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aka_geometry.hh
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/**
* @file aka_geometry.hh
*
* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
*
* @date creation: Fri Jan 04 2013
* @date last modification: Tue Sep 02 2014
*
* @brief geometric operations
*
* @section LICENSE
*
* Copyright (©) 2014 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu 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 Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
#ifndef __AKANTU_GEOMETRY_HH__
#define __AKANTU_GEOMETRY_HH__
#include <iostream>
#include <tuple>
#include "aka_point.hh"
#include "aka_plane.hh"
#include "aka_math.hh"
__BEGIN_AKANTU__
// predicates
// combined tolerance test, from Christer Ericson
template <typename T>
typename std::enable_if<std::is_floating_point<T>::value, bool>::type
equal(T x, T y, T tol = 2*std::numeric_limits<T>::epsilon()) {
T absTol = tol;
T relTol = absTol;
// here both tolerances are equal, but the code is written
// like this so that tolerance values can be assigned indepently
// in the future
return std::abs(x - y) <= std::max(absTol, relTol * std::max(std::abs(x), std::abs(y)));
}
// combined tolerance test, from Christer Ericson
template <typename T>
typename std::enable_if<std::is_integral<T>::value, bool>::type
equal(T x, T y)
{ return x == y; }
Real left_turn(const Point<2>& p, const Point<2>& q, const Point<2>& r);
// closest point computations
//! Computes the closest point laying on a segment to a point
/*! Given segment \c ab and point \c c, computes closest point \c d on ab.
* Also returns \c t for the position of the point: a + t*(b - a)
*/
template <int d, typename T>
Point<d,T> closest_point_to_segment(const Point<d,T>& c,
const Point<d,T>& a,
const Point<d,T>& b) {
Point<d,T> ab = b - a;
// project c onto ab, computing parameterized position d(t) = a + t*(b – a)
T t = (c - a)*ab / sqrt(ab*ab);
// if outside segment, clamp t (and therefore d) to the closest endpoint
if (t < 0.)
t = 0.;
else if (t > 1.)
t = 1.;
// compute projected position from the clamped t
return a + t * ab;
}
//! Predicate that checks if a point has a projection on a line segment
/*! Given segment \c ab and point \c c, checks if the point has a projection in the segment.
*/
template <int d, typename T>
bool has_projection(const Point<d,T>& c,
const Point<d,T>& a,
const Point<d,T>& b) {
Point<d,T> ab = b - a;
// project c onto ab, computing parameterized position d(t) = a + t*(b – a)
T t = (c - a)*ab / (ab*ab);
return t > 0. && t < 1.;
}
//! Tests if a point has a projection to a triangle
/*! This function uses the concept of Voronoi regions to determine
* if a point has a projection within a triangle defined by points
* \c a, \c b, and \c c.
*/
template <typename T>
bool point_has_projection_to_triangle(const Point<3,T>& p,
const Point<3,T>& a,
const Point<3,T>& b,
const Point<3,T>& c) {
typedef Point<3,T> point_type;
// obtain plane of the triangle
Plane pi(a,b,c);
// get point in the plane closest to p
point_type q = closest_point_to_plane(p,pi);
// return if point is within the triangle
if (is_point_in_triangle(q, a, b, c))
return true;
return false;
}
//! Tests if point P lies inside a triangle
/*! The triangle is defined by points \c a, \c b and \c c.
*/
template <typename T>
bool is_point_in_triangle(const Point<3,T>& p,
const Point<3,T>& a,
const Point<3,T>& b,
const Point<3,T>& c) {
typedef Point<3,T> point_type;
point_type v0 = b-a, v1 = c-a, v2 = p-a;
Real d00 = v0*v0;
Real d01 = v0*v1;
Real d11 = v1*v1;
Real d20 = v2*v0;
Real d21 = v2*v1;
Real denom = d00*d11 - d01*d01;
// compute parametric coordinates
Real v = (d11 * d20 - d01 * d21) / denom;
Real w = (d00 * d21 - d01 * d20) / denom;
return v >= 0. && w >= 0. && v + w <= 1.;
}
//! Compute the closest point to a triangle
/*! This function uses the concept of Voronoi regions to determine
* the closest point \c p to a triangle defined by points \c a, \c b
* \c c.
*/
template <typename T>
Point<3,T> closest_point_to_triangle(const Point<3,T>& p,
const Point<3,T>& a,
const Point<3,T>& b,
const Point<3,T>& c) {
typedef Point<3,T> point_type;
// check if P in vertex region outside A
point_type ab = b - a;
point_type ac = c - a;
point_type ap = p - a;
// compute scalar products
T d1 = ab * ap;
T d2 = ac * ap;
if (d1 <= 0. && d2 <= 0.)
return a; // barycentric coordinates (1,0,0)
// check if P in vertex region outside B
point_type bp = p - b;
T d3 = ab * bp;
T d4 = ac * bp;
if (d3 >= 0.0f && d4 <= d3)
return b; // barycentric coordinates (0,1,0)
// check if P in edge region of AB, if so return projection of P onto AB
T vc = d1*d4 - d3*d2;
if (vc <= 0. && d1 >= 0. && d3 <= 0.) {
T v = d1 / (d1 - d3);
return a + v * ab; // barycentric coordinates (1-v,v,0)
}
// check if P in vertex region outside C
point_type cp = p - c;
T d5 = ab * cp;
T d6 = ac * cp;
if (d6 >= 0.0f && d5 <= d6)
return c; // barycentric coordinates (0,0,1)
// check if P in edge region of AC, if so return projection of P onto AC
T vb = d5*d2 - d1*d6;
if (vb <= 0.0f && d2 >= 0.0f && d6 <= 0.0f) {
T w = d2 / (d2 - d6);
return a + w * ac; // barycentric coordinates (1-w,0,w)
}
// Check if P in edge region of BC, if so return projection of P onto BC
T va = d3*d6 - d5*d4;
if (va <= 0.0f && (d4 - d3) >= 0.0f && (d5 - d6) >= 0.0f) {
T w = (d4 - d3) / ((d4 - d3) + (d5 - d6));
return b + w * (c - b); // barycentric coordinates (0,1-w,w)
}
// P inside face region. Compute Q through its barycentric coordinates (u,v,w)
T denom = 1.0f / (va + vb + vc);
T v = vb * denom;
T w = vc * denom;
return a + ab*v + ac*w; // = u*a + v*b + w*c, u = va*denom = 1.0f - v - w
}
template <typename T>
Point<3,T> closest_point_to_plane(const Point<3,T>& q, const Plane& p) {
typedef Point<3,T> point_type;
const point_type& n = p.normal();
T t = (n*q - p.distance()) / (n*n);
return q - t * n;
}
//! Compute the closest point to a triangle
/*! Obtains the plane of the triangle and checks if the point lies inside the
* triangle. If not, it computes the closest point to each of the triangle
* edges.
*/
template <typename T>
Point<3,T> naive_closest_point_to_triangle(const Point<3,T>& p,
const Point<3,T>& a,
const Point<3,T>& b,
const Point<3,T>& c) {
typedef Point<3,T> point_type;
// obtain plane of the triangle
Plane pi(a,b,c);
// get point in the plane closest to p
point_type q = closest_point_to_plane(p,pi);
// return if point is within the triangle
if (is_point_in_triangle(q, a, b, c))
return q;
// else get the closest point taking into account all edges
// first edge
q = closest_point_to_segment(p, a, b);
T d = (q-p).sq_norm();
// second edge
point_type r = closest_point_to_segment(p, b, c);
T d2 = (r-p).sq_norm();
if (d2 < d) {
q = r;
d = d2;
}
// third edge
r = closest_point_to_segment(p,c,a);
d2 = (r-p).sq_norm();
if (d2 < d)
q = r;
// return closest point
return q;
}
// intersect point p with velocity v with plane
// the function returns collision time and point of contact
// this function does not consider acceleration
template <typename T>
std::tuple<Real, Point<3,T> >
moving_point_against_plane(const Point<3,T>& p, const Point<3,T>& v, Plane& pi) {
typedef Point<3,T> point_type;
// compute distance of point to plane
Real dist = pi.normal()*p - pi.distance();
// if point already in the plane
if (std::abs(dist) <= 1e-10)
return std::make_tuple(0., p);
else {
Real denom = pi.normal()*v;
// no intersection as poin moving parallel to or away from plane
if (denom * dist >= 0.)
return std::make_tuple(inf, point_type());
// point moving towards the plane
else {
// point is moving towards the plane
Real t = -dist/denom;
return std::make_tuple(t, p + t*v);
}
}
}
template <int dim, typename T>
std::tuple<Real, Point<dim,T> >
moving_point_against_point(const Point<dim,T>& s1, const Point<dim,T>& s2, /* point centers */
const Point<dim,T>& v1, const Point<dim,T>& v2) /* point velocities */ {
typedef Point<dim,T> point_type;
typedef typename Point<dim,T>::value_type value_type;
// vector between points
point_type s = s2 - s1;
// relative motion of s1 with respect to stationary s0
point_type v = v2 - v1;
value_type c = s*s;
// if points within tolerance
if (equal(s.sq_norm(), value_type()))
return std::make_tuple(value_type(), s1);
value_type epsilon = 2*std::numeric_limits<T>::epsilon();;
value_type a = v*v;
// if points not moving relative to each other
if (a < epsilon)
return std::make_tuple(inf, point_type());
value_type b = v*s;
// if points not moving towards each other
if (b >= 0.)
return std::make_tuple(inf, point_type());
value_type d = b*b - a*c;
// if no real-valued root (d < 0), points do not intersect
if (d >= 0.) {
value_type ts = (-b - sqrt(d))/a;
point_type q = s1+v1*ts;
return std::make_tuple(ts, q);
}
return std::make_tuple(inf, point_type());
}
__END_AKANTU__
#endif /* __AKANTU_GEOMETRY_HH__ */

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