``````
// 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 "e3ga.h"
#include "e3ga_util.h"
#include "gabits.h"

namespace e3ga {

// improved rotorFromVectorToVector due to Allen Cortzen
rotor rotorFromVectorToVector(const vector &v1, const vector &v2) {
if (_Float(scp(v1, v2)) < -0.9f) {
// "near" 180 degree rotation:
//v1 factor in returning blade regardless of any loss of precision
vector w0 = _vector( v1 << (v1^v2) );
double n2 = _Float(norm_e2(w0));
if (n2 == 0.0f){
vector w1 = _vector( v1 << (v1^e1));
vector w2 = _vector( v1 << (v1^e2));
if(_Float(norm_e2(w1)) > _Float(norm_e2(w2))) return _rotor(v1^unit_e(w1));
else return _rotor(v1^unit_e(w2));
}
//v1 replaced with -v1 and additional 180 degree rotation:
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f - _Float(v2 << v1)));
return _rotor((1.0 - v2 * v1) * (1.0f / s)*(v1^unit_e(w0)));
}
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f + _Float(v2 << v1)));
return _rotor((1.0 + v2 * v1) * (1.0f / s));
}

// improved rotorFromVectorToVector due to Allen Cortzen
rotor rotorFromVectorToVector(const vector &v1, const vector &v2, const bivector &rotPlane) {
if (_Float(scp(v1, v2)) < -0.9f) {
//v1 replaced with -v1 and additional 180 degree rotation:
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f - _Float(v2 << v1)));
return _rotor((1.0 - v2 * v1) * (1.0f / s)*unit_e(rotPlane));
}
else {
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f + _Float(v2 << v1)));
return _rotor((1.0 + v2 * v1) * (1.0f / s));
}
}
/*
// old rotorFromVectorToVector()
rotor rotorFromVectorToVector(const vector &v1, const vector &v2) {
if (_Float(scp(v1, v2)) < -0.99999f) {
// (near) 180 degree rotation:
vector tmp = _vector(lcont(v1, op(v1, v2)));
double n2 = _Float(norm_e2(tmp));
if (n2 != 0.0f) tmp = _vector(tmp * (1.0f /  (mv::Float)sqrt(n2)));
else
tmp = (_Float(norm_e2(v1 ^ e1)) > _Float(norm_e2(v1 ^ e2))) ? _vector(e1) : _vector(e2);
return _rotor(unit_e(v1 ^ tmp));
}
else {
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f + _Float(v2 << v1)));
return _rotor((1.0 + v2 * v1) * (1.0f / s));
}
}

// old rotorFromVectorToVector()
rotor rotorFromVectorToVector(const vector &v1, const vector &v2, const bivector &rotPlane) {
if (_Float(scp(v1, v2)) < -0.99999f) {
return _rotor(unit_e(rotPlane));
}
else {
mv::Float s = (mv::Float)sqrt(2.0 * (1.0f + _Float(v2 << v1)));
return _rotor((1.0 + v2 * v1) * (1.0f / s));
}
}
*/
void rotorToMatrix(const rotor &R, mv::Float M[9]) {
mv::Float qw = _Float(R);
mv::Float qx = -R.e2e3();
mv::Float qy = -R.e3e1();
mv::Float qz = -R.e1e2();

M[0 * 3 + 0] = 1.0f - 2.0f * qy * qy - 2.0f * qz * qz;
M[1 * 3 + 0] = 2.0f * (qx * qy + qz * qw);
M[2 * 3 + 0] = 2.0f * (qx * qz - qy * qw);

M[0 * 3 + 1] = 2.0f * (qx * qy - qz * qw);
M[1 * 3 + 1] = 1.0f - 2.0f * qx * qx - 2.0f * qz * qz;
M[2 * 3 + 1] = 2.0f * (qy * qz + qx * qw);

M[0 * 3 + 2] = 2.0f * (qx * qz + qy * qw);
M[1 * 3 + 2] = 2.0f * (qy * qz - qx * qw);
M[2 * 3 + 2] = 1.0f - 2.0f * qx * qx - 2.0f * qy * qy;
}

rotor matrixToRotor(const mv::Float M[9]) {
mv::Float trace = M[0 * 3 + 0] + M[1 * 3 + 1] + M[2 * 3 + 2] + 1.0f;
mv::Float qw; // scalar coordinate
mv::Float qx; // coordinate for -e2^e3
mv::Float qy; // coordinate for -e3^e1
mv::Float qz; // coordinate for -e1^e2
if (trace > 0.00001f) {
mv::Float s = 0.5f / (mv::Float)sqrt(trace);
qw = 0.25f / s;
qw = sqrt(trace) * (0.5f);
qx = (M[2 * 3 + 1] - M[1 * 3 + 2]) * s;
qy = (M[0 * 3 + 2] - M[2 * 3 + 0]) * s;
qz = (M[1 * 3 + 0] - M[0 * 3 + 1]) * s;
}
else {
if (M[0 * 3 + 0] > M[1 * 3 + 1] && M[0 * 3 + 0] > M[2 * 3 + 2]) {
mv::Float s = 2.0f * (mv::Float)sqrt( 1.0f + M[0 * 3 + 0] - M[1 * 3 + 1] - M[2 * 3 + 2]);
qx = 0.25f * s;
qy = (M[0 * 3 + 1] + M[1 * 3 + 0]) / s;
qz = (M[0 * 3 + 2] + M[2 * 3 + 0]) / s;
qw = (M[1 * 3 + 2] - M[2 * 3 + 1]) / s;
}
else if (M[1 * 3 + 1] > M[2 * 3 + 2]) {
mv::Float s = 2.0f * (mv::Float)sqrt( 1.0f + M[1 * 3 + 1] - M[0 * 3 + 0] - M[2 * 3 + 2]);
qx = (M[0 * 3 + 1] + M[1 * 3 + 0]) / s;
qy = 0.25f * s;
qz = (M[1 * 3 + 2] + M[2 * 3 + 1]) / s;
qw = (M[0 * 3 + 2] - M[2 * 3 + 0]) / s;
}
else {
mv::Float s = 2.0f * (mv::Float)sqrt( 1.0f + M[2 * 3 + 2] - M[0 * 3 + 0] - M[1 * 3 + 1] );
qx = (M[0 * 3 + 2] + M[2 * 3 + 0]) / s;
qy = (M[1 * 3 + 2] + M[2 * 3 + 1]) / s;
qz = 0.25f * s;
qw = (M[0 * 3 + 1] - M[1 * 3 + 0]) / s;
}
}

mv::Float s = (mv::Float) sqrt(qw *qw + qx * qx + qy * qy + qz * qz);

return rotor(rotor_scalar_e1e2_e2e3_e3e1, qw / s, -qz / s, -qx / s, -qy / s);
}

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;
}

bivector log(const rotor &R) {
// get the bivector/2-blade part of R
bivector B = _bivector(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 _bivector((float)M_PI * (e1 ^ e2)); // we return a 360 degree rotation in an arbitrary plane
else
return bivector();  // return log(1) = 0
}

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

// special exp for 3D Euclidean bivectors:
rotor exp(const bivector &x) {
// compute the square
mv::Float x2 = _Float(x << x);

// x2 must always be <= 0, but round off error can make it positive:
if (x2 > 0.0f) x2 = 0.0f;

// compute half angle:
mv::Float ha = sqrt(-x2);

if (ha == (mv::Float)0.0) return _rotor((mv::Float)1.0);

// return rotor:
return _rotor((mv::Float)cos(ha) + ((mv::Float)sin(ha) / ha) * x);
}

void reciprocalFrame(const e3ga::vector *IF, e3ga::vector *RF, int nbVectors) {
if (nbVectors == 0) return; // nothing to do
else if (nbVectors == 1) {
// trivial case
if (_Float(norm_r2(IF[0])) == 0.0)
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)) == 0.0)
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] = _vector(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, vector 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[3] = {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] = _vector(unit_e(lcont(lcont(ei, Bc), Bc))); // no inverse(Bc) required, since Bc is always unit

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

// last factor = what is left of the input blade
factor[k-1] = _vector(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 <= 3; 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 = 3, 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[3] =
{
(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 == 3) {
return mv(GRADE_3, 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[8];
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 = I3;
int Ej = 3 - ((ga + gb + gd) >> 1);

// check join excessity
if (Ej == 0) {
m = lcont(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 = lcont(d, I3i);

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

mv e[3] = {
};

for (unsigned int i = 0; i < 3; i++) {
// compute next factor 'c'
mv c = lcont(lcont(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 = lcont(c, ca);
cp = lcont(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) {
m = op(m, cp);
Em--;
if (Em == 0) {
j = op(d, m);
m = unit_e(m);
j = unit_e(j);

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

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

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

// optionally remove 'c' from 's' (do that?)
}

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

} /* end of namespace e3ga */

``````