001// License: GPL. For details, see LICENSE file.
002package org.openstreetmap.josm.data.projection.proj;
003
004import static org.openstreetmap.josm.tools.I18n.tr;
005
006import org.openstreetmap.josm.data.Bounds;
007import org.openstreetmap.josm.data.coor.LatLon;
008import org.openstreetmap.josm.data.projection.ProjectionConfigurationException;
009
010/**
011 * Oblique Mercator Projection. A conformal, oblique, cylindrical projection with the cylinder
012 * touching the ellipsoid (or sphere) along a great circle path (the central line). The
013 * {@linkplain Mercator} and {@linkplain TransverseMercator Transverse Mercator} projections can
014 * be thought of as special cases of the oblique mercator, where the central line is along the
015 * equator or a meridian, respectively. The Oblique Mercator projection has been used in
016 * Switzerland, Hungary, Madagascar, Malaysia, Borneo and the panhandle of Alaska.
017 * <p>
018 * The Oblique Mercator projection uses a (<var>U</var>,<var>V</var>) coordinate system, with the
019 * <var>U</var> axis along the central line. During the forward projection, coordinates from the
020 * ellipsoid are projected conformally to a sphere of constant total curvature, called the
021 * "aposphere", before being projected onto the plane. The projection coordinates are further
022 * convented to a (<var>X</var>,<var>Y</var>) coordinate system by rotating the calculated
023 * (<var>u</var>,<var>v</var>) coordinates to give output (<var>x</var>,<var>y</var>) coordinates.
024 * The rotation value is usually the same as the projection azimuth (the angle, east of north, of
025 * the central line), but some cases allow a separate rotation parameter.
026 * <p>
027 * There are two forms of the oblique mercator, differing in the origin of their grid coordinates.
028 * The Hotine Oblique Mercator (EPSG code 9812) has grid coordinates start at the intersection of
029 * the central line and the equator of the aposphere.
030 * The Oblique Mercator (EPSG code 9815) is the same, except the grid coordinates begin at the
031 * central point (where the latitude of center and central line intersect). ESRI separates these
032 * two case by appending {@code "Natural_Origin"} (for the {@code "Hotine_Oblique_Mercator"}) and
033 * {@code "Center"} (for the {@code "Oblique_Mercator"}) to the projection names.
034 * <p>
035 * Two different methods are used to specify the central line for the oblique mercator:
036 * 1) a central point and an azimuth, east of north, describing the central line and
037 * 2) two points on the central line. The EPSG does not use the two point method,
038 * while ESRI separates the two cases by putting {@code "Azimuth"} and {@code "Two_Point"}
039 * in their projection names. Both cases use the point where the {@code "latitude_of_center"}
040 * parameter crosses the central line as the projection's central point.
041 * The {@linkplain #centralMeridian central meridian} is not a projection parameter,
042 * and is instead calculated as the intersection between the central line and the
043 * equator of the aposphere.
044 * <p>
045 * For the azimuth method, the central latitude cannot be &plusmn;90.0 degrees
046 * and the central line cannot be at a maximum or minimum latitude at the central point.
047 * In the two point method, the latitude of the first and second points cannot be
048 * equal. Also, the latitude of the first point and central point cannot be
049 * &plusmn;90.0 degrees. Furthermore, the latitude of the first point cannot be 0.0 and
050 * the latitude of the second point cannot be -90.0 degrees. A change of
051 * 10<sup>-7</sup> radians can allow calculation at these special cases. Snyder's restriction
052 * of the central latitude being 0.0 has been removed, since the equations appear
053 * to work correctly in this case.
054 * <p>
055 * Azimuth values of 0.0 and &plusmn;90.0 degrees are allowed (and used in Hungary
056 * and Switzerland), though these cases would usually use a Mercator or
057 * Transverse Mercator projection instead. Azimuth values &gt; 90 degrees cause
058 * errors in the equations.
059 * <p>
060 * The oblique mercator is also called the "Rectified Skew Orthomorphic" (RSO). It appears
061 * is that the only difference from the oblique mercator is that the RSO allows the rotation
062 * from the (<var>U</var>,<var>V</var>) to (<var>X</var>,<var>Y</var>) coordinate system to
063 * be different from the azimuth. This separate parameter is called
064 * {@code "rectified_grid_angle"} (or {@code "XY_Plane_Rotation"} by ESRI) and is also
065 * included in the EPSG's parameters for the Oblique Mercator and Hotine Oblique Mercator.
066 * The rotation parameter is optional in all the non-two point projections and will be
067 * set to the azimuth if not specified.
068 * <p>
069 * Projection cases and aliases implemented by the {@link ObliqueMercator} are:
070 * <ul>
071 *   <li>{@code Oblique_Mercator} (EPSG code 9815)<br>
072 *       grid coordinates begin at the central point,
073 *       has {@code "rectified_grid_angle"} parameter.</li>
074 *   <li>{@code Hotine_Oblique_Mercator_Azimuth_Center} (ESRI)<br>
075 *       grid coordinates begin at the central point.</li>
076 *   <li>{@code Rectified_Skew_Orthomorphic_Center} (ESRI)<br>
077 *       grid coordinates begin at the central point,
078 *       has {@code "rectified_grid_angle"} parameter.</li>
079 *
080 *   <li>{@code Hotine_Oblique_Mercator} (EPSG code 9812)<br>
081 *       grid coordinates begin at the interseciton of the central line and aposphere equator,
082 *       has {@code "rectified_grid_angle"} parameter.</li>
083 *   <li>{@code Hotine_Oblique_Mercator_Azimuth_Natural_Origin} (ESRI)<br>
084 *       grid coordinates begin at the interseciton of the central line and aposphere equator.</li>
085 *   <li>{@code Rectified_Skew_Orthomorphic_Natural_Origin} (ESRI)<br>
086 *       grid coordinates begin at the interseciton of the central line and aposphere equator,
087 *       has {@code "rectified_grid_angle"} parameter.</li>
088 *
089 *   <li>{@code Hotine_Oblique_Mercator_Two_Point_Center} (ESRI)<br>
090 *       grid coordinates begin at the central point.</li>
091 *   <li>{@code Hotine_Oblique_Mercator_Two_Point_Natural_Origin} (ESRI)<br>
092 *       grid coordinates begin at the interseciton of the central line and aposphere equator.</li>
093 * </ul>
094 * <p>
095 * This class has been derived from the implementation of the Geotools project;
096 * git 8cbf52d, org.geotools.referencing.operation.projection.ObliqueMercator
097 * at the time of migration.
098 * <p>
099 * Note that automatic calculation of bounds is very limited for this projection,
100 * since the central line can have any orientation.
101 * <p>
102 * <b>References:</b>
103 * <ul>
104 *   <li>{@code libproj4} is available at
105 *       <A HREF="http://members.bellatlantic.net/~vze2hc4d/proj4/">libproj4 Miscellanea</A><br>
106 *       Relevent files are: {@code PJ_omerc.c}, {@code pj_tsfn.c},
107 *       {@code pj_fwd.c}, {@code pj_inv.c} and {@code lib_proj.h}</li>
108 *   <li>John P. Snyder (Map Projections - A Working Manual,
109 *       U.S. Geological Survey Professional Paper 1395, 1987)</li>
110 *   <li>"Coordinate Conversions and Transformations including Formulas",
111 *       EPSG Guidence Note Number 7 part 2, Version 24.</li>
112 *   <li>Gerald Evenden, 2004, <a href="http://members.verizon.net/~vze2hc4d/proj4/omerc.pdf">
113 *       Documentation of revised Oblique Mercator</a></li>
114 * </ul>
115 *
116 * @author Gerald I. Evenden (for original code in Proj4)
117 * @author  Rueben Schulz
118 *
119 * @see <A HREF="http://mathworld.wolfram.com/MercatorProjection.html">Oblique Mercator projection on MathWorld</A>
120 * @see <A HREF="http://www.remotesensing.org/geotiff/proj_list/hotine_oblique_mercator.html">"hotine_oblique_mercator" on RemoteSensing.org</A>
121 * @see <A HREF="http://www.remotesensing.org/geotiff/proj_list/oblique_mercator.html">"oblique_mercator" on RemoteSensing.org</A>
122 */
123public class ObliqueMercator extends AbstractProj implements ICentralMeridianProvider {
124
125    /**
126     * Maximum difference allowed when comparing real numbers.
127     */
128    private static final double EPSILON = 1E-6;
129
130    /**
131     * Maximum difference allowed when comparing latitudes.
132     */
133    private static final double EPSILON_LATITUDE = 1E-10;
134
135    //////
136    //////    Map projection parameters. The following are NOT used by the transformation
137    //////    methods, but are stored in order to restitute them in WKT formatting.  They
138    //////    are made visible ('protected' access) for documentation purpose and because
139    //////    they are user-supplied parameters, not derived coefficients.
140    //////
141
142    /**
143     * The azimuth of the central line passing throught the centre of the projection, in radians.
144     */
145    protected double azimuth;
146
147    /**
148     * The rectified bearing of the central line, in radians. This is equals to the
149     * {@linkplain #azimuth} if the parameter value is not set.
150     */
151    protected double rectifiedGridAngle;
152
153    //////
154    //////    Map projection coefficients computed from the above parameters.
155    //////    They are the fields used for coordinate transformations.
156    //////
157
158    /**
159     * Constants used in the transformation.
160     */
161    private double b, a, e;
162
163    /**
164     * Convenience values equal to {@link #a} / {@link #b},
165     * {@link #a}&times;{@link #b}, and {@link #b} / {@link #a}.
166     */
167    private double arb, ab, bra;
168
169    /**
170     * <var>v</var> values when the input latitude is a pole.
171     */
172    private double vPoleN, vPoleS;
173
174    /**
175     * Sine and Cosine values for gamma0 (the angle between the meridian
176     * and central line at the intersection between the central line and
177     * the Earth equator on aposphere).
178     */
179    private double singamma0, cosgamma0;
180
181    /**
182     * Sine and Cosine values for the rotation between (U,V) and
183     * (X,Y) coordinate systems
184     */
185    private double sinrot, cosrot;
186
187    /**
188     * <var>u</var> value (in (U,V) coordinate system) of the central point. Used in
189     * the oblique mercator case. The <var>v</var> value of the central point is 0.0.
190     */
191    private double uc;
192
193    /**
194     * Central longitude in <u>radians</u>. Default value is 0, the Greenwich meridian.
195     * This is called '<var>lambda0</var>' in Snyder.
196     */
197    protected double centralMeridian;
198
199    /**
200     * A reference point, which is known to be on the central line.
201     */
202    private LatLon referencePoint;
203
204    @Override
205    public String getName() {
206        return tr("Oblique Mercator");
207    }
208
209    @Override
210    public String getProj4Id() {
211        return "omerc";
212    }
213
214    @Override
215    public void initialize(ProjParameters params) throws ProjectionConfigurationException {
216        super.initialize(params);
217        boolean twoPoint = params.alpha == null;
218
219        double latCenter = 0;
220        if (params.lat0 != null) {
221            latCenter = Math.toRadians(params.lat0);
222        }
223
224        final double com = Math.sqrt(1.0 - e2);
225        double sinph0 = Math.sin(latCenter);
226        double cosph0 = Math.cos(latCenter);
227        final double con = 1. - e2 * sinph0 * sinph0;
228        double temp = cosph0 * cosph0;
229        b = Math.sqrt(1.0 + e2 * (temp * temp) / (1.0 - e2));
230        a = b * com / con;
231        final double d = b * com / (cosph0 * Math.sqrt(con));
232        double f = d * d - 1.0;
233        if (f < 0.0) {
234            f = 0.0;
235        } else {
236            f = Math.sqrt(f);
237            if (latCenter < 0.0) {
238                f = -f;
239            }
240        }
241        e = f += d;
242        e = f * Math.pow(tsfn(latCenter, sinph0), b);
243
244        /*
245         * Computes the constants that depend on the "twoPoint" vs "azimuth" case. In the
246         * two points case, we compute them from (LAT_OF_1ST_POINT, LONG_OF_1ST_POINT) and
247         * (LAT_OF_2ND_POINT, LONG_OF_2ND_POINT).  For the "azimuth" case, we compute them
248         * from LONGITUDE_OF_CENTRE and AZIMUTH.
249         */
250        final double gamma0;
251        Double lonCenter = null;
252        if (twoPoint) {
253            if (params.lon1 == null)
254                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lon_1"));
255            if (params.lat1 == null)
256                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lat_1"));
257            if (params.lon2 == null)
258                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lon_2"));
259            if (params.lat2 == null)
260                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lat_2"));
261            referencePoint = new LatLon(params.lat1, params.lat2);
262            double lon1 = Math.toRadians(params.lon1);
263            double lat1 = Math.toRadians(params.lat1);
264            double lon2 = Math.toRadians(params.lon2);
265            double lat2 = Math.toRadians(params.lat2);
266
267            if (Math.abs(lat1 - lat2) <= EPSILON ||
268                Math.abs(lat1) <= EPSILON ||
269                Math.abs(Math.abs(lat1) - Math.PI / 2) <= EPSILON ||
270                Math.abs(Math.abs(latCenter) - Math.PI / 2) <= EPSILON ||
271                Math.abs(Math.abs(lat2) - Math.PI / 2) <= EPSILON) {
272                throw new ProjectionConfigurationException(
273                    tr("Unsuitable parameters ''{0}'' and ''{1}'' for two point method.", "lat_1", "lat_2"));
274            }
275            /*
276             * The coefficients for the "two points" case.
277             */
278            final double h = Math.pow(tsfn(lat1, Math.sin(lat1)), b);
279            final double l = Math.pow(tsfn(lat2, Math.sin(lat2)), b);
280            final double fp = e / h;
281            final double p = (l - h) / (l + h);
282            double j = e * e;
283            j = (j - l * h) / (j + l * h);
284            double diff = lon1 - lon2;
285            if (diff < -Math.PI) {
286                lon2 -= 2.0 * Math.PI;
287            } else if (diff > Math.PI) {
288                lon2 += 2.0 * Math.PI;
289            }
290            centralMeridian = normalizeLonRad(0.5 * (lon1 + lon2) -
291                     Math.atan(j * Math.tan(0.5 * b * (lon1 - lon2)) / p) / b);
292            gamma0 = Math.atan(2.0 * Math.sin(b * normalizeLonRad(lon1 - centralMeridian)) /
293                     (fp - 1.0 / fp));
294            azimuth = Math.asin(d * Math.sin(gamma0));
295            rectifiedGridAngle = azimuth;
296        } else {
297            if (params.lonc == null)
298                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lonc"));
299            if (params.lat0 == null)
300                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "lat_0"));
301            if (params.alpha == null)
302                throw new ProjectionConfigurationException(tr("Parameter ''{0}'' required.", "alpha"));
303            referencePoint = new LatLon(params.lat0, params.lonc);
304
305            lonCenter = Math.toRadians(params.lonc);
306            azimuth = Math.toRadians(params.alpha);
307            if ((azimuth > -1.5*Math.PI && azimuth < -0.5*Math.PI) ||
308                (azimuth >  0.5*Math.PI && azimuth <  1.5*Math.PI)) {
309                throw new ProjectionConfigurationException(
310                        tr("Illegal value for parameter ''{0}'': {1}", "alpha", Double.toString(params.alpha)));
311            }
312            if (params.gamma != null) {
313                rectifiedGridAngle = Math.toRadians(params.gamma);
314            } else {
315                rectifiedGridAngle = azimuth;
316            }
317            gamma0 = Math.asin(Math.sin(azimuth) / d);
318            // Check for asin(+-1.00000001)
319            temp = 0.5 * (f - 1.0 / f) * Math.tan(gamma0);
320            if (Math.abs(temp) > 1.0) {
321                if (Math.abs(Math.abs(temp) - 1.0) > EPSILON) {
322                    throw new ProjectionConfigurationException(tr("error in initialization"));
323                }
324                temp = (temp > 0) ? 1.0 : -1.0;
325            }
326            centralMeridian = lonCenter - Math.asin(temp) / b;
327        }
328
329        /*
330         * More coefficients common to all kind of oblique mercator.
331         */
332        singamma0 = Math.sin(gamma0);
333        cosgamma0 = Math.cos(gamma0);
334        sinrot    = Math.sin(rectifiedGridAngle);
335        cosrot    = Math.cos(rectifiedGridAngle);
336        arb       = a / b;
337        ab        = a * b;
338        bra       = b / a;
339        vPoleN  = arb * Math.log(Math.tan(0.5 * (Math.PI/2.0 - gamma0)));
340        vPoleS  = arb * Math.log(Math.tan(0.5 * (Math.PI/2.0 + gamma0)));
341        boolean hotine = params.no_off != null && params.no_off;
342        if (hotine) {
343            uc = 0.0;
344        } else {
345            if (Math.abs(Math.abs(azimuth) - Math.PI/2.0) < EPSILON_LATITUDE) {
346                // lonCenter == null in twoPoint, but azimuth cannot be 90 here (lat1 != lat2)
347                if (lonCenter == null) {
348                    throw new ProjectionConfigurationException("assertion error");
349                }
350                uc = a * (lonCenter - centralMeridian);
351            } else {
352                double uC = Math.abs(arb * Math.atan2(Math.sqrt(d * d - 1.0), Math.cos(azimuth)));
353                if (latCenter < 0.0) {
354                    uC = -uC;
355                }
356                this.uc = uC;
357            }
358        }
359    }
360
361    private double normalizeLonRad(double a) {
362        return Math.toRadians(LatLon.normalizeLon(Math.toDegrees(a)));
363    }
364
365    @Override
366    public double[] project(double y, double x) {
367        double u, v;
368        if (Math.abs(Math.abs(y) - Math.PI/2.0) > EPSILON) {
369            double q = e / Math.pow(tsfn(y, Math.sin(y)), b);
370            double temp = 1.0 / q;
371            double s = 0.5 * (q - temp);
372            double V = Math.sin(b * x);
373            double U = (s * singamma0 - V * cosgamma0) / (0.5 * (q + temp));
374            if (Math.abs(Math.abs(U) - 1.0) < EPSILON) {
375                v = 0; // this is actually an error and should be reported to the caller somehow
376            } else {
377                v = 0.5 * arb * Math.log((1.0 - U) / (1.0 + U));
378            }
379            temp = Math.cos(b * x);
380            if (Math.abs(temp) < EPSILON_LATITUDE) {
381                u = ab * x;
382            } else {
383                u = arb * Math.atan2(s * cosgamma0 + V * singamma0, temp);
384            }
385        } else {
386            v = y > 0 ? vPoleN : vPoleS;
387            u = arb * y;
388        }
389        u -= uc;
390        x = v * cosrot + u * sinrot;
391        y = u * cosrot - v * sinrot;
392        return new double[] {x, y};
393    }
394
395    @Override
396    public double[] invproject(double x, double y) {
397        double v = x * cosrot - y * sinrot;
398        double u = y * cosrot + x * sinrot + uc;
399        double qp = Math.exp(-bra * v);
400        double temp = 1.0 / qp;
401        double sp = 0.5 * (qp - temp);
402        double vp = Math.sin(bra * u);
403        double up = (vp * cosgamma0 + sp * singamma0) / (0.5 * (qp + temp));
404        if (Math.abs(Math.abs(up) - 1.0) < EPSILON) {
405            x = 0.0;
406            y = up < 0.0 ? -Math.PI / 2.0 : Math.PI / 2.0;
407        } else {
408            y = Math.pow(e / Math.sqrt((1. + up) / (1. - up)), 1.0 / b);  //calculate t
409            y = cphi2(y);
410            x = -Math.atan2(sp * cosgamma0 - vp * singamma0, Math.cos(bra * u)) / b;
411        }
412        return new double[] {y, x};
413    }
414
415    @Override
416    public Bounds getAlgorithmBounds() {
417        // The central line of this projection can be oriented in any direction.
418        // Moreover, the projection doesn't work too well very far off the central line.
419        // This makes it hard to choose proper bounds automatically.
420        //
421        // We return a small box around a reference point. This default box is
422        // probably too small for most applications. If this is the case, the
423        // bounds should be configured explicitly.
424        double lat = referencePoint.lat();
425        double dLat = 3.0;
426        double lon = referencePoint.lon() - Math.toDegrees(centralMeridian);
427        double dLon = 3.0;
428        return new Bounds(lat - dLat, lon - dLon, lat + dLat, lon + dLon, false);
429    }
430
431    @Override
432    public double getCentralMeridian() {
433        return Math.toDegrees(centralMeridian);
434    }
435}