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ardupilot/libraries/AP_Math/AP_GeodesicGrid.cpp

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/*
* Copyright (C) 2016 Intel Corporation. All rights reserved.
*
* This file is free software: you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by the
* Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This file 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, see <http://www.gnu.org/licenses/>.
*/
/*
* This comment section explains the basic idea behind the implementation.
*
* Vectors difference notation
* ===========================
* Let v and w be vectors. For readability purposes, unless explicitly
* otherwise noted, the notation vw will be used to represent w - v.
*
* Relationship between a vector and a triangle in 3d space
* ========================================================
* Vector in the area of a triangle
* --------------------------------
* Let T = (a, b, c) be a triangle, where a, b and c are also vectors and
* linearly independent. A vector inside that triangle can be written as one of
* its vertices plus the sum of the positively scaled vectors from that vertex
* to the other ones. Taking a as the first vertex, a vector p in the area
* formed by T can be written as:
*
* p = a + w_ab * ab + w_ac * ac
*
* It's fairly easy to see that if p is in the area formed by T, then w_ab >= 0
* and w_ac >= 0. That vector p can also be written as:
*
* p = b + w_ba * ba + w_bc * bc
*
* It's easy to check that the triangle formed by (a + w_ab * ab, b + w_ba *
* ba, p) is similar to T and, with the correct algebraic manipulations, we can
* come to the conclusion that:
*
* w_ba = 1 - w_ab - w_ac
*
* Since we know that w_ba >= 0, then w_ab + w_ac <= 1. Thus:
*
* ----------------------------------------------------------
* | p = a + w_ab * ab + w_ac * ac is in the area of T iff: |
* | w_ab >= 0 and w_ac >= 0 and w_ab + w_ac <= 1 |
* ----------------------------------------------------------
*
* Proving backwards shouldn't be difficult.
*
* Vector p can also be written as:
*
* p = (1 - w_ab - w_ba) * a + w_ab * b + w_ba * c
*
*
* Vector that crosses a triangle
* ------------------------------
* Let T be the same triangle discussed above and let v be a vector such that:
*
* v = x * a + y * b + z * c
* where x >= 0, y >= 0, z >= 0, and x + y + z > 0.
*
* It's geometrically easy to see that v crosses the triangle T. But that can
* also be verified analytically.
*
* The vector v crosses the triangle T iff there's a positive alpha such that
* alpha * v is in the area formed by T, so we need to prove that such value
* exists. To find alpha, we solve the equation alpha * v = p, which will lead
* us to the system, for the variables alpha, w_ab and w_ac:
*
* alpha * x = 1 - w_ab - w_ac
* alpha * y = w_ab
* alpha * z = w_ac,
* where w_ab >= 0 and w_ac >= 0 and w_ab + w_ac <= 1
*
* That will lead to alpha = 1 / (x + y + z), w_ab = y / (x + y + b) and
* w_ac = z / (x + y + z) and the following holds:
* - alpha does exist because x + y + z > 0.
* - w_ab >= 0 and w_ac >= 0 because y >= 0 and z >= 0 and x + y + z > 0.
* - 0 <= 1 - w_ab - w_ac <= 1 because 0 <= (y + z) / (x + y + z) <= 1.
*
* Thus:
*
* ----------------------------------------------------------
* | v = x * a + y * b + z * c crosses T = (a, b, c), where |
* | a, b and c are linearly independent, iff: |
* | x >= 0, y >= 0, z >= 0 and x + y + z > 0 |
* ----------------------------------------------------------
*
* Moreover:
* - if one of the coefficients is zero, then v crosses the edge formed by the
* vertices multiplied by the non-zero coefficients.
* - if two of the coefficients are zero, then v crosses the vertex multiplied
* by the non-zero coefficient.
*/
#include <assert.h>
#include "AP_GeodesicGrid.h"
/* This was generated with
* libraries/AP_Math/tools/geodesic_grid/geodesic_grid.py */
const struct AP_GeodesicGrid::neighbor_umbrella
AP_GeodesicGrid::_neighbor_umbrellas[3]{
{{ 9, 8, 7, 12, 14}, 1, 2, 0, 0, 2},
{{ 1, 2, 4, 5, 3}, 0, 0, 2, 2, 0},
{{16, 15, 13, 18, 17}, 2, 2, 0, 2, 1},
};
/* This was generated with
* libraries/AP_Math/tools/geodesic_grid/geodesic_grid.py */
const Matrix3f AP_GeodesicGrid::_inverses[10]{
{{-0.309017f, 0.500000f, 0.190983f},
{ 0.000000f, 0.000000f, -0.618034f},
{-0.309017f, -0.500000f, 0.190983f}},
{{-0.190983f, 0.309017f, -0.500000f},
{-0.500000f, -0.190983f, 0.309017f},
{ 0.309017f, -0.500000f, -0.190983f}},
{{-0.618034f, 0.000000f, 0.000000f},
{ 0.190983f, -0.309017f, -0.500000f},
{ 0.190983f, -0.309017f, 0.500000f}},
{{-0.500000f, 0.190983f, -0.309017f},
{ 0.000000f, -0.618034f, 0.000000f},
{ 0.500000f, 0.190983f, -0.309017f}},
{{-0.190983f, -0.309017f, -0.500000f},
{-0.190983f, -0.309017f, 0.500000f},
{ 0.618034f, 0.000000f, 0.000000f}},
{{-0.309017f, -0.500000f, -0.190983f},
{ 0.190983f, 0.309017f, -0.500000f},
{ 0.500000f, -0.190983f, 0.309017f}},
{{ 0.309017f, -0.500000f, 0.190983f},
{ 0.000000f, 0.000000f, -0.618034f},
{ 0.309017f, 0.500000f, 0.190983f}},
{{ 0.190983f, -0.309017f, -0.500000f},
{ 0.500000f, 0.190983f, 0.309017f},
{-0.309017f, 0.500000f, -0.190983f}},
{{ 0.500000f, -0.190983f, -0.309017f},
{ 0.000000f, 0.618034f, 0.000000f},
{-0.500000f, -0.190983f, -0.309017f}},
{{ 0.309017f, 0.500000f, -0.190983f},
{-0.500000f, 0.190983f, 0.309017f},
{-0.190983f, -0.309017f, -0.500000f}},
};
/* This was generated with
* libraries/AP_Math/tools/geodesic_grid/geodesic_grid.py */
const Matrix3f AP_GeodesicGrid::_mid_inverses[10]{
{{-0.000000f, 1.000000f, -0.618034f},
{ 0.000000f, -1.000000f, -0.618034f},
{-0.618034f, 0.000000f, 1.000000f}},
{{-1.000000f, 0.618034f, -0.000000f},
{-0.000000f, -1.000000f, 0.618034f},
{ 0.618034f, -0.000000f, -1.000000f}},
{{-0.618034f, -0.000000f, -1.000000f},
{ 1.000000f, -0.618034f, -0.000000f},
{-0.618034f, 0.000000f, 1.000000f}},
{{-1.000000f, -0.618034f, -0.000000f},
{ 1.000000f, -0.618034f, 0.000000f},
{-0.000000f, 1.000000f, -0.618034f}},
{{-1.000000f, -0.618034f, 0.000000f},
{ 0.618034f, 0.000000f, 1.000000f},
{ 0.618034f, 0.000000f, -1.000000f}},
{{-0.618034f, -0.000000f, -1.000000f},
{ 1.000000f, 0.618034f, -0.000000f},
{ 0.000000f, -1.000000f, 0.618034f}},
{{ 0.000000f, -1.000000f, -0.618034f},
{ 0.000000f, 1.000000f, -0.618034f},
{ 0.618034f, -0.000000f, 1.000000f}},
{{ 1.000000f, -0.618034f, -0.000000f},
{ 0.000000f, 1.000000f, 0.618034f},
{-0.618034f, 0.000000f, -1.000000f}},
{{ 1.000000f, 0.618034f, -0.000000f},
{-1.000000f, 0.618034f, 0.000000f},
{ 0.000000f, -1.000000f, -0.618034f}},
{{-0.000000f, 1.000000f, 0.618034f},
{-1.000000f, -0.618034f, -0.000000f},
{ 0.618034f, 0.000000f, -1.000000f}},
};
int AP_GeodesicGrid::section(const Vector3f &v, bool inclusive)
{
int i = _triangle_index(v, inclusive);
if (i < 0) {
return -1;
}
int j = _subtriangle_index(i, v, inclusive);
if (j < 0) {
return -1;
}
return 4 * i + j;
}
int AP_GeodesicGrid::_neighbor_umbrella_component(int idx, int comp_idx)
{
if (idx < 3) {
return _neighbor_umbrellas[idx].components[comp_idx];
}
return (_neighbor_umbrellas[idx % 3].components[comp_idx] + 10) % 20;
}
int AP_GeodesicGrid::_from_neighbor_umbrella(int idx,
const Vector3f &v,
const Vector3f &u,
bool inclusive)
{
/* The following comparisons between the umbrella's first and second
* vertices' coefficients work for this algorithm because all vertices'
* vectors are of the same length. */
if (is_equal(u.x, u.y)) {
/* If the coefficients of the first and second vertices are equal, then
* v crosses the first component or the edge formed by the umbrella's
* pivot and forth vertex. */
int comp = _neighbor_umbrella_component(idx, 0);
auto w = _inverses[comp % 10] * v;
if (comp > 9) {
w = -w;
}
float x0 = w[_neighbor_umbrellas[idx % 3].v0_c0];
if (is_zero(x0)) {
if (!inclusive) {
return -1;
}
return comp;
} else if (x0 < 0) {
if (!inclusive) {
return -1;
}
return _neighbor_umbrella_component(idx, u.x < u.y ? 3 : 2);
}
return comp;
}
if (u.y > u.x) {
/* If the coefficient of the second vertex is greater than the first
* one's, then v crosses the first, second or third component. */
int comp = _neighbor_umbrella_component(idx, 1);
auto w = _inverses[comp % 10] * v;
if (comp > 9) {
w = -w;
}
float x1 = w[_neighbor_umbrellas[idx % 3].v1_c1];
float x2 = w[_neighbor_umbrellas[idx % 3].v2_c1];
if (is_zero(x1)) {
if (!inclusive) {
return -1;
}
return _neighbor_umbrella_component(idx, x1 < 0 ? 2 : 1);
} else if (x1 < 0) {
return _neighbor_umbrella_component(idx, 2);
}
if (is_zero(x2)) {
if (!inclusive) {
return -1;
}
return _neighbor_umbrella_component(idx, x2 > 0 ? 1 : 0);
} else if (x2 < 0) {
return _neighbor_umbrella_component(idx, 0);
}
return comp;
} else {
/* If the coefficient of the second vertex is lesser than the first
* one's, then v crosses the first, fourth or fifth component. */
int comp = _neighbor_umbrella_component(idx, 4);
auto w = _inverses[comp % 10] * v;
if (comp > 9) {
w = -w;
}
float x4 = w[_neighbor_umbrellas[idx % 3].v4_c4];
float x0 = w[_neighbor_umbrellas[idx % 3].v0_c4];
if (is_zero(x4)) {
if (!inclusive) {
return -1;
}
return _neighbor_umbrella_component(idx, x4 < 0 ? 0 : 4);
} else if (x4 < 0) {
return _neighbor_umbrella_component(idx, 0);
}
if (is_zero(x0)) {
if (!inclusive) {
return -1;
}
return _neighbor_umbrella_component(idx, x0 > 0 ? 4 : 3);
} else if (x0 < 0) {
return _neighbor_umbrella_component(idx, 3);
}
return comp;
}
}
int AP_GeodesicGrid::_triangle_index(const Vector3f &v, bool inclusive)
{
/* w holds the coordinates of v with respect to the basis comprised by the
* vectors of T_i */
auto w = _inverses[0] * v;
int zero_count = 0;
int balance = 0;
int umbrella = -1;
if (is_zero(w.x)) {
zero_count++;
} else if (w.x > 0) {
balance++;
} else {
balance--;
}
if (is_zero(w.y)) {
zero_count++;
} else if (w.y > 0) {
balance++;
} else {
balance--;
}
if (is_zero(w.z)) {
zero_count++;
} else if (w.z > 0) {
balance++;
} else {
balance--;
}
switch (balance) {
case 3:
/* All coefficients are positive, thus return the first triangle. */
return 0;
case -3:
/* All coefficients are negative, which means that the coefficients for
* -w are positive, thus return the first triangle's opposite. */
return 10;
case 2:
/* Two coefficients are positive and one is zero, thus v crosses one of
* the edges of the first triangle. */
return inclusive ? 0 : -1;
case -2:
/* Analogous to the previous case, but for the opposite of the first
* triangle. */
return inclusive ? 10 : -1;
case 1:
/* There are two possible cases when balance is 1:
*
* 1) Two coefficients are zero, which means v crosses one of the
* vertices of the first triangle.
*
* 2) Two coefficients are positive and one is negative. Let a and b be
* vertices with positive coefficients and c the one with the negative
* coefficient. That means that v crosses the triangle formed by a, b
* and -c. The vector -c happens to be the 3-th vertex, with respect to
* (a, b), of the first triangle's neighbor umbrella with respect to a
* and b. Thus, v crosses one of the components of that umbrella. */
if (zero_count == 2) {
return inclusive ? 0 : -1;
}
if (!is_zero(w.x) && w.x < 0) {
umbrella = 1;
} else if (!is_zero(w.y) && w.y < 0) {
umbrella = 2;
} else {
umbrella = 0;
}
break;
case -1:
/* Analogous to the previous case, but for the opposite of the first
* triangle. */
if (zero_count == 2) {
return inclusive ? 10 : -1;
}
if (!is_zero(w.x) && w.x > 0) {
umbrella = 4;
} else if (!is_zero(w.y) && w.y > 0) {
umbrella = 5;
} else {
umbrella = 3;
}
w = -w;
break;
case 0:
/* There are two possible cases when balance is 1:
*
* 1) The vector v is the null vector, which doesn't cross any section.
*
* 2) One coefficient is zero, another is positive and yet another is
* negative. Let a, b and c be the respective vertices for those
* coefficients, then the statements in case (2) for when balance is 1
* are also valid here.
*/
if (zero_count == 3) {
return -1;
}
if (!is_zero(w.x) && w.x < 0) {
umbrella = 1;
} else if (!is_zero(w.y) && w.y < 0) {
umbrella = 2;
} else {
umbrella = 0;
}
break;
}
assert(umbrella >= 0);
switch (umbrella % 3) {
case 0:
w.z = -w.z;
break;
case 1:
w(w.y, w.z, -w.x);
break;
case 2:
w(w.z, w.x, -w.y);
break;
}
return _from_neighbor_umbrella(umbrella, v, w, inclusive);
}
int AP_GeodesicGrid::_subtriangle_index(const unsigned int triangle_index,
const Vector3f &v,
bool inclusive)
{
/* w holds the coordinates of v with respect to the basis comprised by the
* vectors of the middle triangle of T_i where i is triangle_index */
auto w = _mid_inverses[triangle_index % 10] * v;
if (triangle_index > 9) {
w = -w;
}
if ((is_zero(w.x) || is_zero(w.y) || is_zero(w.z)) && !inclusive) {
return -1;
}
/* At this point, we know that v crosses the icosahedron triangle pointed
* by triangle_index. Thus, we can geometrically see that if v doesn't
* cross its middle triangle, then one of the coefficients will be negative
* and the other ones positive. Let a and b be the non-negative
* coefficients and c the negative one. In that case, v will cross the
* triangle with vertices (a, b, -c). Since we know that v crosses the
* icosahedron triangle and the only sub-triangle that contains the set of
* points (seen as vectors) that cross the triangle (a, b, -c) is the
* middle triangle's neighbor with respect to a and b, then that
* sub-triangle is the one crossed by v. */
if (!is_zero(w.x) && w.x < 0) {
return 3;
}
if (!is_zero(w.y) && w.y < 0) {
return 1;
}
if (!is_zero(w.z) && w.z < 0) {
return 2;
}
/* If x >= 0 and y >= 0 and z >= 0, then v crosses the middle triangle. */
return 0;
}
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