GSkyRegionRectangle.cpp

Go to the documentation of this file.

/***************************************************************************
 *           GSkyRegionRectangle.cpp - Rectangular sky region class        *
 * ----------------------------------------------------------------------- *
 *  copyright (C) 2019-2021 by Andreas Specovius                           *
 * ----------------------------------------------------------------------- *
 *                                                                         *
 *  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 3 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, see <http://www.gnu.org/licenses/>.  *
 *                                                                         *
 ***************************************************************************/
/**
 * @file GSkyRegionRectangle.hpp
 * @brief Rectangular sky region implementation
 * @author Andreas Specovius
 */

/* __ Includes ___________________________________________________________ */
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
#include "GTools.hpp"
#include "GSkyRegionCircle.hpp"
#include "GSkyRegionRectangle.hpp"
#include "GSkyRegionMap.hpp"

/* __ Method name definitions ____________________________________________ */
#define G_WIDTH                         "GSkyRegionRectangle::width(double&)"
#define G_HEIGHT                       "GSkyRegionRectangle::height(double&)"
#define G_READ                      "GSkyRegionRectangle::read(std::string&)"
#define G_CONTAINS               "GSkyRegionRectangle::contains(GSkyRegion&)"
#define G_OVERLAPS               "GSkyRegionRectangle::overlaps(GSkyRegion&)"
#define G_CORNER                          "GSkyRegionRectangle::corner(int&)"

/* __ Macros _____________________________________________________________ */

/* __ Coding definitions _________________________________________________ */

/* __ Debug definitions __________________________________________________ */

/* __ Prototype __________________________________________________________ */


/*==========================================================================
 =                                                                         =
 =                         Constructors/destructors                        =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Void constructor
 ***************************************************************************/
GSkyRegionRectangle::GSkyRegionRectangle(void) : GSkyRegion()
{
    // Initialise members
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Sky direction constructor
 *
 * @param[in] centre Centre of rectangle.
 * @param[in] width Region width (degrees).
 * @param[in] height Region height (degrees).
 * @param[in] posang Position angle (degrees).
 ***************************************************************************/
GSkyRegionRectangle::GSkyRegionRectangle(const GSkyDir& centre,
                                         const double&  width,
                                         const double&  height,
                                         const double&  posang) : GSkyRegion()
{
    // Initialise members
    init_members();

    // Set members
    this->centre(centre);
    this->width(width);
    this->height(height);
    this->posang(posang);

    // Compute solid angle
    compute_solid_angle();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Right Ascension and Declination constructor
 *
 * @param[in] ra Right Ascension of region centre (degrees).
 * @param[in] dec Declination of region centre (degrees).
 * @param[in] width Region width (degrees).
 * @param[in] height Region height (degrees).
 * @param[in] posang Position angle (degrees).
 ***************************************************************************/
GSkyRegionRectangle::GSkyRegionRectangle(const double& ra,
                                         const double& dec,
                                         const double& width,
                                         const double& height,
                                         const double& posang) : GSkyRegion()
{
    // Initialise members
    init_members();

    // Set members
    this->centre(ra, dec);
    this->width(width);
    this->height(height);
    this->posang(posang);

    // Compute solid angle
    compute_solid_angle();

    // Return
    return;
}


/***********************************************************************//**
 * @brief String constructor
 *
 * @param[in] line DS9 region file line.
 *
 * Constructs region from a DS9 region file line.
 ***************************************************************************/
GSkyRegionRectangle::GSkyRegionRectangle(const std::string& line) : GSkyRegion()
{
     // Initialise members
     init_members();

     // Read information from DS9 region file line
     read(line);

     // Return
     return;

}


/***********************************************************************//**
 * @brief Copy constructor
 *
 * @param[in] region Rectangular sky region.
 ***************************************************************************/
GSkyRegionRectangle::GSkyRegionRectangle(const GSkyRegionRectangle& region) :
                     GSkyRegion(region)
{
    // Initialise members
    init_members();

    // Copy members
    copy_members(region);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Destructor
 ***************************************************************************/
GSkyRegionRectangle::~GSkyRegionRectangle(void)
{
    // Free members
    free_members();

    // Return
    return;
}


/*==========================================================================
 =                                                                         =
 =                               Operators                                 =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Assignment operator
 *
 * @param[in] region Rectangular sky region.
 * @return Rectangular sky region.
 ***************************************************************************/
GSkyRegionRectangle& GSkyRegionRectangle::operator=(const GSkyRegionRectangle& region)
{
    // Execute only if object is not identical
    if (this != &region) {

        // Copy base class members
        this->GSkyRegion::operator=(region);

        // Free members
        free_members();

        // Initialise private members for clean destruction
        init_members();

        // Copy members
        copy_members(region);

    } // endif: object was not identical

    // Return this object
    return *this;
}


/*==========================================================================
 =                                                                         =
 =                              Public methods                             =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Clear instance
 ***************************************************************************/
void GSkyRegionRectangle::clear(void)
{
    // Free members
    free_members();
    this->GSkyRegion::free_members();

    // Initialise private members
    this->GSkyRegion::init_members();
    init_members();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Clone rectangular sky region
 *
 * @return Deep copy to rectangular sky region.
 ***************************************************************************/
GSkyRegionRectangle* GSkyRegionRectangle::clone(void) const
{
    // Clone rectangular sky region
    return new GSkyRegionRectangle(*this);
}


/***********************************************************************//**
 * @brief Set width of rectangular region
 *
 * @param[in] width Rectangle width (degrees).
 *
 * @exception GException::invalid_argument
 *            Width value is less than 0.
 *
 * Sets the @p width of the rectangular sky region. Only non-negative widths
 * are allowed.
 ***************************************************************************/
void GSkyRegionRectangle::width(const double& width)
{
    // Throw an exception if the width is less than zero
    if (width < 0.0) {
        std::string msg =
            "A negative width of "+gammalib::str(width)+" degrees has been "
            "specified for a rectangular sky region. Please specify a "
            "non-negative width.";
        throw GException::invalid_argument(G_WIDTH, msg);
    }

    // Set radius value
    m_width = width;

    // Compute the solid angle
    compute_solid_angle();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Set height of rectangular region
 *
 * @param[in] height Rectangle height (degrees).
 *
 * @exception GException::invalid_argument
 *            Height value is less than 0.
 *
 * Sets the @p height of the rectangular sky region. Only non-negative
 * heights are allowed.
 *
 * Note that for a position angle of zero, the height axis is pointing to
 * celestial north.
 ***************************************************************************/
void GSkyRegionRectangle::height(const double& height)
{
    if (height < 0.0) {
        std::string msg =
            "A negative height of "+gammalib::str(height)+" degrees has been "
            "specified for a rectangular sky region. Please specify a "
            "non-negative height.";
        throw GException::invalid_argument(G_HEIGHT, msg);
    }

    // Set radius value
    m_height = height;

    // Compute the solid angle
    compute_solid_angle();

    // Return
    return;
}


/***********************************************************************//**
 * @brief Read region from DS9 string
 *
 * @param[in] line String in DS9 format.
 *
 * @exception GException::invalid_value
 *            Invalid value found in DS9 format string.
 ***************************************************************************/
void GSkyRegionRectangle::read(const std::string& line)
{
    // Clear the current instance
    clear();

    // Split the string into 2 parts seperated by #
    std::vector<std::string> substrings = gammalib::split(line,"#");
    std::string region_def = (!substrings.empty())   ? substrings[0] : "";
    std::string comment    = (substrings.size() > 1) ? substrings[1] : "";

    // Finding the box
    if (region_def.find("box") == std::string::npos) {
        std::string msg =
            "Unable to find the key word \"box\" in provided string "
            "\""+line+"\". The \"box\" key word is mandatory.";
        throw GException::invalid_value(G_READ, msg);
    }

    // Get the the coordinate system of the values
    std::string system = gammalib::split(region_def, ";")[0];

    // Get the substring of the important values
    unsigned    pos          = region_def.find("box(");
    unsigned    end          = region_def.find(")");
    std::string regionstring = region_def.substr(pos+4, end);
    regionstring.erase(regionstring.find(")"), 1);

    // Get the values of the region x,y,and radius
    std::vector<std::string> values = gammalib::split(regionstring,",");
    if (values.size() != 5) {
        std::string msg =
            "Invalid number of "+gammalib::str(values.size())+" arguments "
            "after the \"box\" key word in provided string \""+line+"\". "
            "Exactly 5 arguments are expected.";
        throw GException::invalid_value(G_READ, msg);
    }
    double x      = gammalib::todouble(values[0]);
    double y      = gammalib::todouble(values[1]);
    double width  = gammalib::todouble(values[2]);
    double height = gammalib::todouble(values[3]);
    double posang = gammalib::todouble(values[4]);

    // Get radius units
    if (gammalib::contains(values[2], "'")) {
        width /= 60.0;
    }
    else if (gammalib::contains(values[2], "\"")) {
        width /= 3600.0;
    }
    if (gammalib::contains(values[3], "'")) {
        height /= 60.0;
    }
    else if (gammalib::contains(values[3], "\"")) {
        height /= 3600.0;
    }
    if (gammalib::contains(values[4], "'")) {
        posang /= 60.0;
    }
    else if (gammalib::contains(values[4], "\"")) {
        posang /= 3600.0;
    }

    // Initialise centre direction
    GSkyDir centre = GSkyDir();

    // Set the values in the correct system
    if (system == "fk5" || system == "icrs") {
        centre.radec_deg(x,y);
    }
    else if (system == "galactic") {
        centre.lb_deg(x,y);
    }
    else {
        std::string msg =
            "Unsupported coordinate system \""+system+"\" in provided string "
            "\""+line+"\". Only the following coordinate systems are "
            "supported: \"fk5\", \"icrs\" and \"galactic\".";
        throw GException::invalid_value(G_READ, msg);
    }

    // Set members
    this->centre(centre);
    this->width(width);
    this->height(height);
    this->posang(posang);

    // Compute solid angle
    compute_solid_angle();

    // Check if there is a given name for the region and set it
    std::vector<std::string>comments = gammalib::split(comment, " ");
    for (int i = 0; i < comments.size(); ++i) {
        if (gammalib::contains(comments[i], "text")) {
            std::vector<std::string> attributes = gammalib::split(comments[i], "=");
            if (attributes.size() < 2) {
                std::string msg =
                    "Invalid character sequence encountered in provided "
                    "string \""+line+"\". An attribute of the type "
                    "\"text=Name\" is expected.";
                throw GException::invalid_value(G_READ, msg);
            }
            m_name = attributes[1];
        }
    }

    // Return
    return;
}


/***********************************************************************//**
 * @brief Write region into a string
 *
 * @return String to be written in a DS9 region file
 *
 * Writes a DS9 region into a string. The region name is only written if it
 * is defined.
 ***************************************************************************/
std::string GSkyRegionRectangle::write(void) const
{
    // Allocate string
    std::string result;

    // Set string
    result.append("fk5;box(");
    result.append(gammalib::str(m_centre.ra_deg()));
    result.append(",");
    result.append(gammalib::str(m_centre.dec_deg()));
    result.append(",");
    result.append(gammalib::str(m_width));
    result.append(",");
    result.append(gammalib::str(m_height));
    result.append(",");
    result.append(gammalib::str(m_posang));
    result.append(")");

    // Optionally add region name
    if (m_name.length() > 0) {
        result.append(" # text=");
        result.append(m_name);
    }

    // Return string
    return result;
}


/***********************************************************************//**
 * @brief Checks if sky direction lies within region
 *
 * @param[in] dir Sky direction.
 * @return True if sky direction is contained in region.
 *
 * A sky direction lies within a region when its distance to the region
 * centre is not larger than the region extension in both axes directions.
 ***************************************************************************/
bool GSkyRegionRectangle::contains(const GSkyDir& dir) const
{
    // Transform sky direction to local coordinate system
    GSkyPixel local = dir_to_local(dir);

    // Check containment using local coordinate system
    bool contains = this->contains(local);

    // Return containment flag
    return contains;
}


/***********************************************************************//**
 * @brief Checks if region is fully contained within this region
 *
 * @param[in] reg Sky region.
 * @return True if region is contained in region.
 *
 * @exception GException::feature_not_implemented
 *            Not all region types supported currently.
 ***************************************************************************/
bool GSkyRegionRectangle::contains(const GSkyRegion& reg) const
{
    // Initialise return value
    bool contains = false;

    // If the other region is a circle then check whether it is fully
    // contained within the rectangle
    if (reg.type() == "Circle") {

        // Create circular region from reg
        const GSkyRegionCircle* circle =
              static_cast<const GSkyRegionCircle*>(&reg);

        // Transform circle center to local coordinate system
        GSkyPixel local = dir_to_local(circle->centre());
        double    x_deg = local.x() * gammalib::rad2deg;
        double    y_deg = local.y() * gammalib::rad2deg;

        // Get circle radius in degrees
        double radius = circle->radius();

        // Check whether circle is contained within rectangle
        if ((std::abs(y_deg) + radius) <= 0.5 * m_height) {
            if ((std::abs(x_deg) + radius) <= 0.5 * m_width) {
                contains = true;
            }
        }

    } // endif: other region was of type "Circle"

    // ... otherwise, if the other region is a rectangle then check whether
    // all corners fall within the rectangle
    else if (reg.type() == "Rectangle") {

        // Create rectangular region from reg
        const GSkyRegionRectangle* rect =
              static_cast<const GSkyRegionRectangle*>(&reg);

        // Initialise containment flag to true
        contains = true;

        // Loop over the four corners of the rectangle :regrect:
        for (int i = 0; i < 4; ++i) {

            // Get sky direction of current corner
            GSkyDir corner = rect->corner(i);

            // If corner is not contained then set containment flag to false
            // and break
            if (!this->contains(corner)) {
                contains = false;
                break;
            }

        } // endfor: looped over the four corners

    } // endif: other region was a rectangle

    // ... otherwise, if the other region is a map then check whether it
    // is contained within the rectangle
    else if (reg.type() == "Map") {

        // Create map from reg
        const GSkyRegionMap* map = static_cast<const GSkyRegionMap*>(&reg);

        // Get non-zero indices
        const std::vector<int>& indices = map->nonzero_indices();

        // Initialise containment flag to true
        contains = true;

        // Loop over all indices
        for (int i = 0; i < indices.size(); ++i) {

            // Get sky direction of map pixel
            GSkyDir dir = map->map().inx2dir(indices[i]);

            // If pixel is not contained then set containment flag to false
            // and break
            if (!this->contains(dir)) {
                contains = false;
                break;
            }

        } // endfor: looped over non-zero map indices

    } // endif: other region was of type "Map"

    // ... otherwise throw an exception
    else {
        throw GException::feature_not_implemented(G_CONTAINS,
              "Cannot compare to region type \""+reg.type()+"\" yet.");
    }

    // Return containment flag
    return contains;
}


/***********************************************************************//**
 * @brief Checks if region is overlapping with this region
 *
 * @param[in] reg Sky region.
 * @return True if region overlaps with region.
 *
 * @exception GException::feature_not_implemented
 *            Regions differ in type.
 *
 * Checks whether a region overlaps with the rectangle.
 *
 * @todo Improve implementation for rectangle-rectangle
 ***************************************************************************/
bool GSkyRegionRectangle::overlaps(const GSkyRegion& reg) const
{
    // Initialise return value
    bool overlap = false;

    // If the region is a circle then check overlap between circle and
    // rectangle using the alogrithm given under #292 at
    // https://stackoverflow.com/questions/401847/circle-rectangle-collision-detection-intersection
    if (reg.type() == "Circle") {

        // Create circular region from reg
        const GSkyRegionCircle* circle =
              static_cast<const GSkyRegionCircle*>(&reg);

        // Transform circle center to local coordinate system
        GSkyPixel local = dir_to_local(circle->centre());
        double    x_deg = std::abs(local.x() * gammalib::rad2deg);
        double    y_deg = std::abs(local.y() * gammalib::rad2deg);

        // Get circle radius in degrees
        double radius = circle->radius();

        // Compute half width and height
        double half_width  = 0.5 * width();
        double half_height = 0.5 * height();

        // Assess overlap
        if (x_deg > (half_width + radius)) {
            overlap = false;
        }
        else if (y_deg > (half_height + radius)) {
            overlap = false;
        }
        else if (x_deg <= half_width) {
            overlap = true;
        }
        else if (y_deg <= half_height) {
            overlap = true;
        }
        else {
            double dx = x_deg - half_width;
            double dy = y_deg - half_height;
            double d2 = dx * dx + dy * dy;
            if (d2 <= radius*radius) {
                overlap = true;
            }
        }

    } // endif: region was of type "Circle"

    // ... otherwise if region is a rectangle then check overlap between
    // two rectangles
    else if (reg.type() == "Rectangle") {

        // Create rectangular region from reg
        const GSkyRegionRectangle* rect =
              static_cast<const GSkyRegionRectangle*>(&reg);

        // Dirty kludge: compare with map
        GSkyRegionMap map = GSkyRegionMap(rect);
        overlap           = map.overlaps(*this);

    } // endif: region was a rectangle

    // ... otherwise if region is map then check overlap with map
    else if (reg.type() == "Map") {

        // Create map from reg
        const GSkyRegionMap* map = static_cast<const GSkyRegionMap*>(&reg);

        // Check overlap with circle
        overlap = map->overlaps(*this);

    } // endif: region was map

    // ... otherwise throw an exception
    else {
        throw GException::feature_not_implemented(G_CONTAINS,
              "Cannot compare to region type \""+reg.type()+"\" yet.");
    }

    // Return value
    return overlap;
}


/***********************************************************************//**
 * @brief Return sky direction of one corner of the rectangle
 *
 * @param[in] index Corner index [0,3]
 * @return Sky direction of corner.
 *
 * @exception GException::out_of_range
 *            Corner index is not in [0,3].
 *
 * Returns the sky direction of one corner of the rectangle. The corner
 * @p index is counted counterclockwise from the positive Right Ascension
 * and Declination directions:
 *
 *  * `index = 0` : \f$(\alpha + \frac{\rm width}{2}, \delta + \frac{\rm height}{2})\f$
 *  * `index = 1` : \f$(\alpha + \frac{\rm width}{2}, \delta - \frac{\rm height}{2})\f$
 *  * `index = 2` : \f$(\alpha - \frac{\rm width}{2}, \delta + \frac{\rm height}{2})\f$
 *  * `index = 3` : \f$(\alpha - \frac{\rm width}{2}, \delta - \frac{\rm height}{2})\f$
 *
 * where \f$\alpha\f$ and \f$\delta\f$ are the Right Ascension and
 * Declination of the rectangle centre, and \f${\rm width}\f$ and
 * \f${\rm height}\f$ are the width and height of the rectangle.
 ***************************************************************************/
GSkyDir GSkyRegionRectangle::corner(const int& index) const
{
    // Assert index is in [0,3]
    if ((index < 0) || (index > 3)) {
        throw GException::out_of_range(G_CORNER, "Corner index", index, 3);
    }

    // Compute the offset to the corners in degrees
    double dx = 0.5 * m_width  * (1 - 2*int(index >= 2));
    double dy = 0.5 * m_height * (1 - 2*int((index==1) || (index==2)));

    // Convert offset into radians
    dx *= gammalib::deg2rad;
    dy *= gammalib::deg2rad;

    // Define corner in local coordinates
    GSkyPixel local(dx,dy);

    // Transform to global sky direction
    GSkyDir corner = local_to_dir(local);

    // Return
    return corner;
}


/***********************************************************************//**
 * @brief Print rectangular region
 *
 * @param[in] chatter Chattiness
 * @return String containing region information.
 ***************************************************************************/
std::string GSkyRegionRectangle::print(const GChatter& chatter) const
{
    // Initialise result string
    std::string result;;

    // Continue only if chatter is not silent
    if (chatter != SILENT) {

        // Append header
        result.append("=== GSkyRegionRectangle ===");

        // Append sky region information
        result.append("\n"+gammalib::parformat("Right Ascension of centre"));
        result.append(gammalib::str(ra())+" deg");
        result.append("\n"+gammalib::parformat("Declination of centre"));
        result.append(gammalib::str(dec())+" deg");
        result.append("\n"+gammalib::parformat("Width"));
        result.append(gammalib::str(width())+" deg");
        result.append("\n"+gammalib::parformat("Height"));
        result.append(gammalib::str(height())+" deg");
        result.append("\n"+gammalib::parformat("PA"));
        result.append(gammalib::str(posang())+" deg");

    } // endif: chatter was not silent

    // Return result
    return result;
}


/*==========================================================================
 =                                                                         =
 =                             Private methods                             =
 =                                                                         =
 ==========================================================================*/

/***********************************************************************//**
 * @brief Initialise class members
 ***************************************************************************/
void GSkyRegionRectangle::init_members(void)
{
    // Initialise members
    m_type   = "Rectangle";
    m_centre.clear();
    m_width  = 0.0;
    m_height = 0.0;
    m_posang = 0.0;

    //Return
    return;
}


/***********************************************************************//**
 * @brief Copy class members
 *
 * @param[in] region Rectangular sky region.
 ***************************************************************************/
void GSkyRegionRectangle::copy_members(const GSkyRegionRectangle& region)
{
    // Copy attributes
    m_centre = region.m_centre;
    m_width  = region.m_width;
    m_height = region.m_height;
    m_posang = region.m_posang;

    // Return
    return;
}


/***********************************************************************//**
 * @brief Delete class members
 ***************************************************************************/
void GSkyRegionRectangle::free_members(void)
{
    // Return
    return;
}


/***********************************************************************//**
 * @brief Compute solid angle of rectangle
 *
 * This method computes the solid angle of the rectangle in steradians and
 * stores the result in the `m_solid` member.
 *
 * The solid angle is computed using
 *
 * \f[
 *    \Omega = 2 {\rm width} \sin \left( \frac{\rm height}{2} \right)
 * \f]
 *
 * where \f${\rm width}\f$ and \f${\rm height}\f$ are the width and the
 * height of the rectangle expressed in radians.
 ***************************************************************************/
void GSkyRegionRectangle::compute_solid_angle(void)
{
    // Compute solid angle
    m_solid = 2.0 * m_width * gammalib::deg2rad * sin(0.5 * m_height * gammalib::deg2rad);

    // Return
    return;
}


/***********************************************************************//**
 * @brief Checks if local direction lies within region
 *
 * @param[in] local Sky direction in local coordinate system.
 *
 * A local direction lies within a region when its distance to the region
 * centre is not larger than the region extension in both axes directions.
 ***************************************************************************/
bool GSkyRegionRectangle::contains(const GSkyPixel& local) const
{
    // Initialise return value
    bool contains = false;

    // Get local y coordinate in degrees
    double y = local.y() * gammalib::rad2deg;

    // Continue if the y coordinate is within half of the height
    if ((std::abs(y) - 1.0e-10) <= 0.5 * m_height) {

        // Get local x coordinate in degrees
        double x = local.x() * gammalib::rad2deg;

        // If the x coordinate is within half of the width the local
        // coordinate is contained within the rectangle
        if ((std::abs(x) - 1.0e-10) <= 0.5 * m_width) {
            contains = true;
        }

    } // endif: y coordinate is within half of the height

    // Return containment
    return contains;
}


/***********************************************************************//**
 * @brief Transform sky direction to local rectangle coordinates
 *
 * @param[in] dir Sky direction.
 * @return Local cartesian region object coordinates.
 *
 * Transform the sky direction @p dir to the local cartesian coordinate
 * system. The local coordinates \f$(x,y)\f$ are defined by
 *
 * \f[
 *    x = d \sin(\phi)
 * \f]
 *
 * and
 *
 * \f[
 *    y = d \cos(\phi)
 * \f]
 *
 * where \f$d\f$ is the angular distance between the sky direction @p dir
 * and the centre of the rectangle in radians, and \f$\phi\f$ is the
 * position angle of the sky direction @p dir with respect to the position
 * angle of the rectangle, computed counter clockwise.
 ***************************************************************************/
GSkyPixel GSkyRegionRectangle::dir_to_local(const GSkyDir& dir) const
{
    // Get distance and polar angle relative to rectangle centre
    double dist = m_centre.dist(dir);
    double pa   = m_centre.posang(dir) - m_posang * gammalib::deg2rad;

    // Compute corresponding x and y coordinate from polar coords
    double x = dist * std::sin(pa);
    double y = dist * std::cos(pa);

    // Create local coordinates
    GSkyPixel local(x,y);

    // Return local coordinates
    return local;
}


/***********************************************************************//**
 * @brief Transform local rectangle coordinates to sky direction
 *
 * @param[in] local Local rectangle coordinates.
 * @return Sky direction.
 *
 * Transform the local coordinates @p local to a sky direction. See
 * GSkyRegionRectangle::dir_to_local for the relation between local rectangle
 * coordinates and the sky direction.
 ***************************************************************************/
GSkyDir GSkyRegionRectangle::local_to_dir(const GSkyPixel& local) const
{
    // Get local coordinates (radians)
    double x = local.x();
    double y = local.y();

    // Compute dist and position angle in polar coords
    double dist = std::sqrt(x*x + y*y);
    double pa   = gammalib::pihalf - std::atan2(y, x) +
                  m_posang  * gammalib::deg2rad;

    // Create output global sky direction
    GSkyDir dir(m_centre);
    dir.rotate(pa, dist);

    // Return sky direction
    return dir;
}