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Spherical Coordinates in XNA

OTD uses a globe based board. As such, spherical coordinates can be useful. The aim is to eventually be able to load a terrain map from a 2D image file. While I have a ways to go before that, I do have a utility class for managing spherical coordinates. Taking from DirectX’s Vector2 and Vector3, I settled on Polar3 as a short yet descriptive stuct name.

Spherical coordinates need three basic elements - the azimuth phi, Φ; elevation theta θ; and a radius, R. Φ runs between 0 and 2π. θ however presents some choices. We can set one of the poles to be 0 and the other π, or set the equator to 0 and the poles to be +/- π/2. I haven’t gotten far enough to determine which is the best way, leaving the north pole at 0 appears to be the easiest option when dealing with rotations. We’ll see in practice though. Two properties for latitude and longitude can be set up that handle as expected.

Another thing to remember is which direction is up. The equations over at wikipedia and elsewhere place up as the Z axis. However, the default in XNA is Y axis, so Cartesian conversions need to account for this.

/// <summary>
/// Spherical Coordinate
/// </summary>
/// <see cref="http://en.wikipedia.org/wiki/Spherical_coordinates"/>
/// <see cref="http://www.radicalcartography.net/?projectionref"/>
public struct Polar3 : IEquatable<Polar3>
{
    /// <summary>
    /// Angle in the Right, Forward plane
    /// </summary>
    public float Phi;

    /// <summary>
    /// Angle around the Up axis
    /// </summary>
    public float Theta;

    /// <summary>
    /// Radius
    /// </summary>
    public float R;

    #region Latitude and Longitude Properties

    /// <summary>
    /// Latitude in radians
    /// -pi/2 <= theta <= pi/2
    /// </summary>
    public float Latitude
    {
        get
        {
            return MathHelper.PiOver2 - Theta;
        }
    }

    /// <summary>
    /// Longitude in radians
    /// 0 <= phi < 2pi
    /// </summary>
    public float Longitude
    {
        get
        {
            return Phi + (float)Math.PI;
        }
    }

    #endregion;


    #region Constructors

    public Polar3(float theta, float phi, float r)
    {
        Theta = theta;
        Phi = phi;
        R = r;
    }

    #endregion;

    #region Operators

    /// <summary>
    /// Equals
    /// </summary>
    /// <param name="value1"></param>
    /// <param name="value2"></param>
    /// <returns></returns>
    public bool Equals(Polar3 other)
    {
        return ((this.Theta == other.Theta) &&
            (this.Phi == other.Phi) &&
            (this.R == other.R));
    }

    /// <summary>
    /// Equality Operator
    /// </summary>
    /// <param name="value1"></param>
    /// <param name="value2"></param>
    /// <returns></returns>
    public static bool operator ==(Polar3 value1, Polar3 value2)
    {
        return value1.Equals(value2);
    }

    /// <summary>
    /// Inequality Operator
    /// </summary>
    /// <param name="value1"></param>
    /// <param name="value2"></param>
    /// <returns></returns>
    public static bool operator !=(Polar3 value1, Polar3 value2)
    {
        return !value1.Equals(value2);
    }

    /// <summary>
    /// Get the object hashcode
    /// </summary>
    /// <returns></returns>
    public override int GetHashCode()
    {
        return Theta.GetHashCode() + Phi.GetHashCode() + R.GetHashCode();
    }

    #endregion;

    #region 2D Map Projections

    /// <summary>
    /// Returns the xy Miller projection
    /// </summary>
    /// <see cref="http://en.wikipedia.org/wiki/Miller_cylindrical_projection"/>
    /// <returns></returns>
    public Vector2 ProjectMiller()
    {
        Vector2 xy = new Vector2();

        xy.X = Longitude;

        /*
         * (5/4) * ln( tan( pi/4 + 2theta/5 ) )
         */
        xy.Y = Latitude * 2 / 5;
        xy.Y = xy.Y + MathHelper.PiOver4;
        xy.Y = (float)Math.Log( Math.Tan(xy.Y) );
        xy.Y = xy.Y * 1.25f;

        return xy;
    }

    #endregion;

    /// <summary>
    /// Create a polar coordinate from a cartesian vector.
    /// </summary>
    /// <param name="vector"></param>
    /// <returns></returns>
    public static Polar3 CreateFromVector(Vector3 vector)
    {
        /*
         * DirectX defines up as (0,1,0), so we need to move some coords around.
         * Theta becomes the vertical angle against the Y axis
         * Phi becomes the horizontal angle in the XZ plane.
         */
        Polar3 coordinate = new Polar3(
            (float)Math.Acos(vector.Y / vector.Length()),
            (float)Math.Atan2(vector.X, vector.Z),
            vector.Length()
            );
        return coordinate;
    }

    /// <summary>
    /// To String
    /// </summary>
    /// <returns></returns>
    public override string ToString()
    {
        return (String.Format("(Theta:{0}, Phi:{1}, R:{2}, Lat:{3}, Long:{4})", Theta, Phi, R, Latitude, Longitude));
    }
}

Map Projections

ProjectMiller() returns 2D coordinate when using a Miller projection. It needs a little bit of work still, but the coordinates appear to be sound (just needs some offsetting) to map to a 2D image.

Other projections such as Mercator and Mollweide could be added as well.

Notes

Currently all the available values (Phi, Theta) and properties (Latitude, Longitude) are in radians. Given that all other C# functions use radians, this seemed the simplest.