math.cpp

Go to the documentation of this file.

#include "linden_common.h"
#include "lltut.h"

#include "llcrc.h"
#include "llline.h"
#include "llmath.h"
#include "llrand.h"
#include "llsphere.h"
#include "lluuid.h"
#include "v3math.h"

namespace tut
{
        struct math_data
        {
        };
        typedef test_group<math_data> math_test;
        typedef math_test::object math_object;
        tut::math_test tm("basic_linden_math");

        template<> template<>
        void math_object::test<1>()
        {
                S32 val = 89543;
                val = llabs(val);
                ensure("integer absolute value 1", (89543 == val));
                val = -500;
                val = llabs(val);
                ensure("integer absolute value 2", (500 == val));
        }

        template<> template<>
        void math_object::test<2>()
        {
                F32 val = -2583.4f;
                val = llabs(val);
                ensure("float absolute value 1", (2583.4f == val));
                val = 430903.f;
                val = llabs(val);
                ensure("float absolute value 2", (430903.f == val));
        }

        template<> template<>
        void math_object::test<3>()
        {
                F64 val = 387439393.987329839;
                val = llabs(val);
                ensure("double absolute value 1", (387439393.987329839 == val));
                val = -8937843.9394878;
                val = llabs(val);
                ensure("double absolute value 2", (8937843.9394878 == val));
        }

        template<> template<>
        void math_object::test<4>()
        {
                F32 val = 430903.9f;
                S32 val1 = lltrunc(val);
                ensure("float truncate value 1", (430903 == val1));
                val = -2303.9f;
                val1 = lltrunc(val);
                ensure("float truncate value 2", (-2303  == val1));
        }

        template<> template<>
        void math_object::test<5>()
        {
                F64 val = 387439393.987329839 ;
                S32 val1 = lltrunc(val);
                ensure("float truncate value 1", (387439393 == val1));
                val = -387439393.987329839;
                val1 = lltrunc(val);
                ensure("float truncate value 2", (-387439393  == val1));
        }

        template<> template<>
        void math_object::test<6>()
        {
                F32 val = 430903.2f;
                S32 val1 = llfloor(val);
                ensure("float llfloor value 1", (430903 == val1));
                val = -430903.9f;
                val1 = llfloor(val);
                ensure("float llfloor value 2", (-430904 == val1));
        }

        template<> template<>
        void math_object::test<7>()
        {
                F32 val = 430903.2f;
                S32 val1 = llceil(val);
                ensure("float llceil value 1", (430904 == val1));
                val = -430903.9f;
                val1 = llceil(val);
                ensure("float llceil value 2", (-430903 == val1));
        }

        template<> template<>
        void math_object::test<8>()
        {
                F32 val = 430903.2f;
                S32 val1 = llround(val);
                ensure("float llround value 1", (430903 == val1));
                val = -430903.9f;
                val1 = llround(val);
                ensure("float llround value 2", (-430904 == val1));
        }

        template<> template<>
        void math_object::test<9>()
        {
                F32 val = 430905.2654f, nearest = 100.f;
                val = llround(val, nearest);
                ensure("float llround value 1", (430900 == val));
                val = -430905.2654f, nearest = 10.f;
                val = llround(val, nearest);
                ensure("float llround value 1", (-430910 == val));
        }

        template<> template<>
        void math_object::test<10>()
        {
                F64 val = 430905.2654, nearest = 100.0;
                val = llround(val, nearest);
                ensure("double llround value 1", (430900 == val));
                val = -430905.2654, nearest = 10.0;
                val = llround(val, nearest);
                ensure("double llround value 1", (-430910.00000 == val));
        }

        template<> template<>
        void math_object::test<11>()
        {
                const F32       F_PI            = 3.1415926535897932384626433832795f;
                F32 angle = 3506.f;
                angle =  llsimple_angle(angle);
                ensure("llsimple_angle  value 1", (angle <=F_PI && angle >= -F_PI));
                angle = -431.f;
                angle =  llsimple_angle(angle);
                ensure("llsimple_angle  value 1", (angle <=F_PI && angle >= -F_PI));
        }
}

namespace tut
{
        struct uuid_data
        {
                LLUUID id;
        };
        typedef test_group<uuid_data> uuid_test;
        typedef uuid_test::object uuid_object;
        tut::uuid_test tu("uuid");

        template<> template<>
        void uuid_object::test<1>()
        {
                ensure("uuid null", id.isNull());
                id.generate();
                ensure("generate not null", id.notNull());
                id.setNull();
                ensure("set null", id.isNull());
        }

        template<> template<>
        void uuid_object::test<2>()
        {
                id.generate();
                LLUUID a(id);
                ensure_equals("copy equal", id, a);
                a.generate();
                ensure_not_equals("generate not equal", id, a);
                a = id;
                ensure_equals("assignment equal", id, a);
        }

        template<> template<>
        void uuid_object::test<3>()
        {
                id.generate();
                LLUUID copy(id);
                LLUUID mask;
                mask.generate();
                copy ^= mask;
                ensure_not_equals("mask not equal", id, copy);
                copy ^= mask;
                ensure_equals("mask back", id, copy);
        }

        template<> template<>
        void uuid_object::test<4>()
        {
                id.generate();
                std::string id_str = id.asString();
                LLUUID copy(id_str.c_str());
                ensure_equals("string serialization", id, copy);
        }
        
}

namespace tut
{
        struct crc_data
        {
        };
        typedef test_group<crc_data> crc_test;
        typedef crc_test::object crc_object;
        tut::crc_test tc("crc");

        template<> template<>
        void crc_object::test<1>()
        {
                /* Test buffer update and individual char update */
                const char TEST_BUFFER[] = "hello &#$)$&Nd0";
                LLCRC c1, c2;
                c1.update((U8*)TEST_BUFFER, sizeof(TEST_BUFFER) - 1);
                char* rh = (char*)TEST_BUFFER;
                while(*rh != '\0')
                {
                        c2.update(*rh);
                        ++rh;
                }
                ensure_equals("crc update 1", c1.getCRC(), c2.getCRC());
        }

        template<> template<>
        void crc_object::test<2>()
        {
                /* Test mixing of buffer and individual char update */
                const char TEST_BUFFER1[] = "Split Buffer one $^%$%#@$";
                const char TEST_BUFFER2[] = "Split Buffer two )(8723#5dsds";
                LLCRC c1, c2;
                c1.update((U8*)TEST_BUFFER1, sizeof(TEST_BUFFER1) - 1);
                char* rh = (char*)TEST_BUFFER2;
                while(*rh != '\0')
                {
                        c1.update(*rh);
                        ++rh;
                }

                rh = (char*)TEST_BUFFER1;
                while(*rh != '\0')
                {
                        c2.update(*rh);
                        ++rh;
                }
                c2.update((U8*)TEST_BUFFER2, sizeof(TEST_BUFFER2) - 1);

                ensure_equals("crc update 2", c1.getCRC(), c2.getCRC());
        }
}

namespace tut
{
        struct sphere_data
        {
        };
        typedef test_group<sphere_data> sphere_test;
        typedef sphere_test::object sphere_object;
        tut::sphere_test tsphere("LLSphere");

        template<> template<>
        void sphere_object::test<1>()
        {
                // test LLSphere::contains() and ::overlaps()
                S32 number_of_tests = 10;
                for (S32 test = 0; test < number_of_tests; ++test)
                {
                        LLVector3 first_center(1.f, 1.f, 1.f);
                        F32 first_radius = 3.f;
                        LLSphere first_sphere( first_center, first_radius );
        
                        F32 half_millimeter = 0.0005f;
                        LLVector3 direction( ll_frand(2.f) - 1.f, ll_frand(2.f) - 1.f, ll_frand(2.f) - 1.f);
                        direction.normalize();
        
                        F32 distance = ll_frand(first_radius - 2.f * half_millimeter);
                        LLVector3 second_center = first_center + distance * direction;
                        F32 second_radius = first_radius - distance - half_millimeter;
                        LLSphere second_sphere( second_center, second_radius );
                        ensure("first sphere should contain the second", first_sphere.contains(second_sphere));
                        ensure("first sphere should overlap the second", first_sphere.overlaps(second_sphere));
        
                        distance = first_radius + ll_frand(first_radius);
                        second_center = first_center + distance * direction;
                        second_radius = distance - first_radius + half_millimeter;
                        second_sphere.set( second_center, second_radius );
                        ensure("first sphere should NOT contain the second", !first_sphere.contains(second_sphere));
                        ensure("first sphere should overlap the second", first_sphere.overlaps(second_sphere));
        
                        distance = first_radius + ll_frand(first_radius) + half_millimeter;
                        second_center = first_center + distance * direction;
                        second_radius = distance - first_radius - half_millimeter;
                        second_sphere.set( second_center, second_radius );
                        ensure("first sphere should NOT contain the second", !first_sphere.contains(second_sphere));
                        ensure("first sphere should NOT overlap the second", !first_sphere.overlaps(second_sphere));
                }
        }

        template<> template<>
        void sphere_object::test<2>()
        {
                // test LLSphere::getBoundingSphere()
                S32 number_of_tests = 100;
                S32 number_of_spheres = 10;
                F32 sphere_center_range = 32.f;
                F32 sphere_radius_range = 5.f;

                for (S32 test = 0; test < number_of_tests; ++test)
                {
                        // gegnerate a bunch of random sphere
                        std::vector< LLSphere > sphere_list;
                        for (S32 sphere_count=0; sphere_count < number_of_spheres; ++sphere_count)
                        {
                                LLVector3 direction( ll_frand(2.f) - 1.f, ll_frand(2.f) - 1.f, ll_frand(2.f) - 1.f);
                                direction.normalize();
                                F32 distance = ll_frand(sphere_center_range);
                                LLVector3 center = distance * direction;
                                F32 radius = ll_frand(sphere_radius_range);
                                LLSphere sphere( center, radius );
                                sphere_list.push_back(sphere);
                        }

                        // compute the bounding sphere
                        LLSphere bounding_sphere = LLSphere::getBoundingSphere(sphere_list);

                        // make sure all spheres are inside the bounding sphere
                        {
                                std::vector< LLSphere >::const_iterator sphere_itr;
                                for (sphere_itr = sphere_list.begin(); sphere_itr != sphere_list.end(); ++sphere_itr)
                                {
                                        ensure("sphere should be contained by the bounding sphere", bounding_sphere.contains(*sphere_itr));
                                }
                        }

                        // TODO -- improve LLSphere::getBoundingSphere() to the point where
                        // we can reduce the 'expansion' in the two tests below to about 
                        // 2 mm or less

                        F32 expansion = 0.005f;
                        // move all spheres out a little bit 
                        // and count how many are NOT contained
                        {
                                std::vector< LLVector3 > uncontained_directions;
                                std::vector< LLSphere >::iterator sphere_itr;
                                for (sphere_itr = sphere_list.begin(); sphere_itr != sphere_list.end(); ++sphere_itr)
                                {
                                        LLVector3 direction = sphere_itr->getCenter() - bounding_sphere.getCenter();
                                        direction.normalize();
        
                                        sphere_itr->setCenter( sphere_itr->getCenter() + expansion * direction );
                                        if (! bounding_sphere.contains( *sphere_itr ) )
                                        {
                                                uncontained_directions.push_back(direction);
                                        }
                                }
                                ensure("when moving spheres out there should be at least two uncontained spheres", 
                                                uncontained_directions.size() > 1);

                                /* TODO -- when the bounding sphere algorithm is improved we can open up this test
                                 * at the moment it occasionally fails when the sphere collection is tight and small
                                 * (2 meters or less)
                                if (2 == uncontained_directions.size() )
                                {
                                        // if there were only two uncontained spheres then
                                        // the two directions should be nearly opposite
                                        F32 dir_dot = uncontained_directions[0] * uncontained_directions[1];
                                        ensure("two uncontained spheres should lie opposite the bounding center", dir_dot < -0.95f);
                                }
                                */
                        }

                        // compute the new bounding sphere
                        bounding_sphere = LLSphere::getBoundingSphere(sphere_list);

                        // increase the size of all spheres a little bit
                        // and count how many are NOT contained
                        {
                                std::vector< LLVector3 > uncontained_directions;
                                std::vector< LLSphere >::iterator sphere_itr;
                                for (sphere_itr = sphere_list.begin(); sphere_itr != sphere_list.end(); ++sphere_itr)
                                {
                                        LLVector3 direction = sphere_itr->getCenter() - bounding_sphere.getCenter();
                                        direction.normalize();
        
                                        sphere_itr->setRadius( sphere_itr->getRadius() + expansion );
                                        if (! bounding_sphere.contains( *sphere_itr ) )
                                        {
                                                uncontained_directions.push_back(direction);
                                        }
                                }
                                ensure("when boosting sphere radii there should be at least two uncontained spheres", 
                                                uncontained_directions.size() > 1);

                                /* TODO -- when the bounding sphere algorithm is improved we can open up this test
                                 * at the moment it occasionally fails when the sphere collection is tight and small
                                 * (2 meters or less)
                                if (2 == uncontained_directions.size() )
                                {
                                        // if there were only two uncontained spheres then
                                        // the two directions should be nearly opposite
                                        F32 dir_dot = uncontained_directions[0] * uncontained_directions[1];
                                        ensure("two uncontained spheres should lie opposite the bounding center", dir_dot < -0.95f);
                                }
                                */
                        }
                }
        }
}

namespace tut
{
        F32 SMALL_RADIUS = 1.0f;
        F32 MEDIUM_RADIUS = 5.0f;
        F32 LARGE_RADIUS = 10.0f;

        struct line_data
        {
        };
        typedef test_group<line_data> line_test;
        typedef line_test::object line_object;
        tut::line_test tline("LLLine");

        template<> template<>
        void line_object::test<1>()
        {
                // this is a test for LLLine::intersects(point) which returns TRUE 
                // if the line passes within some tolerance of point

                // these tests will have some floating point error, 
                // so we need to specify how much error is ok
                F32 allowable_relative_error = 0.00001f;
                S32 number_of_tests = 100;
                for (S32 test = 0; test < number_of_tests; ++test)
                {
                        // generate some random point to be on the line
                        LLVector3 point_on_line( ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f);
                        point_on_line.normalize();
                        point_on_line *= ll_frand(LARGE_RADIUS);

                        // generate some random point to "intersect"
                        LLVector3 random_direction ( ll_frand(2.f) - 1.f, 
                                                                                 ll_frand(2.f) - 1.f, 
                                                                                 ll_frand(2.f) - 1.f);
                        random_direction.normalize();

                        LLVector3 random_offset( ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f);
                        random_offset.normalize();
                        random_offset *= ll_frand(SMALL_RADIUS);

                        LLVector3 point = point_on_line + MEDIUM_RADIUS * random_direction
                                + random_offset;
        
                        // compute the axis of approach (a unit vector between the points)
                        LLVector3 axis_of_approach = point - point_on_line;
                        axis_of_approach.normalize();
        
                        // compute the direction of the the first line (perp to axis_of_approach)
                        LLVector3 first_dir( ll_frand(2.f) - 1.f, 
                                                                 ll_frand(2.f) - 1.f, 
                                                                 ll_frand(2.f) - 1.f);
                        first_dir.normalize();
                        F32 dot = first_dir * axis_of_approach;         
                        first_dir -= dot * axis_of_approach;    // subtract component parallel to axis
                        first_dir.normalize();
        
                        // construct the line
                        LLVector3 another_point_on_line = point_on_line + ll_frand(LARGE_RADIUS) * first_dir;
                        LLLine line(another_point_on_line, point_on_line);
        
                        // test that the intersection point is within MEDIUM_RADIUS + SMALL_RADIUS
                        F32 test_radius = MEDIUM_RADIUS + SMALL_RADIUS;
                        test_radius += (LARGE_RADIUS * allowable_relative_error);
                        ensure("line should pass near intersection point", line.intersects(point, test_radius));

                        test_radius = allowable_relative_error * (point - point_on_line).length();
                        ensure("line should intersect point used to define it", line.intersects(point_on_line, test_radius));
                }
        }

        template<> template<>
        void line_object::test<2>()
        {
                // this is a test for LLLine::nearestApproach(LLLIne) method
                // which computes the point on a line nearest another line

                // these tests will have some floating point error, 
                // so we need to specify how much error is ok
                // TODO -- make nearestApproach() algorithm more accurate so
                // we can tighten the allowable_error.  Most tests are tighter
                // than one milimeter, however when doing randomized testing
                // you can walk into inaccurate cases.
                F32 allowable_relative_error = 0.001f;
                S32 number_of_tests = 100;
                for (S32 test = 0; test < number_of_tests; ++test)
                {
                        // generate two points to be our known nearest approaches
                        LLVector3 some_point( ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f);
                        some_point.normalize();
                        some_point *= ll_frand(LARGE_RADIUS);

                        LLVector3 another_point( ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f);
                        another_point.normalize();
                        another_point *= ll_frand(LARGE_RADIUS);

                        // compute the axis of approach (a unit vector between the points)
                        LLVector3 axis_of_approach = another_point - some_point;
                        axis_of_approach.normalize();
        
                        // compute the direction of the the first line (perp to axis_of_approach)
                        LLVector3 first_dir( ll_frand(2.f) - 1.f, 
                                                                 ll_frand(2.f) - 1.f, 
                                                                 ll_frand(2.f) - 1.f);
                        F32 dot = first_dir * axis_of_approach;         
                        first_dir -= dot * axis_of_approach;            // subtract component parallel to axis
                        first_dir.normalize();                                          // normalize
        
                        // compute the direction of the the second line
                        LLVector3 second_dir( ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f);
                        dot = second_dir * axis_of_approach;            
                        second_dir -= dot * axis_of_approach;
                        second_dir.normalize();

                        // make sure the lines aren't too parallel, 
                        dot = fabsf(first_dir * second_dir);
                        if (dot > 0.99f)
                        {
                                // skip this test, we're not interested in testing 
                                // the intractible cases
                                continue;
                        }
        
                        // construct the lines
                        LLVector3 first_point = some_point + ll_frand(LARGE_RADIUS) * first_dir;
                        LLLine first_line(first_point, some_point);
        
                        LLVector3 second_point = another_point + ll_frand(LARGE_RADIUS) * second_dir;
                        LLLine second_line(second_point, another_point);
        
                        // compute the points of nearest approach
                        LLVector3 some_computed_point = first_line.nearestApproach(second_line);
                        LLVector3 another_computed_point = second_line.nearestApproach(first_line);
        
                        // compute the error
                        F32 first_error = (some_point - some_computed_point).length();
                        F32 scale = llmax((some_point - another_point).length(), some_point.length());
                        scale = llmax(scale, another_point.length());
                        scale = llmax(scale, 1.f);
                        F32 first_relative_error = first_error / scale;

                        F32 second_error = (another_point - another_computed_point).length();
                        F32 second_relative_error = second_error / scale;

                        //if (first_relative_error > allowable_relative_error)
                        //{
                        //      std::cout << "first_error = " << first_error 
                        //              << "  first_relative_error = " << first_relative_error 
                        //              << "  scale = " << scale 
                        //              << "  dir_dot = " << (first_dir * second_dir)
                        //              << std::endl;
                        //}
                        //if (second_relative_error > allowable_relative_error)
                        //{
                        //      std::cout << "second_error = " << second_error 
                        //              << "  second_relative_error = " << second_relative_error 
                        //              << "  scale = " << scale 
                        //              << "  dist = " << (some_point - another_point).length()
                        //              << "  dir_dot = " << (first_dir * second_dir)
                        //              << std::endl;
                        //}

                        // test that the errors are small
                        ensure("first line should accurately compute its closest approach", 
                                        first_relative_error <= allowable_relative_error);
                        ensure("second line should accurately compute its closest approach", 
                                        second_relative_error <= allowable_relative_error);
                }
        }

        F32 ALMOST_PARALLEL = 0.99f;
        template<> template<>
        void line_object::test<3>()
        {
                // this is a test for LLLine::getIntersectionBetweenTwoPlanes() method

                // first some known tests
                LLLine xy_plane(LLVector3(0.f, 0.f, 2.f), LLVector3(0.f, 0.f, 3.f));
                LLLine yz_plane(LLVector3(2.f, 0.f, 0.f), LLVector3(3.f, 0.f, 0.f));
                LLLine zx_plane(LLVector3(0.f, 2.f, 0.f), LLVector3(0.f, 3.f, 0.f));

                LLLine x_line;
                LLLine y_line;
                LLLine z_line;

                bool x_success = LLLine::getIntersectionBetweenTwoPlanes(x_line, xy_plane, zx_plane);
                bool y_success = LLLine::getIntersectionBetweenTwoPlanes(y_line, yz_plane, xy_plane);
                bool z_success = LLLine::getIntersectionBetweenTwoPlanes(z_line, zx_plane, yz_plane);

                ensure("xy and zx planes should intersect", x_success);
                ensure("yz and xy planes should intersect", y_success);
                ensure("zx and yz planes should intersect", z_success);

                LLVector3 direction = x_line.getDirection();
                ensure("x_line should be parallel to x_axis", fabs(direction.mV[VX]) == 1.f
                                                                      && 0.f == direction.mV[VY]
                                                                      && 0.f == direction.mV[VZ] );
                direction = y_line.getDirection();
                ensure("y_line should be parallel to y_axis", 0.f == direction.mV[VX]
                                                                                                          && fabs(direction.mV[VY]) == 1.f
                                                                      && 0.f == direction.mV[VZ] );
                direction = z_line.getDirection();
                ensure("z_line should be parallel to z_axis", 0.f == direction.mV[VX]
                                                                      && 0.f == direction.mV[VY]
                                                                                                          && fabs(direction.mV[VZ]) == 1.f );

                // next some random tests
                F32 allowable_relative_error = 0.0001f;
                S32 number_of_tests = 20;
                for (S32 test = 0; test < number_of_tests; ++test)
                {
                        // generate the known line
                        LLVector3 some_point( ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f, 
                                                                  ll_frand(2.f) - 1.f);
                        some_point.normalize();
                        some_point *= ll_frand(LARGE_RADIUS);
                        LLVector3 another_point( ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f, 
                                                                         ll_frand(2.f) - 1.f);
                        another_point.normalize();
                        another_point *= ll_frand(LARGE_RADIUS);
                        LLLine known_intersection(some_point, another_point);

                        // compute a plane that intersect the line
                        LLVector3 point_on_plane( ll_frand(2.f) - 1.f, 
                                                                          ll_frand(2.f) - 1.f, 
                                                                          ll_frand(2.f) - 1.f);
                        point_on_plane.normalize();
                        point_on_plane *= ll_frand(LARGE_RADIUS);
                        LLVector3 plane_normal = (point_on_plane - some_point) % known_intersection.getDirection();
                        plane_normal.normalize();
                        LLLine first_plane(point_on_plane, point_on_plane + plane_normal);

                        // compute a different plane that intersect the line
                        LLVector3 point_on_different_plane( ll_frand(2.f) - 1.f, 
                                                                                                ll_frand(2.f) - 1.f, 
                                                                                                ll_frand(2.f) - 1.f);
                        point_on_different_plane.normalize();
                        point_on_different_plane *= ll_frand(LARGE_RADIUS);
                        LLVector3 different_plane_normal = (point_on_different_plane - another_point) % known_intersection.getDirection();
                        different_plane_normal.normalize();
                        LLLine second_plane(point_on_different_plane, point_on_different_plane + different_plane_normal);

                        if (fabs(plane_normal * different_plane_normal) > ALMOST_PARALLEL)
                        {
                                // the two planes are approximately parallel, so we won't test this case
                                continue;
                        }

                        LLLine measured_intersection;
                        bool success = LLLine::getIntersectionBetweenTwoPlanes(
                                        measured_intersection,
                                        first_plane,
                                        second_plane);

                        ensure("plane intersection should succeed", success);

                        F32 dot = fabs(known_intersection.getDirection() * measured_intersection.getDirection());
                        ensure("measured intersection should be parallel to known intersection",
                                        dot > ALMOST_PARALLEL);

                        ensure("measured intersection should pass near known point",
                                        measured_intersection.intersects(some_point, LARGE_RADIUS * allowable_relative_error));
                }
        }
}

---