
}
return result;
}

bivectorE3GA log(const rotor &R) {
// get the bivector/2-blade part of R
bivectorE3GA B = _bivectorE3GA(R);

// compute the 'reverse norm' of the bivector part of R:
mv::Float R2 = _Float(norm_r(B));

// check to avoid divide-by-zero (and also below zero due to FP roundoff):
if (R2 <= 0.0) {
if (_Float(R) < 0)  // this means the user ask for log(-1):
return _bivectorE3GA((float)M_PI * (e1 ^ e2)); // we return a 360 degree rotation in an arbitrary plane
else
return bivectorE3GA();  // return log(1) = 0
}

// return the log:
return _bivectorE3GA(B * ((float)atan2(R2, _Float(R)) / R2));
}

dualLine log(const TRversor &V) {
// isolate rotation, translation part:
rotor R = _rotor(-no << (V * ni));
vectorE3GA t = _vectorE3GA(-2.0f * (no << V) * inverse(R));

const float EPSILON = 1e-6f;

if (_Float(norm_e2(_bivectorE3GA(R))) < EPSILON * EPSILON) {
// special cases:
if (_Float(R) < 0.0f)
{// R = -1
// The versor has a rotation over 360 degrees.
bivectorE3GA I; // Get a rotation plane 'I', depending on 't'
if (_Float(norm_e2(t)) >  EPSILON * EPSILON)
I = _bivectorE3GA(unit_e(t << I3));
else I = _bivectorE3GA(e1^e2); // when t = 0, any plane will do

// return translation plus 360 degree rotation:
return _dualLine(0.5f *(I * 2.0f * (float)M_PI - (t^ni)));
}
else
{ // R = 1;
// return translation :
return _dualLine(-0.5f *(t^ni));
}
}
else { // regular case

// compute logarithm of rotation part
bivectorE3GA Iphi = _bivectorE3GA(-2.0f * log(R));

// determine rotation plane:
rotor I = _rotor(unit_e(Iphi));

// compose log of V:
return _dualLine(
0.5f * (
-(t ^ I) * inverse(I) * ni +
inverse(1.0f - R * R) * (t << Iphi) * ni -
Iphi));
}
}

TRSversorLog log(const TRSversor &V) {
// get rotor part:
rotor X = _rotor(- no << (V * ni));

const float EPSILON = 1e-6f;

// get scaling part
mv::Float gamma = (mv::Float)::log(_Float(X * reverse(X)));
mv::Float gammaPrime;
if (fabs(gamma) < EPSILON) gammaPrime = 1.0f;
else gammaPrime = gamma / (::exp(gamma) - 1);
scalor S = exp(_noni_t(0.5f * gamma * noni));

// get rotation part
rotor R = _rotor(::exp(-0.5f * gamma) * X);

// get translation part:
translator T = _translator(V * reverse(S) * reverse(R));
vectorE3GA t = _vectorE3GA(-2.0f * (no << T));

if (_Float(norm_e2(_bivectorE3GA(R))) < EPSILON * EPSILON) {
if (_Float(R) > 0.0f) { // R = 1
// no rotation, so no rotation plane
return _TRSversorLog(0.5f * (-gammaPrime * (t ^ ni) + gamma * noni));
}
else { // R = -1
// We need to add a 360 degree rotation to the result.
// Take it perperdicular to 't':
bivectorE3GA I; // Get a rotation plane 'I', depending on 't'
if (_Float(norm_e2(t)) >  EPSILON * EPSILON)
I = _bivectorE3GA(unit_e(t << I3));
else I = _bivectorE3GA(e1^e2); // when t = 0, any plane will do

return _TRSversorLog(0.5f * (
-gammaPrime * (t ^ ni) +
gamma * noni +
2.0f * (float)M_PI * I));
}
}
else {
// get rotation plane, angle
bivectorE3GA I = _bivectorE3GA(R);
mv::Float sR2 = _Float(norm_e(I));
I = _bivectorE3GA(I * (1.0f / sR2));
mv::Float phi = -2.0f * (mv::Float)atan2(sR2, _Float(R));

// form bivector log of versor:
normalizedTranslator Tv = _normalizedTranslator(
1.0f - 0.5f * inverse(1.0f - (mv::Float)::exp(gamma) * R * R) * (t << I) * reverse(I) * ni);
return _TRSversorLog(0.5f * (-gammaPrime * (t^I) * reverse(I) * ni + Tv * (-phi * I + gamma * noni) * reverse(Tv)));
}
}

// this is the version that can not handle negative scaling . . .
TRSversor matrix4x4ToVersorPS(const mv::Float _M[4 * 4], bool transpose /*= false*/) {
mv::Float M[4 * 4];
if (transpose) {
// transpose & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 0] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 0] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 0] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 1] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 2] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 3] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}
else {
// copy & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[0 * 4 + 1] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[0 * 4 + 2] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[0 * 4 + 3] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[1 * 4 + 0] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[2 * 4 + 0] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[3 * 4 + 0] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}

/*  printf("Matrix:\n");
printf("%f %f %f %f\n", M[0 * 4 + 0], M[0 * 4 + 1], M[0 * 4 + 2], M[0 * 4 + 3]);
printf("%f %f %f %f\n", M[1 * 4 + 0], M[1 * 4 + 1], M[1 * 4 + 2], M[1 * 4 + 3]);
printf("%f %f %f %f\n", M[2 * 4 + 0], M[2 * 4 + 1], M[2 * 4 + 2], M[2 * 4 + 3]);
printf("%f %f %f %f\n", M[3 * 4 + 0], M[3 * 4 + 1], M[3 * 4 + 2], M[3 * 4 + 3]);*/

// determine translation:
vectorE3GA t(vectorE3GA_e1_e2_e3, M[0 * 4 + 3], M[1 * 4 + 3], M[2 * 4 + 3]);

// initialize images of Euclidean basis vectors (the columns of the matrix)
vectorE3GA imageOfE1(vectorE3GA_e1_e2_e3, M[0 * 4 + 0], M[1 * 4 + 0], M[2 * 4 + 0]);
vectorE3GA imageOfE2(vectorE3GA_e1_e2_e3, M[0 * 4 + 1], M[1 * 4 + 1], M[2 * 4 + 1]);
vectorE3GA imageOfE3(vectorE3GA_e1_e2_e3, M[0 * 4 + 2], M[1 * 4 + 2], M[2 * 4 + 2]);

// get scale of the 3x3 part (e1, e2, e3)
mv::Float scale = _Float(norm_e(imageOfE1) + norm_e(imageOfE2) + norm_e(imageOfE3))  / 3.0f;

// compute determinant of matrix (if negative, flip 3rd column)
float sc3 = 1.0f; // sc3 = scale column 3
if ((imageOfE1 ^ imageOfE2 ^ imageOfE3).e1e2e3() < 0.0f) sc3 = -1.0f;

// initialize 3x3 'rotation' matrix, call e3ga::matrixToRotor
float RM[3 * 3] = {
M[0 * 4 + 0] / scale, M[0 * 4 + 1] / scale, sc3 * M[0 * 4 + 2] / scale,
M[1 * 4 + 0] / scale, M[1 * 4 + 1] / scale, sc3 * M[1 * 4 + 2] / scale,
M[2 * 4 + 0] / scale, M[2 * 4 + 1] / scale, sc3 * M[2 * 4 + 2] / scale
};
e3ga::rotor tmpR = e3ga::matrixToRotor(RM);

// convert e3ga rotor to c3ga rotor:
c3ga::rotor R(rotor_scalar_e1e2_e2e3_e3e1,
tmpR.getC(e3ga::rotor_scalar_e1e2_e2e3_e3e1));

// get log of scale:
mv::Float logScale = (mv::Float) ::log(scale);

// return full versor:
TRSversor result = _TRSversor(
exp(_freeVector(-0.5f * (t ^ ni))) *  // translation
R * // rotation
exp(_noni_t(0.5f * logScale * noni)) // scaling
);

/*  {
mv V = result;
mv Vi = inverse(V);
// compute images of basis vectors:
flatPoint imageOfE1NI = _flatPoint(V * e1ni * Vi);
flatPoint imageOfE2NI = _flatPoint(V * e2ni * Vi);
flatPoint imageOfE3NI = _flatPoint(V * e3ni * Vi);
flatPoint imageOfNONI = _flatPoint(V * noni * Vi);

// create matrix representation:
omFlatPoint OM(imageOfE1NI, imageOfE2NI, imageOfE3NI, imageOfNONI);

printf("Reconstructed Matrix:\n");
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 0], OM.m_c[1 * 4 + 0], OM.m_c[2 * 4 + 0], OM.m_c[3 * 4 + 0]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 1], OM.m_c[1 * 4 + 1], OM.m_c[2 * 4 + 1], OM.m_c[3 * 4 + 1]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 2], OM.m_c[1 * 4 + 2], OM.m_c[2 * 4 + 2], OM.m_c[3 * 4 + 2]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 3], OM.m_c[1 * 4 + 3], OM.m_c[2 * 4 + 3], OM.m_c[3 * 4 + 3]);
}*/

return result;
}

TRSversor matrix4x4ToVersor(const mv::Float _M[4 * 4], bool transpose /*= false*/) {
mv::Float M[4 * 4];
if (transpose) {
// transpose & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 0] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 0] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 0] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 1] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 2] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 3] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}
else {
// copy & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[0 * 4 + 1] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[0 * 4 + 2] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[0 * 4 + 3] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[1 * 4 + 0] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[2 * 4 + 0] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[3 * 4 + 0] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}

// determine translation:
vectorE3GA t(vectorE3GA_e1_e2_e3, M[0 * 4 + 3], M[1 * 4 + 3], M[2 * 4 + 3]);

// initialize images of Euclidean basis vectors (the columns of the matrix)
vectorE3GA imageOfE1(vectorE3GA_e1_e2_e3, M[0 * 4 + 0], M[1 * 4 + 0], M[2 * 4 + 0]);
vectorE3GA imageOfE2(vectorE3GA_e1_e2_e3, M[0 * 4 + 1], M[1 * 4 + 1], M[2 * 4 + 1]);
vectorE3GA imageOfE3(vectorE3GA_e1_e2_e3, M[0 * 4 + 2], M[1 * 4 + 2], M[2 * 4 + 2]);

// get scale of the 3x3 part (e1, e2, e3)
mv::Float scale = _Float(norm_e(imageOfE1) + norm_e(imageOfE2) + norm_e(imageOfE3))  / 3.0f;

// compute determinant of matrix (if negative, negate all matrix elements)
mv::Float n = 1.0f; // used to negate the matrix
scalor negScale = _scalor(1.0f);
if ((imageOfE1 ^ imageOfE2 ^ imageOfE3).e1e2e3() < 0.0f) {
n = -1.0f;
negScale = noni; // no^ni provides the negative scaling in the final versor
}

// initialize 3x3 'rotation' matrix RM, call e3ga::matrixToRotor
mv::Float si = n / scale;
mv::Float RM[3 * 3] = {
M[0 * 4 + 0] * si, M[0 * 4 + 1] * si, M[0 * 4 + 2] * si,
M[1 * 4 + 0] * si, M[1 * 4 + 1] * si, M[1 * 4 + 2] * si,
M[2 * 4 + 0] * si, M[2 * 4 + 1] * si, M[2 * 4 + 2] * si
};
e3ga::rotor tmpR = e3ga::matrixToRotor(RM);

// convert e3ga rotor to c3ga rotor:
c3ga::rotor R(rotor_scalar_e1e2_e2e3_e3e1,
tmpR.getC(e3ga::rotor_scalar_e1e2_e2e3_e3e1));

// get log of scale:
mv::Float logScale = (mv::Float) ::log(scale);

// return full versor:
TRSversor result = _TRSversor(
exp(_freeVector(-0.5f * (t ^ ni))) *  // translation
R *                                                   // rotation
exp(_noni_t(0.5f * logScale * noni)) * // scaling
negScale                                       // negative scaling
);

return result;
}

#ifdef RIEN
mv matrix4x4ToVersor(const mv::Float _M[4 * 4], bool transpose /*= false*/) {
/*  printf("Raw Matrix:\n");
printf("%f %f %f %f\n", _M[0 * 4 + 0], _M[0 * 4 + 1], _M[0 * 4 + 2], _M[0 * 4 + 3]);
printf("%f %f %f %f\n", _M[1 * 4 + 0], _M[1 * 4 + 1], _M[1 * 4 + 2], _M[1 * 4 + 3]);
printf("%f %f %f %f\n", _M[2 * 4 + 0], _M[2 * 4 + 1], _M[2 * 4 + 2], _M[2 * 4 + 3]);
printf("%f %f %f %f\n", _M[3 * 4 + 0], _M[3 * 4 + 1], _M[3 * 4 + 2], _M[3 * 4 + 3]);*/

mv::Float M[4 * 4];
if (transpose) {
// transpose & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 0] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 0] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 0] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 1] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 2] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[0 * 4 + 3] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}
else {
// copy & normalize
M[0 * 4 + 0] = _M[0 * 4 + 0] / _M[3 * 4 + 3]; M[0 * 4 + 1] = _M[0 * 4 + 1] / _M[3 * 4 + 3]; M[0 * 4 + 2] = _M[0 * 4 + 2] / _M[3 * 4 + 3]; M[0 * 4 + 3] = _M[0 * 4 + 3] / _M[3 * 4 + 3];
M[1 * 4 + 0] = _M[1 * 4 + 0] / _M[3 * 4 + 3]; M[1 * 4 + 1] = _M[1 * 4 + 1] / _M[3 * 4 + 3]; M[1 * 4 + 2] = _M[1 * 4 + 2] / _M[3 * 4 + 3]; M[1 * 4 + 3] = _M[1 * 4 + 3] / _M[3 * 4 + 3];
M[2 * 4 + 0] = _M[2 * 4 + 0] / _M[3 * 4 + 3]; M[2 * 4 + 1] = _M[2 * 4 + 1] / _M[3 * 4 + 3]; M[2 * 4 + 2] = _M[2 * 4 + 2] / _M[3 * 4 + 3]; M[2 * 4 + 3] = _M[2 * 4 + 3] / _M[3 * 4 + 3];
M[3 * 4 + 0] = _M[3 * 4 + 0] / _M[3 * 4 + 3]; M[3 * 4 + 1] = _M[3 * 4 + 1] / _M[3 * 4 + 3]; M[3 * 4 + 2] = _M[3 * 4 + 2] / _M[3 * 4 + 3]; M[3 * 4 + 3] = _M[3 * 4 + 3] / _M[3 * 4 + 3];
}

/*  printf("Input Matrix:\n");
printf("%f %f %f %f\n", M[0 * 4 + 0], M[0 * 4 + 1], M[0 * 4 + 2], M[0 * 4 + 3]);
printf("%f %f %f %f\n", M[1 * 4 + 0], M[1 * 4 + 1], M[1 * 4 + 2], M[1 * 4 + 3]);
printf("%f %f %f %f\n", M[2 * 4 + 0], M[2 * 4 + 1], M[2 * 4 + 2], M[2 * 4 + 3]);
printf("%f %f %f %f\n", M[3 * 4 + 0], M[3 * 4 + 1], M[3 * 4 + 2], M[3 * 4 + 3]);*/

// determine translation:
vectorE3GA t(vectorE3GA_e1_e2_e3, M[0 * 4 + 3], M[1 * 4 + 3], M[2 * 4 + 3]);

// initialize images of Euclidean basis vectors (the columns of the matrix)
vectorE3GA imageOfE1(vectorE3GA_e1_e2_e3, M[0 * 4 + 0], M[1 * 4 + 0], M[2 * 4 + 0]);
vectorE3GA imageOfE2(vectorE3GA_e1_e2_e3, M[0 * 4 + 1], M[1 * 4 + 1], M[2 * 4 + 1]);
vectorE3GA imageOfE3(vectorE3GA_e1_e2_e3, M[0 * 4 + 2], M[1 * 4 + 2], M[2 * 4 + 2]);

// get scale of the 3x3 part (e1, e2, e3)
mv::Float scale = _Float(norm_e(imageOfE1) + norm_e(imageOfE2) + norm_e(imageOfE3))  / 3.0f;

// compute determinant of matrix (if negative, flip 3rd column)
float sc3 = 1.0f; // sc3 = scale column 3
mv reflectionPlane = 1;
if ((imageOfE1 ^ imageOfE2 ^ imageOfE3).e1e2e3() < 0.0f) {
sc3 = -1.0f;
reflectionPlane = e3;
}

// initialize 3x3 'rotation' matrix, call e3ga::matrixToRotor
float RM[3 * 3] = {
M[0 * 4 + 0] / scale, M[0 * 4 + 1] / scale, sc3 * M[0 * 4 + 2] / scale,
M[1 * 4 + 0] / scale, M[1 * 4 + 1] / scale, sc3 * M[1 * 4 + 2] / scale,
M[2 * 4 + 0] / scale, M[2 * 4 + 1] / scale, sc3 * M[2 * 4 + 2] / scale
};
e3ga::rotor tmpR = e3ga::matrixToRotor(RM);

// convert e3ga rotor to c3ga rotor:
c3ga::rotor R(rotor_scalar_e1e2_e2e3_e3e1,
tmpR.getC(e3ga::rotor_scalar_e1e2_e2e3_e3e1));

// get log of scale:
mv::Float logScale = (mv::Float) ::log(scale);

// return full versor:
mv result = exp(_freeVector(-0.5f * (t ^ ni))) *  // translation
R * // rotation
exp(_noni_t(0.5f * logScale * noni)) * // scaling
reflectionPlane;

/*  {
mv V = result;
mv Vi = inverse(V);
// compute images of basis vectors:
flatPoint imageOfE1NI = _flatPoint(V * e1ni * Vi);
flatPoint imageOfE2NI = _flatPoint(V * e2ni * Vi);
flatPoint imageOfE3NI = _flatPoint(V * e3ni * Vi);
flatPoint imageOfNONI = _flatPoint(V * noni * Vi);

// create matrix representation:
omFlatPoint OM(imageOfE1NI, imageOfE2NI, imageOfE3NI, imageOfNONI);

printf("Reconstructed Matrix:\n");
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 0], OM.m_c[1 * 4 + 0], OM.m_c[2 * 4 + 0], OM.m_c[3 * 4 + 0]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 1], OM.m_c[1 * 4 + 1], OM.m_c[2 * 4 + 1], OM.m_c[3 * 4 + 1]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 2], OM.m_c[1 * 4 + 2], OM.m_c[2 * 4 + 2], OM.m_c[3 * 4 + 2]);
printf("%f %f %f %f\n", OM.m_c[0 * 4 + 3], OM.m_c[1 * 4 + 3], OM.m_c[2 * 4 + 3], OM.m_c[3 * 4 + 3]);

}*/

return result;
}
#endif

void reciprocalFrame(const dualSphere *IF, dualSphere *RF, int nbVectors) {
if (nbVectors == 0) return; // nothing to do
else if (nbVectors == 1) {
// trivial case
if (_Float(norm_r2(IF[0])) == 0.0f)
throw std::string("reciprocalFrame(): null vector");
RF[0] = inverse(IF[0]);
return;
}
else {
// compute pseudoscalar 'I' of space spanned by input frame:
mv I = IF[0];
for (int i = 1; i < nbVectors; i++) I ^= IF[i];
if (_Float(norm_r2(I)) == 0.0f)
throw std::string("reciprocalFrame(): vectors are not independent");

// compute inverse of 'I':
mv Ii = inverse(I);

// compute the vectors of the reciprocal framevector
for (int i = 0; i < nbVectors; i++) {
// compute outer product of all vectors except IF[i]
mv P = (i & 1) ? -1.0f : 1.0f; // = pow(-1, i)
for (int j = 0; j < nbVectors; j++)
if (j != i) P ^= IF[j];

// compute reciprocal vector 'i':
RF[i] = _dualSphere(P << Ii);
}
return;
}
}

/// factors blade into vectors (euclidean unit length), returns  scale (or throws exception when non-blade is passed)
mv::Float factorizeBlade(const mv &X, dualSphere factor[], int gradeOfX /* = -1 */) {
//  printf("X = %s;\n", X.c_str());
if (k < 0) {
mvType T(X);
}
mv::Float s = (k == 0) ? _Float(X): _Float(norm_e(X));

if (k < 0) throw -1;

// set scale of output, no matter what:
mv::Float scale = s;

if ((s == 0.0) || (k == 0))
return scale;

// get largest basis blade, basis vectors
unsigned int E;
int Eidx = 0;

// setup the 'current input blade'
mv Bc = unit_e(X);

mv::Float coords[5] = {0.0f, 0.0f, 0.0f, 0.0f, 0.0f};
for (int i = 0; i < (k-1); i++) {
// get next basisvector
while (!(E&1)) {
coords[Eidx] = 0.0;
E >>= 1;    Eidx++;
}
coords[Eidx] = 1.0;
E ^= 1;

// project basis vector ei, normalize projection:
factor[i] = _dualSphere(unit_e(lcontEM(lcontEM(ei, Bc), Bc))); // no inverse(Bc) required, since Bc is always unit

// remove f[i] from Bc
Bc = lcontEM(factor[i], Bc);
}

// last factor = what is left of the input blade
factor[k-1] = _dualSphere(unit_e(Bc)); // already normalized, but renormalize to remove any FP round-off error

return scale;
}

// todo: integrate into G2
int nbBits = bitCount16(X.gu()); // bitCount16 also goes into Gaigen 2
if (nbBits == 1) {
return X;
}
else if (nbBits == 0) {
return mv(0.0f);
}
else {
// if grade is present: sum + square coordinates
//                               if larger than current -> keep pointer
mv::Float largestG = 0.0f;
int largestGidx = -1;
const mv::Float *largestC = NULL;
const mv::Float *C = X.m_c;

for (int i = 0; i <= 5; i++) {
if ((X.gu() & (1 << i)) == 0) continue;
else {
// square, sum
mv::Float l = C[0] * C[0];
for (int j = 1; j < s; j++) l = C[j] * C[j];

// check if larger
if (l > largestG) {
largestC = C;
largestG = l;
largestGidx = i;
}

}
}
if (largestC)
return mv((unsigned int)(1 << largestGidx), largestC);
else {
return mv(0.0f);
}
}
}

// todo: integrate into G2
/**
The returned integer is a bitwise or of
constants.
*/
mv grade(const mv &X, float epsilon /* = 1e-7 */);

// todo: integrate into G2
mv highestGradePart(const mv &X, float epsilon /* = 1e-7 */, int *gradeIdx /* = NULL*/) {
int g = 5, gu = X.gu(), iX = mv_size[gu], size, i;
const float *cptr = NULL;
do {
if (gu & (1 << g)) {
iX -= size;
cptr = X.m_c + iX;
for (i = 0; i < size; i++)
if ((cptr[i] > epsilon) || (-cptr[i] > epsilon)) {
return mv((unsigned int)(1 << g), cptr);
}
}
} while ((--g)>= 0);

return mv(0.0f);
}

// todo: integrate into G2
int gua;

// determine what the grage usage 'gu' of the result should be:
if (gradeUsageBitmap = ((gua = X.gu()) & gradeUsageBitmap)) { // only execute if any grade will be present in the result
mv::Float C[8];
mv::Float *bc;
const mv::Float *ac;

bc = C; ac = X.m_c; // pointers to the coordinates of source (ac) and result (bc)
for (int i = 1; i <= gradeUsageBitmap; i = i << 1) { // for each grade that is possibly in the result
if (gua & i) { // determine if grade is present in source
int s = mv_size[i]; // get the size of grade
if (gradeUsageBitmap & i) { // determine if grade is present in result
// copy coordinates
for (int j = 0; j < s; j++) bc[j] = ac[j];
bc += s; // increment pointer to result
}
ac += s; // increment pointer to source
}
}
}
else return mv(0.0f);
}

// todo: integrate into G2
mv deltaProduct(const mv &X, const mv &Y, float epsilon /* = 1e-7 */, int *gradeIdx /* = NULL*/) {
}

inline mv randomVector() {
float c[5] =
{
(float)(rand() - RAND_MAX / 2),
(float)(rand() - RAND_MAX / 2),
(float)(rand() - RAND_MAX / 2),
(float)(rand() - RAND_MAX / 2),
(float)(rand() - RAND_MAX / 2)
};
}

/**
Returns a random blade of 'grade' with a (Euclidean) size in range [0, 1.0].

Currently, rand() is used to generate the blade
Todo: use Mersenne twister or something (license issues?)
*/

return mv(size * (-1.0f + 2.0f * (float)rand() / (float)RAND_MAX));
}
else if (grade == 5) {
return mv(GRADE_5, size * (-1.0f + 2.0f * (float)rand() / (float)RAND_MAX));
}
else {
mv result = randomVector();
for (int g = 1; g < grade; g++) {
result ^= randomVector();
}
result = (-1.0f + 2.0f * (float)rand() / (float)RAND_MAX) * size * unit_e(result); // todo: random factor
return result;
}
}

mv::Float C[32];
for (int i = 0; i < s; i++) {
C[i] = -1.0f + 2.0f * (float)rand() * size / (float)RAND_MAX;
}
}

void meetJoin(const mv  &a, const mv &b, mv &m, mv &j, mv::Float smallEpsilon, mv::Float largeEpsilon) {
mv::Float la = a.largestCoordinate();
mv::Float lb = b.largestCoordinate();

// step one: check for near-zero input
if ((la < smallEpsilon) || (lb < smallEpsilon)) {
// set both meet and join to 0;
m.set();
j.set();
return; // done
}

int ga, gb;

// normalize (approximatelly) and swap (optionally)
mv ca, cb;
if (ga <= gb) {
ca = op(_ca, 1.0f / la);
cb = op(_cb, 1.0f / lb);
}
else {
ca = op(_cb, 1.0f / lb);
cb = op(_ca, 1.0f / la);
int tempg = ga;
ga = gb;
gb = tempg;
}

// compute delta product & 'normalize'
mv d,_d;
int gd;
_d = deltaProduct(a, b, smallEpsilon, &gd);
mv::Float ld = _d.largestCoordinate();
d = op(_d, 1.0f / ld);

// if delta product is scalar, we're done:
if (gd == 0) {
// a == b (up to scalar)
m = ca;
j  = ca;
// todo: largest coordinate positive?
return;
}

// if grade of delta product is equal to ga + gb, we're done, too
if (gd == ga + gb) {
// a and b entirely disjoint
m.set(1.0f);
j = unit_e(op(ca, cb));
// todo: largest coordinate positive?
return;
}

// init join
j = I5;
int Ej = 5 - ((ga + gb + gd) >> 1);

// check join excessity
if (Ej == 0) {
m = lcontEM(d, j);
// todo: largest coordinate positive?
return;
}

// init meet
m = 1.0f;
int Em = ((ga + gb - gd) >> 1);

// init s, the dual of the delta product:
mv s = lcontEM(d, I5i);

// precompute inverse of ca
mv cai = inverseEM(ca);

mv e[5] = {
mv(GRADE_1, 1.0f, 0.0f, 0.0f, 0.0f, 0.0f),
mv(GRADE_1, 0.0f, 1.0f, 0.0f, 0.0f, 0.0f),
mv(GRADE_1, 0.0f, 0.0f, 1.0f, 0.0f, 0.0f),
mv(GRADE_1, 0.0f, 0.0f, 0.0f, 1.0f, 0.0f),
mv(GRADE_1, 0.0f, 0.0f, 0.0f, 0.0f, 1.0f)
};

//  printf("d = %s\n", d.c_str_e());

for (unsigned int i = 0; i < 5; i++) {
// compute next factor 'c'
mv c = lcontEM(lcontEM(e[i], s), s);

// check if 'c' is OK to use:
if (c.largestCoordinate() < largeEpsilon)
continue;

// compute projection, rejection of 'c' wrt to 'ca'
mv cp, cr; // c projected, c rejected
mv tmpc = lcontEM(c, ca);
cp = lcontEM(tmpc, cai); // use correct inverse because otherwise cr != c - cp
cr = subtract(c, cp);

// if 'c' has enough of it in 'ca', then add to meet
if (cp.largestCoordinate() > largeEpsilon) {
//          printf("m%d = %s\n", Em, cp.c_str_e());
m = op(m, cp);
Em--;
if (Em == 0) {
j = op(d, m); // ???? here error?
m = unit_e(m);
j = unit_e(j);

// todo: largest coordinate positive?
return;
}
}

if (cr.largestCoordinate() > largeEpsilon) {
j = lcontEM(cr, j);
Ej--;
if (Ej == 0) {
m = lcontEM(d, j);
m = unit_e(m);
j = unit_e(j);

// todo: largest coordinate positive?
return;
}
}

// optionally remove 'c' from 's' (do that?)
// put this also in other meet/join algos?
s = lcontEM(c, s);
}

throw std::string("Error while computing meet & join!");
}

} /* end of namespace c3ga */