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Mapping Images on Spherical Surfaces Using C#

, 9 Apr 2013 GPL3
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Mapping images on spherical surfaces using C#

Introduction

This article describes how to map a flat 2D image (JPG, BMP, or GIF) on a sphere by using basic algebra.

The process is very simple where the x axis of the image will be mapped on sphere longitudes and the y axis of the image will be mapped on sphere latitudes.

The process of mapping is similar to proportion equations x-x0/y-y0 = px-x0/py-y0

public static double MapCoordinate(double i1, double i2, double w1,
    double w2, double p)
{
    return ((p - i1) / (i2 - i1)) * (w2 - w1) + w1;
}

Screenshot - worldmap4.gif
Original image

Screenshot - mapping.png
Resulting image

Background

A Sphere Can Be Represented by Spherical Coordinates in R3

  • radius
  • phi (latitude angle)
  • theta (longitude angle)

Image 2

  • Where radius is a constant, phi=[-PI/2,PI/2], and theta=[0,2*PI]

To Find the Cartesian Coordinates from Spherical Coordinates

  • x = radius * sin(phi) * cos(theta)
  • y = radius * sin(phi) * sin(theta)
  • z = radius * cos(theta)
double phi0 = 0.0;
double phi1 = Math.PI;
double theta0 = 0.0;
double theta1 = 2.0*Math.PI;

The Code

At first we code the image loading

System.Drawing.Image image1 = new Bitmap(Server.MapPath(
    "./images/worldmap4.gif"));
Bitmap imgBitmap = new Bitmap(image1);

Now we make a loop through the 2 dimensions of the image, map phi and theta angles from image coordinates, get the cartesian 3D coordinates from phi and theta, provide some rotation to the obtained 3D points and plot them with respective image color:

for (int i = 0; i < imgBitmap.Width; i++)
     {
     for (int j = 0; j < imgBitmap.Height; j++)
          {
          // map the angles from image coordinates
          double theta = Algebra.MapCoordinate(0.0, imgBitmap.Width - 1,
              theta1, theta0, i);
          double phi = Algebra.MapCoordinate( 0.0, imgBitmap.Height - 1,phi0,
              phi1, j);
          // find the cartesian coordinates
          double x = radius * Math.Sin(phi) * Math.Cos(theta);
          double y = radius * Math.Sin(phi) * Math.Sin(theta);
          double z = radius * Math.Cos(phi);
          // apply rotation around X and Y axis to reposition the sphere
          RotX(1.5, ref y, ref z);
          RotY(-2.5, ref x, ref z);
          // plot only positive points
          if (z > 0)
               {
               Color color = imgBitmap.GetPixel(i, j);
               Brush brs = new SolidBrush(color);
               int ix = (int)x + 100;
               int iy = (int)y + 100;
               graphics.FillRectangle(brs, ix, iy, 1, 1);
               brs.Dispose();
              }
          }
     }

The Rotation Functions [almost forgot]

Actually I made a 3D Math class, but here you will need only these functions

public static void RotX(double angle, ref double y, ref double z)
     {
     double y1 = y * System.Math.Cos(angle) - z * System.Math.Sin(angle);
     double z1 = y * System.Math.Sin(angle) + z * System.Math.Cos(angle);
     y = y1;
     z = z1;
     }
public static void RotY(double angle, ref double x, ref double z)
     {
     double x1 = x * System.Math.Cos(angle) - z * System.Math.Sin(angle);
     double z1 = x * System.Math.Sin(angle) + z * System.Math.Cos(angle);
     x = x1;
     z = z1;
     }
public static void RotZ(double angle, ref double x, ref double y)
     {
     double x1 = x * System.Math.Cos(angle) - y * System.Math.Sin(angle);
     double y1 = x * System.Math.Sin(angle) + y * System.Math.Cos(angle);
     x = x1;
     y = y1;
     }

See sample

License

This article, along with any associated source code and files, is licensed under The GNU General Public License (GPLv3)

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About the Author

andalmeida
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Brazil Brazil
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Comments and Discussions

 
GeneralRe: Why? PinmemberPatrick Sears31-Jul-07 11:17 
GeneralRe: Why? PinmemberWerdna13-Aug-07 5:57 
Jokesweet PinmemberBen Daniel23-Jul-07 19:10 
AnswerRe: sweet Pinmemberandalmeida24-Jul-07 4:53 
GeneralRe: sweet PinmemberN.L. Neilson2-Aug-07 12:01 
GeneralRe: sweet Pinmemberandalmeida2-Aug-07 12:03 
GeneralRe: sweet Pinmemberandalmeida2-Aug-07 12:17 
GeneralRe: sweet PinmemberN.L. Neilson2-Aug-07 12:51 
That is an interesting link, it has been many years since I took calculus.
I have tinkered with the Vincenty Formula considerably for the geodesic distance and bearings.
I have not tried 2D pictures on a sphere, that is why your post was interesting.
 
NASA WorldWind 1.4 in .net C# and WWJ in Java is using a sphereoid rather than an ellipsoid. If the math could be worked out it would be a real improvement.
 
nlneilson

GeneralRe: sweet Pinmemberandalmeida2-Aug-07 12:51 
GeneralRe: sweet PinmemberN.L. Neilson2-Aug-07 14:48 

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