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search.cpp
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2812 lines (2319 loc) · 49.8 KB
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//
// search.cpp:
// Implementation of the functions and classes defined
// in search.hpp.
//
// V2 - bug fixed in
// partial_pf (
// const convex_polygon_pf&, cyclic_uint,
// cyclic_uint, eff_perimeter*);
//
// Build options
//
// The SEARCH_CPP_THROW_RANGE parameter controls the way
// the functions react to the wrong arguments:
//
// If SEARCH_CPP_THROW_RANGE is defined, the functions will throw
// (1) std::invalid_argument if any of the arguments is NaN;
// (2) std::out_of_range if any of the arguments is a finite
// number of +/- infinity but is out of the specified range.
//
// If SEARCH_CPP_THROW_RANGE is not defined, the functions will
// return NaN in all of the above cases.
//
#define SEARCH_CPP_THROW_RANGE
#include <limits>
#include <string>
#include "search.hpp"
namespace search
{
//
// Some internal definitions only used in the
// implementation of namespace search
//
//
// Name of the namespace used to report errors
//
static const std::string name_of_namespace ("search::");
//
// Positive infinity
//
static const double pos_infinity (std::numeric_limits<double>::infinity());
//
// Quiet NaN
// Note: this initialization is due to the STLport bug.
// Replace when possible with
// const double qnan (std::numeric_limits<double>::quiet_nan());
//
static const double qnan (pos_infinity/pos_infinity);
//
// Auxiliary function used by **pf_circle functions.
// It calculates
// aux_1(beta) = beta - tan(beta) + (pi/2 - beta)*tan^2(beta).
//
double aux_1 (double beta);
//
// Auxiliary function used by **pf_sphere_3d functions.
// It calculates
// aux_2(beta) = 1 - cos(beta).
//
double aux_2 (double beta);
#ifdef SEARCH_CPP_THROW_RANGE
//
// Range check for inbound arguments
//
bool out_of_range (bool cond, const std::string& name_of_fun)
{
if (!cond)
{
std::string what (name_of_namespace);
what += name_of_fun;
throw std::out_of_range (what);
}
else
{
return false;
}
}
//
// Check for NaN for inbound arguments (3 overloads)
//
bool is_nan (double x, const std::string& name_of_fun)
{
if (_isnan (x))
{
std::string what (name_of_namespace);
what += name_of_fun;
throw std::invalid_argument (what);
}
else
{
return false;
}
}
bool is_nan (double x, double y, const std::string& name_of_fun)
{
if (_isnan (x) || _isnan (y))
{
std::string what (name_of_namespace);
what += name_of_fun;
throw std::invalid_argument (what);
}
else
{
return false;
}
}
bool is_nan (double x, double y, double z, const std::string& name_of_fun)
{
if (_isnan (x) || _isnan (y) || _isnan (z))
{
std::string what (name_of_namespace);
what += name_of_fun;
throw std::invalid_argument (what);
}
else
{
return false;
}
}
#else // SEARCH_CPP_THROW_RANGE
//
// Range check for inbound arguments
//
bool out_of_range (bool cond, const std::string& name_of_fun)
{
return !cond;
}
//
// Check for NaN for inbound arguments (3 overloads)
//
bool is_nan (double x, const std::string& name_of_fun)
{
return _isnan (x) != 0;
}
bool is_nan (double x, double y, const std::string& name_of_fun)
{
return _isnan (x) || _isnan (y);
}
bool is_nan (double x, double y, double z, const std::string& name_of_fun)
{
return _isnan (x) || _isnan (y) || _isnan (z);
}
#endif // SEARCH_CPP_THROW_RANGE
} // namespace search
//
// The implementation of gain functions and perimeter functions
// follows, and it is pretty self-explanatory. Some comments
// are given when necessary. For more info, refer to theory of
// guaranteed search.
//
double search::f (double w, double r)
{
static const std::string name_of_fun ("f(double,double)");
if (is_nan (w, r, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= w && w <= 1.0, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= r, name_of_fun))
{
return qnan;
}
return 2.0*r*(w*(pi - acos (w)) + sqrt (1.0 - w*w));
}
double search::g (double w, double r)
{
static const std::string name_of_fun ("g(double,double)");
if (is_nan (w, r, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= w && w <= 1.0, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= r, name_of_fun))
{
return qnan;
}
const double tmp (1.0 + w);
return pi*r*r*tmp*tmp;
}
double search::h (double w, double r, double a)
{
static const std::string name_of_fun ("h(double,double,double)");
if (is_nan (w, r, a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= w && w <= 1.0, name_of_fun))
{
return qnan;
}
// this check may be taken away if needed:
// the function will work on any value of r,
// provided that r/a doesn't evaluate to infinity
if (out_of_range (0.0 <= r && r <= pi*a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 < a, name_of_fun))
{
return qnan;
}
r /= a;
const double r_abs (fabs (r));
const double r_limit (1.0e-10);
if (w == 1.0 && r_abs <= r_limit)
{
// A very special case: w=1 and |r|<<1.
// The general algorithm is in trouble when
// r^4 evaluates to 0, that is, |r^4| ~ 1e-310.
// We'll use
// h(1,r) = 4/sqrt(3) + 2*pi*r + O(r^3)
// in the vicinity of r = 0 where this approximation
// is accurate enough.
const double lim (4.0/sqrt (3.0));
if (r > 0.0)
{
return a*(lim + 2.0*pi*r);
}
else
if (r == 0.0)
{
// For h(1,0) we'll return 0, while, strictly
// speaking, h(1,0) is not defined:
// lim h(w,0) = 0 as w -> 1, but
// lim h(1,r) = 4/sqrt(3) as r -> +0.
// We must return something, and I prefer
// to return 0 because h(w,r) is odd in respect to r.
return 0.0;
}
else // r < 0
{
return a*(-lim + 2.0*pi*r);
}
}
// depending on the relation between w and r,
// different algorithms will be used
const double r_limit_1 (1.0e-5);
const double r_limit_2 (2.0e-5);
const double r_limit_3 (pi/2.0);
int wr_case;
if (r_abs > r_limit_3)
{
wr_case = 1;
}
else
if (r_abs > r_limit_2)
{
wr_case = 2;
}
else
if (r_abs*r_abs > (1.0 - w)/100.0) // 100 is empirical
{
wr_case = 2;
}
else
if (r_abs > r_limit_1)
{
wr_case = 3;
}
else
{
// wr_case = 4; can handle this case immediately:
return a*f (w, r);
}
// intermediate values
const double w2 (w*w);
const double r2 (r*r);
const double c (cos (r));
const double s (sin (r));
const double ws (w*s);
const double ws_r (r==0.0 ? w : ws/r);
double x; // x = 1 - (w*sin(r)/r)^2
double y; // y = cos(r) - w*sin(r)/r
double z; // z = cos(r) - w^2*sin(r)/r
if (wr_case == 1)
{
x = 1.0 - ws_r*ws_r;
y = c - ws_r;
z = c - ws_r*w;
}
else // wr_case is 2 or 3
{
// Expand into power series to ensure
// uniform accuracy over the range -pi/2 <= r <= pi/2.
// I used Mathematica 2.2 to get the coefficients.
double p[12]; // even powers of r
p[0] = r2;
for (int ind = 1; ind < 12; ++ind)
{
p[ind] = p[ind-1]*r2;
}
x = (1.0 - w)*(1.0 + w) + w2*(
+ p[0] / 3.0
- p[1] * 2.0 / 45.0
+ p[2] / 315.0
- p[3] * 2.0 / 14175.0
+ p[4] * 2.0 / 467775.0
- p[5] * 4.0 / 42567525.0
+ p[6] / 638512875.0
- p[7] * 2.0 / 97692469875.0
+ p[8] * 2.0 / 9280784638125.0
- p[9] * 4.0 / 2143861251406875.0
+ p[10]* 2.0 / 147926426347074375.0
- p[11]* 4.0 / 48076088562799171875.0
);
y = (1.0 - w)
- p[0] * (3.0 - w) / 6.0
+ p[1] * (5.0 - w) / 120.0
- p[2] * (7.0 - w) / 5040.0
+ p[3] * (9.0 - w) / 362880.0
- p[4] * (11.0 - w)/ 39916800.0
+ p[5] * (13.0 - w)/ 6227020800.0
- p[6] * (15.0 - w)/ 1307674368000.0
+ p[7] * (17.0 - w)/ 355687428096000.0
- p[8] * (19.0 - w)/ 121645100408832000.0
+ p[9] * (21.0 - w)/ 51090942171709440000.0;
z = (1.0 - w2)
- p[0] * (3.0 - w2) / 6.0
+ p[1] * (5.0 - w2) / 120.0
- p[2] * (7.0 - w2) / 5040.0
+ p[3] * (9.0 - w2) / 362880.0
- p[4] * (11.0 - w2)/ 39916800.0
+ p[5] * (13.0 - w2)/ 6227020800.0
- p[6] * (15.0 - w2)/ 1307674368000.0
+ p[7] * (17.0 - w2)/ 355687428096000.0
- p[8] * (19.0 - w2)/ 121645100408832000.0
+ p[9] * (21.0 - w2)/ 51090942171709440000.0;
}
const double h1 (2.0*ws*(pi - acos (ws_r)));
const double h2 ((2.0*y*z + 2.0*r2*x*x) /
(sqrt (x)*sqrt (z*z + r2*x*x)));
const double h3 ( -2.0*y*c/(sqrt (x)));
double result (h1 + (h2 + h3)/r);
if (wr_case == 3)
{
result = (
result * (r_limit_2 - r_abs) +
f (w, r)* (r_abs - r_limit_1)
) /
(r_limit_2 - r_limit_1);
}
return a*result;
}
double search::pf_plane (double z)
{
static const std::string name_of_fun ("pf_plane(double)");
if (is_nan (z, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= z, name_of_fun))
{
return qnan;
}
return 2.0*sqrt (pi)*sqrt (z);
}
double search::ipf_plane (double p)
{
static const std::string name_of_fun ("ipf_plane(double)");
if (is_nan (p, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= p, name_of_fun))
{
return qnan;
}
return (p/(4.0*pi))*p;
}
double search::pf_angle (double z, double theta)
{
static const std::string name_of_fun ("pf_angle(double,double)");
if (is_nan (z, theta, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= z, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 < theta && theta < 2.0*pi, name_of_fun))
{
return qnan;
}
return sqrt (2.0*std::min (theta, pi))*sqrt (z);
}
double search::ipf_angle (double p, double theta)
{
static const std::string name_of_fun ("ipf_angle(double,double)");
if (is_nan (p, theta, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= p, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 < theta && theta < 2.0*pi, name_of_fun))
{
return qnan;
}
return p/(2.0*std::min (theta, pi))*p;
}
double search::pf_sphere (double z, double a)
{
static const std::string name_of_fun ("pf_sphere(double,double)");
if (is_nan (z, a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= z && z <= 4.0*pi*a*a && z < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 < a, name_of_fun))
{
return qnan;
}
return 2.0*sqrt (z)*sqrt (pi - z/(4.0*a*a));
}
double search::ipf_sphere (double p, double a)
{
static const std::string name_of_fun ("ipf_sphere(double,double)");
if (is_nan (p, a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= p && p <= 2.0*pi*a && p < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 < a, name_of_fun))
{
return qnan;
}
const double tmp (p/(2.0*pi*a));
return (2.0*pi)*(a*(1.0 - sqrt (1.0 - tmp*tmp))*a);
}
double search::aux_1 (double beta)
{
const int size (22);
double b[size];
b[1] = beta;
for (int ind = 2; ind < size; ++ind)
{
b[ind] = b[ind-1]*beta;
}
const double num (
+ b[2] * pi / 2.0
- b[3] * 4.0 / 3.0
- b[4] * pi / 6.0
+ b[5] * 8.0 / 15.0
+ b[6] * pi / 45.0
- b[7] * 8.0 / 105.0
- b[8] * pi / 630.0
+ b[9] * 16.0 / 2835.0
+ b[10]* pi / 14175.0
- b[11]* 8.0 / 31185.0
- b[12]* pi / 467775.0
+ b[13]* 16.0 / 2027025.0
+ b[14]* pi * 2.0/ 42567525.0
- b[15]* 16.0 / 91216125.0
- b[16]* pi / 1277025750.0
+ b[17]* 32.0 / 10854718875.0
+ b[18]* pi / 97692469875.0
- b[19]* 8.0 / 206239658625.0
- b[20]* pi / 9280784638125.0
+ b[21]* 16.0 / 38979295480125.0
);
const double c (cos (beta));
return num/(c*c);
}
//
// Note: in the next 4 functions **pf_circle
// the following system is solved:
//
// (1) p/a = +/- (pi - 2*beta)*tan(beta),
// (2) z/a^2 = beta - tan(beta) + (pi/2 - beta)*tan^2(beta).
//
// In perimeter functions, z is known and p is sought.
// In inverse perimeter functions, p is known and z is sought.
// In some cases, the substitute beta = alpha - pi/2 is used.
//
double search::pf_circle (double z, double a)
{
static const std::string name_of_fun ("pf_circle(double,double)");
if (is_nan (z, a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= z && z <= pi*a*a && z < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= a, name_of_fun))
{
return qnan;
}
if (a == 0.0)
{
return 0.0;
}
double z_norm (z/a);
z_norm /= a;
if (z_norm == 0.0)
{
return sqrt (2.0*pi)*sqrt (z);
}
z = z_norm;
if (z > pi/2.0)
{
z = pi - z;
}
if (z == 0.0)
{
return 0.0;
}
double beta_left (0.0);
double beta_right (pi/2.0);
double beta;
double beta_left_old;
double beta_right_old;
double result;
if (z < pi/2.0 - 1.0)
{
for (;;)
{
beta = (beta_left + beta_right)/2.0;
beta_left_old = beta_left;
beta_right_old = beta_right;
if (aux_1 (beta) > z)
{
beta_right = beta;
if (beta_right >= beta_right_old)
{
break;
}
}
else
{
beta_left = beta;
if (beta_left <= beta_left_old)
{
break;
}
}
}
result = (pi - 2.0*beta)*sin (beta)/cos (beta);
}
else
{
for (;;)
{
beta = (beta_left + beta_right)/2.0;
beta_left_old = beta_left;
beta_right_old = beta_right;
if (beta == pi/2.0)
{
break;
}
const double t (sin (beta)/cos (beta));
if (beta - t + t*(pi/2.0 - beta)*t > z)
{
beta_right = beta;
if (beta_right >= beta_right_old)
{
break;
}
}
else
{
beta_left = beta;
if (beta_left <= beta_left_old)
{
break;
}
}
}
const double c (sin (pi/2.0 - beta));
if (c == 0.0)
{
result = 2.0;
}
else
{
result = (pi - 2.0*beta)*sin (beta)/c;
}
}
return a*result;
}
double search::ipf_circle (double p, double a)
{
static const std::string name_of_fun ("ipf_circle(double,double)");
if (is_nan (p, a, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= p && p <= 2.0*a && p < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= a, name_of_fun))
{
return qnan;
}
if (a == 0.0)
{
return 0.0;
}
const double p_norm (p/a);
if (p_norm == 0.0)
{
return (p/(2.0*pi))*p;
}
p = p_norm;
double beta_left (0.0);
double beta_right (pi/2.0);
double beta;
double beta_left_old;
double beta_right_old;
double result;
for (;;)
{
beta = (beta_left + beta_right)/2.0;
beta_left_old = beta_left;
beta_right_old = beta_right;
if (beta == pi/2.0)
{
break;
}
if ((pi - 2.0*beta)*sin (beta)/cos (beta) > p)
{
beta_right = beta;
if (beta_right >= beta_right_old)
{
break;
}
}
else
{
beta_left = beta;
if (beta_left <= beta_left_old)
{
break;
}
}
}
if (beta < pi/4.0)
{
result = aux_1 (beta);
}
else
{
if (beta == pi/2.0)
{
result = pi/2.0;
}
else
{
const double t (sin (beta)/cos (beta));
result = beta - t + t*(pi/2.0 - beta)*t;
}
}
return a*result*a;
}
double search::opf_circle (double z, double a)
{
static const std::string name_of_fun ("opf_circle(double,double)");
if (is_nan (z, a, name_of_fun))
{
return qnan;
}
if (out_of_range (z < pos_infinity || a < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= z, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= a, name_of_fun))
{
return qnan;
}
if (z == pos_infinity)
{
return pos_infinity;
}
if (a == 0.0)
{
return 2.0*sqrt (pi)*sqrt (z);
}
double z_norm (z/a);
z_norm /= a;
if (z_norm == 0.0)
{
return sqrt (2.0*pi)*sqrt(z);
}
if (z_norm == pos_infinity)
{
return 2.0*(sqrt (pi)*sqrt (z) - a);
}
z = z_norm;
double result;
if (z < pi/2.0 - 1.0)
{
double beta_left (-pi/2.0);
double beta_right (0.0);
double beta;
double beta_left_old;
double beta_right_old;
for (;;)
{
beta = (beta_left + beta_right)/2.0;
beta_left_old = beta_left;
beta_right_old = beta_right;
if (aux_1 (beta) < z)
{
beta_right = beta;
if (beta_right >= beta_right_old)
{
break;
}
}
else
{
beta_left = beta;
if (beta_left <= beta_left_old)
{
break;
}
}
}
result = (2.0*beta - pi)*sin (beta)/cos (beta);
}
else
{
double alpha_left (0.0);
double alpha_right (pi/2.0);
double alpha;
double alpha_left_old;
double alpha_right_old;
for (;;)
{
alpha = (alpha_left + alpha_right)/2.0;
alpha_left_old = alpha_left;
alpha_right_old = alpha_right;
const double t (sin (alpha)/cos (alpha));
const double tt (t*t);
if (tt == 0.0)
{
alpha = 0.0;
break;
}
if (-pi/2.0 + alpha + 1/t + (pi - alpha)/(t*t) < z)
{
alpha_right = alpha;
if (alpha_right >= alpha_right_old)
{
break;
}
}
else
{
alpha_left = alpha;
if (alpha_left <= alpha_left_old)
{
break;
}
}
}
result = 2.0*(pi - alpha)*cos (alpha)/sin (alpha);
}
return a*result;
}
double search::iopf_circle (double p, double a)
{
static const std::string name_of_fun ("iopf_circle(double,double)");
if (is_nan (p, a, name_of_fun))
{
return qnan;
}
if (out_of_range (p < pos_infinity || a < pos_infinity, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= p, name_of_fun))
{
return qnan;
}
if (out_of_range (0.0 <= a, name_of_fun))
{
return qnan;
}
if (p == pos_infinity)
{
return pos_infinity;
}
if (a == 0.0)
{
return (p/(4.0*pi))*p;
}
double p_norm (p/a);
if (p_norm == 0.0)
{
return (p/(2.0*pi))*p;
}
if (p_norm == pos_infinity)
{
const double tmp (p/2.0 + a);
return tmp*(1.0/pi)*tmp;
}
p = p_norm;
double result;