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LocationInterpolation.cpp
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LocationInterpolation.cpp
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/*
* Copyright (c) 2019 Eric Christoffersen (impolexg@outlook.com)
*
* This program 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 2 of the License, or (at your option)
* any later version.
*
* 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., 51
* Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <assert.h>
#include "LocationInterpolation.h"
#include "BlinnSolver.h"
static double radianstodegrees(double r) { return r * (360.0 / (2 * M_PI)); }
static double degreestoradians(double d) { return d * (2 * M_PI / 360.0); }
xyz geolocation::toxyz() const
{
// Approach developed by:
//
// (Olson, D.K. (1996).
// "Converting earth-Centered, Earth-Fixed Coordinates to Geodetic Coordinates,"
// IEEE Transactions on Aerospace and Electronic Systems, Vol. 32, No. 1, January 1996, pp. 473 - 476).
//
//
// Java implementation:
//
// D. Rose (2014).
// "Converting between Earth-Centered, Earth Fixed and Geodetic Coordinates"
//
//Convert Lat, Lon, Altitude to Earth-Centered-Earth-Fixed (ECEF)
//Input is a three element array containing lat, lon (rads) and alt (m)
//Returned array contains x, y, z in meters
static const double a = 6378137.0; //WGS-84 semi-major axis
static const double e2 = 6.6943799901377997e-3; //WGS-84 first eccentricity squared
double lat = degreestoradians(Lat());
double lon = degreestoradians(Long());
double alt = Alt();
double n = a / sqrt(1 - e2 * sin(lat)*sin(lat));
double dx = ((n + alt)*cos(lat)*cos(lon)); //ECEF x
double dy = ((n + alt)*cos(lat)*sin(lon)); //ECEF y
double dz = ((n*(1 - e2) + alt)*sin(lat)); //ECEF z
return(xyz(dx, dy, dz)); //Return x, y, z in ECEF
}
geolocation xyz::togeolocation() const
{
// Approach developed by:
// (Olson, D.K. (1996).
// "Converting earth-Centered, Earth-Fixed Coordinates to Geodetic Coordinates,"
// IEEE Transactions on Aerospace and Electronic Systems, Vol. 32, No. 1, January 1996, pp. 473 - 476).
//
// Java implementation:
//
// D. Rose (2014).
// "Converting between Earth-Centered, Earth Fixed and Geodetic Coordinates"
//
//Convert Earth-Centered-Earth-Fixed (ECEF) to lat, Lon, Altitude
//Input is a three element array containing x, y, z in meters
//Returned array contains lat and lon in radians, and altitude in meters
static const double a = 6378137.0; //WGS-84 semi-major axis
static const double e2 = 6.6943799901377997e-3; //WGS-84 first eccentricity squared
static const double a1 = 4.2697672707157535e+4; //a1 = a*e2
static const double a2 = 1.8230912546075455e+9; //a2 = a1*a1
static const double a3 = 1.4291722289812413e+2; //a3 = a1*e2/2
static const double a4 = 4.5577281365188637e+9; //a4 = 2.5*a2
static const double a5 = 4.2840589930055659e+4; //a5 = a1+a3
static const double a6 = 9.9330562000986220e-1; //a6 = 1-e2
double x = xyz::x();
double y = xyz::y();
double z = xyz::z();
double zp = std::abs(z);
double w2 = x * x + y * y;
double w = sqrt(w2);
double r2 = w2 + z * z;
double r = sqrt(r2);
double retLong;
double retLat;
double retAlt;
retLong = atan2(y, x); // Lon (final)
double s2 = z * z / r2;
double c2 = w2 / r2;
double u = a2 / r;
double v = a3 - a4 / r;
double s, ss, c;
if (c2 > 0.3) {
s = (zp / r)*(1.0 + c2 * (a1 + u + s2 * v) / r);
retLat = asin(s); //Lat
ss = s * s;
c = sqrt(1.0 - ss);
}
else {
c = (w / r)*(1.0 - s2 * (a5 - u - c2 * v) / r);
retLat = acos(c); //Lat
ss = 1.0 - c * c;
s = sqrt(ss);
}
double g = 1.0 - e2 * ss;
double rg = a / sqrt(g);
double rf = a6 * rg;
u = w - rg * c;
v = zp - rf * s;
double f = c * u + s * v;
double m = c * v - s * u;
double p = m / (rf / g + f);
retLat = retLat + p; //Lat
retAlt = f + m * p / 2.0; //Altitude
if (z < 0.0) {
retLat = retLat * -1.0; //Lat
}
return geolocation(radianstodegrees(retLat), radianstodegrees(retLong), retAlt);
}
xyz Slerper::Slerp(double frac)
{
if (m_sin_angle != 0.0)
{
double scale = sin(frac * m_angle) / m_sin_angle;
return m_x0_norm.scale(sin((1 - frac) * m_angle) / m_sin_angle).add(m_x1_norm.scale(scale));
}
return m_x0_norm;
}
// Precompute invariant values needed to geoslerp
Slerper::Slerper(geolocation g0, geolocation g1) :
m_g0(g0),
m_g1(g1),
m_x0(m_g0.toxyz()),
m_x1(m_g1.toxyz()),
m_x0_norm(m_x0.normalize()),
m_x1_norm(m_x1.normalize()),
m_x0_magnitude(m_x0.magnitude())
{
m_angle = atan2(m_x0_norm.cross(m_x1_norm).magnitude(), m_x0_norm.dot(m_x1_norm));
m_sin_angle = sin(m_angle);
m_altitude_delta = (m_g1.Alt() - m_g0.Alt());
}
// Interpolate fractional arc between two wgs84 vectors.
// This models the earths ellipsoid.
geolocation Slerper::GeoSlerp(double frac)
{
// Generate normalized ECEF vector
xyz slerp = Slerp(frac);
// Scale ecef unit vector so its roughly at earth surface
double interpaltitudestep = m_altitude_delta * frac;
slerp = slerp.scale(m_x0_magnitude + interpaltitudestep);
// Convert ecef back to wgs84
geolocation slerpgeo = slerp.togeolocation();
// Set linear interpolated altitude on slerped wgs84 vector
double interpaltitude = m_altitude_delta * frac;
slerpgeo.Alt() = m_g0.Alt() + interpaltitude;
return slerpgeo;
}
xyz LinearTwoPointInterpolator::InterpolateNext(xyz p0, xyz p1)
{
xyz delta = p1.subtract(p0);
xyz retval = p1.add(delta);
return retval;
}
xyz SphericalTwoPointInterpolator::InterpolateNext(xyz p0, xyz p1)
{
Slerper slerper(p0.togeolocation(), p1.togeolocation());
geolocation geo = slerper.GeoSlerp(2.0);
return geo.toxyz();
}
void UnitCatmullRomInterpolator::Init(double pm1, double p0, double p1, double p2)
{
std::get<0>(m_p) = pm1;
std::get<1>(m_p) = p0;
std::get<2>(m_p) = p1;
std::get<3>(m_p) = p2;
}
UnitCatmullRomInterpolator::UnitCatmullRomInterpolator()
{
Init(0.0, 0.0, 0.0, 0.0);
}
UnitCatmullRomInterpolator::UnitCatmullRomInterpolator(double pm1, double p0, double p1, double p2)
{
Init(pm1, p0, p1, p2);
}
double UnitCatmullRomInterpolator::T()
{
// Control curvature:
// 0 is standard (straigtest)
// 0.5 is called centripetal
// 1 is chordal (loopiest)
double t = 0.5;
return t;
}
double UnitCatmullRomInterpolator::Location(double u) const
{
double t = T();
double p0 = std::get<0>(m_p);
double p1 = std::get<1>(m_p);
double p2 = std::get<2>(m_p);
double p3 = std::get<3>(m_p);
// Curvature-parameterized CatmullRom equation courtesy of wolfram alpha:
// [1, u, u ^ 2, u ^ 3] * [[0, 1, 0, 0], [-t, 0, t, 0], [2 * t, t - 3, 3 - 2t, -t], [-t, 2 - t, t - 2, t]] * [p0, p1, p2, p3]
double retval = p0 * u * (u * (2 * t - t * u) - t) +
u * (u * (u * (p1 * (-t) + 2 * p1 + p2 * (t - 2) + p3 * t) + p1 * t - 3 * p1 + p2 * (3 - 2 * t) - p3 * t) + p2 * t) + p1;
return retval;
}
double UnitCatmullRomInterpolator::Tangent(double u) const
{
double t = T();
double p0 = std::get<0>(m_p);
double p1 = std::get<1>(m_p);
double p2 = std::get<2>(m_p);
double p3 = std::get<3>(m_p);
// Curvature-parameterized CatmullRom equation courtesy of wolfram alpha:
// [1, u, u ^ 2, u ^ 3] * [[0, 1, 0, 0], [-t, 0, t, 0], [2 * t, t - 3, 3 - 2t, -t], [-t, 2 - t, t - 2, t]] * [p0, p1, p2, p3]
// d f(u)/du
// Closed form slope for cubic at point u in 0..1.
double retval = 3 * (u *u) * ((p3 + p2 - p1 - p0) * t - 2 * p2 + 2 * p1) +
2 * (u) * ((-p3 - 2 * p2 + p1 + 2 * p0)*t + 3 * p2 - 3 * p1) +
(p2 - p0) * t;
return retval;
}
// Given interpolated value r, provides u that would yield r.
// Only successful if spline is invertable (which is true for
// the distance spline which is monotonic, etc.)
bool UnitCatmullRomInterpolator::Inverse(double r, double &u) const
{
const double t = T();
double p0 = std::get<0>(m_p);
double p1 = std::get<1>(m_p);
double p2 = std::get<2>(m_p);
double p3 = std::get<3>(m_p);
// Normalized form of equation from Location.
double a = ((p3 + p2 - p1 - p0) * t - 2 * p2 + 2 * p1);
double b = ((-p3 - 2 * p2 + p1 + 2 * p0)*t + 3 * p2 - 3 * p1);
double c = (p2 - p0) * t;
double d = p1 - r; // "- r" because root finder expects coefficients for expression that equals zero.
Roots roots = BlinnCubicSolver(a, b, c, d);
// There are 0, 1, 2 or 3 roots.
// In general it is possible that there are multiple roots in range... but should never happen
// for monotonic distance mapping.
bool ret = false;
for (unsigned i = 0; i < roots.resultcount(); i++) {
double r = roots.result(i).x / roots.result(i).w;
// Take the first root we find in range 0..1.
if (r >= 0. && r <= 1.) {
u = r;
ret = true;
break;
}
}
return ret;
}
void UnitCatmullRomInterpolator3D::Init(xyz pm1, xyz p0, xyz p1, xyz p2)
{
x.Init(pm1.x(), p0.x(), p1.x(), p2.x());
y.Init(pm1.y(), p0.y(), p1.y(), p2.y());
z.Init(pm1.z(), p0.z(), p1.z(), p2.z());
}
UnitCatmullRomInterpolator3D::UnitCatmullRomInterpolator3D(xyz pm1, xyz p0, xyz p1, xyz p2)
{
Init(pm1, p0, p1, p2);
}
xyz UnitCatmullRomInterpolator3D::Location(double frac) const
{
return xyz(x.Location(frac), y.Location(frac), z.Location(frac));
}
xyz UnitCatmullRomInterpolator3D::Tangent(double frac) const
{
return xyz(x.Tangent(frac), y.Tangent(frac), z.Tangent(frac));
}
geolocation GeoPointInterpolator::Location(double distance)
{
xyz l0xyz = DistancePointInterpolator<SphericalTwoPointInterpolator>::Location(distance);
geolocation l0 = l0xyz.togeolocation();
return l0;
}
geolocation GeoPointInterpolator::Location(double distance, double &slope)
{
xyz tangentVector;
xyz l0xyz = DistancePointInterpolator<SphericalTwoPointInterpolator>::Location(distance, tangentVector);
// First step, construct 2 geo locations that are separated by a unit-length tangent veocity vector.
geolocation l1 = xyz(l0xyz.add(tangentVector.normalize())).togeolocation();
geolocation l0 = l0xyz.togeolocation();
// - Route distance is independent of geometric distance.
// - Route distance is fixed in stone because it must match video
// synchronization files.
// - Geometric distance is also fixed since it is a fact of gps location.
//
// - Altitude is always in route distance units (so is computable
// using gradient and route distance.)
//
// - Path length on spline is gps-based so uses geometric units.
//
// - Catmull-Rom spline is non-uniform, meaning speed of interpolated point can
// vary across bracket which distorts the tangent vector.
// Vertical velocity of unit tangent vector (altitude is always kept in in route distance units.)
double rise = l1.Alt() - l0.Alt();
// Tangent vector's magnitude is velocity in terms of distance spline.
// Will be unit vector when distance spline has constant rate.
double hyp = tangentVector.magnitude();
// Compute adjacent speed.
double run = sqrt(fabs((hyp * hyp) - (rise * rise)));
// Gradient.
slope = run ? rise / run : 0;
// No matter what we still don't permit slopes above 40%.
slope = std::min(slope, .4);
slope = std::max(slope, -.4);
// Clear out location if we are an altitude only spline.
if (!HasLocation()) {
l0.Lat() = 0;
l0.Long() = 0;
}
return l0;
}
void GeoPointInterpolator::Push(double distance, geolocation point)
{
if (m_locationState == Unset)
m_locationState = YesLocation;
else
assert(m_locationState == YesLocation);
DistancePointInterpolator<SphericalTwoPointInterpolator>::Push(distance, point.toxyz());
}
// Special form for case where altitude exists but no location.
void GeoPointInterpolator::Push(double distance, double altitude)
{
if (m_locationState == Unset)
m_locationState = NoLocation;
else
assert(m_locationState == NoLocation);
geolocation geo(0, 0, altitude);
xyz point = geo.toxyz();
point.y() = distance;
DistancePointInterpolator<SphericalTwoPointInterpolator>::Push(distance, point);
}