## Introduction

In part one of this series, I described how to write an interpreter for raw GPS NMEA data. Part two described how to monitor and enforce GPS precision data to develop commercial-quality software. The articles includes source code in C# and VB.NET which harness the power of GPS satellites to determine the current location, synchronize the computer clock to atomic time, and point to a satellite on a cloudy day. Yet, even with all of this code, most developers still need a way to display GPS information along with other geographic features. With the help of my colleague Phil Smith, a lead developer of our “GIS.NET” mapping component and the “Geodesy.NET” coordinate and projection library, this article will teach you how to generate your own maps.

## The Rule of Threes

In order to understand the technology behind mapping, it’s necessary to have a solid understanding of three coordinate systems: *geographic*, *projected*, and *pixel*. Each system serves an important role when displaying a map, and transformations from one system to another are essential. Developers typically start with a *geographic* coordinate (expressed as latitude and longitude). Then, it is transformed from Earth’s oblate spheroid (roughly spherical) shape to a plane, resulting in a *projected* coordinate: a truly flat, two-dimensional coordinate. A projected coordinate is an easting/northing pair describing a distance East of a “central meridian” (a line of longitude) and a distance North of a “central parallel” (a line of latitude). The method which transforms a geographic coordinate to and from a projected easting/northing coordinate is called a *projection*. Finally, projected coordinates are translated and scaled so that they align to a specific place on-screen, resulting in *pixel* coordinates. Pixel coordinates are the same coordinates that you’ve already used to align controls on a Form.

Figure 1.1: Geographic coordinates are converted from 3D to 2D using a map projection. Then, a viewport scales and translates a portion of the map to display Florida.

Let’s take a closer look at each of the three coordinate systems and how to convert between them to produce a map.

## The Earth is Better Flat

In parts one and two of this article, we discussed how GPS devices report your location as a latitude and longitude. These pairs are often referred to as “geographic coordinates.” The problem with geographic coordinates, however, is that they represent coordinates on the Earth’s surface, which is a spheroid (roughly spherical) shape. Since our computer monitors are flat, we need a way to “unfold” the Earth into a perfectly flat shape before we try to display it. This technique is known as *projection*, and it is essential for displaying maps. There are over a hundred map projections in use around the world today, and each projection serves a specific purpose. For example, the Mercator projection is widely used by boats and ships because it produces a map in which lines of constant bearing are a straight line, which greatly simplifies navigation. However, a side-effect of this projection is that it distorts the size of everything as you get closer to the North or South Poles, making this projection unsuitable for other purposes.

Figure 1.2: Countries of the world are displayed using two projections, Mercator and Polyconic, to demonstrate how projections can produce widely-differing views of the same data.

As you can see, map projections can make the same geographic features appear in completely different sizes and shapes, but each is perfectly valid. In fact, some projections such as the “Orthographic” projection are flat, but produce the illusion of a 3D image on a flat monitor. This projection is a contemporary example which is widely used by many 3D applications, including modern 3D game engines. Regardless of the shape and size, projected coordinates flatten 3D coordinates, and this greatly simplifies the task of mapping.

## Here Comes the Science

The mathematics behind map projections can be somewhat intimidating. For example, the “Van der Grinten” projection uses the following formula (where A, G, P, and Q are mapping parameters):

For this article, we’ll be writing the code for a much simpler projection known as “Equidistant Cylindrical” or “Plate Carée,” which, because of its simplicity and speed, is the default projection used by our GIS.NET 3.0 component, which includes a library of twenty-five other projections:

The formulas for map projection are easier to work with when geographic coordinates are expressed as radians. Radians are straightforward to calculate, and can be applied to either a latitude or longitude:

double degrees = 90.0;
double radians = degrees * (Math.Pi / 180.0);
degrees = radians / (Math.PI / 180.0);

All map projection source code is divided into two methods. The first method, referred to as a *forward* projection, will convert a geographic coordinate into a projected coordinate. The second method is exactly the opposite, converting a projected coordinate back into a geographic coordinate. This is referred to as a *reverse projection* or *de-projection*. Here’s how the two methods look for our example Plate Carée projection:

using System.Drawing;
public class PlateCaree
{
public PointF Project(PointF geographicCoordinate)
{
double radianX = geographicCoordinate.X * (Math.PI / 180);
double radianY = geographicCoordinate.Y * (Math.PI / 180);
PointF result = new PointF();
result.X = (float)(radianX * Math.Cos(0));
result.Y = (float)radianY;
return result;
}
public PointF Deproject(PointF projectedCoordinate)
{
PointF result = new PointF();
result.X = (float)(projectedCoordinate.X / Math.Cos(0) /
(Math.PI / 180.0));
result.Y = (float)(projectedCoordinate.Y / (Math.PI / 180.0));
return result;
}
}

With this class, we can now produce projected coordinates for any location on Earth:

PointF myLocation = new PointF();
myLocation.Y = 39.0;
myLocation.X = -105.0;
PlateCaree projection = new PlateCaree();
PointF myProjectedLocation = projection.Project(myLocation);
myLocation = projection.Deproject(myProjectedLocation);

... this process is then repeated for each geographic coordinate, until all data can be represented in projected coordinates. Once this has been done, only one step remains to convert these coordinates into *pixel* coordinates which can be painted on the screen.

## Paint the Planet

If we were creating a map to display on a wall, our task would be easy because we could paint all of the data once and be done with it. However, mapping software should let users pan and zoom a map so that they can explore any part of it in greater detail. To do this, we must imagine a rectangle (which we refer to as a “viewport” in GIS.NET 3.0) which represents the portion of the map we actually want to see. Once this is known, math is applied a third time to convert projected coordinates into pixel coordinates. In other words, we must make the upper-left of our viewport match up to (0,0) in our `Form`

.

Figure 1.4: A viewport is used to see a portion of all the projected coordinates. In this case, a viewport displays the continent of Africa.

.NET developers are already familiar with pixel coordinates. These are the same coordinates which you’ve used to place controls onto a `Form`

, so there’s nothing new to explain here. But, we need a way to convert projected coordinates into pixel coordinates. To do this, projected coordinates must be scaled and translated to make the viewport align with the pixel size of the `Form`

. Translation is performed by applying the negative value of the X coordinate, then the Y coordinates of the upper-left corner of the viewport. Horizontal scale is calculated by dividing the pixel width of the area to paint by the projected width of the viewport, and similarly to calculate vertical scale.

Fortunately, on desktops, we can make use of the `Matrix`

class to do all of the heavy lifting for this task. `Matrix`

objects can rotate, translate, and scale an array of coordinates in the form of a `PointF`

array. The resulting code will look something like this:

Matrix transform = new Matrix();
transform.Translate(-viewport.X, viewport.Y, MatrixOrder.Append);
transform.Scale(this.Width / viewport.Width,
this.Height / -viewport.Height, MatrixOrder.Append);

You may have noticed how, for the vertical scale, a negative sign is used. This is because projected coordinate systems have a Y-axis which is the opposite of pixel coordinate systems. In other words, greater Y values travel up in projected coordinates, whereas greater Y values in pixel coordinates travel down. A negative sign here prevents the image from being displayed upside-down.

Since we’re using GDI+ for this example, all painting is done using a `Graphics`

class, typically during an `OnPaint`

method. Thankfully, we can apply our transformation and scale easily by assigning the `Transform`

property of the `Graphics`

object to our `Matrix`

. With this in place, we can now call paint methods such as `DrawLine`

using projected coordinates! As a result, painting objects becomes rather trivial:

protected override void OnPaint(PaintEventArgs e)
{
e.Graphics.Transform = transform;
e.Graphics.FillPolygon(Brushes.Green, projectedCoordinates);
e.Graphics.DrawPolygon(Pens.Black, projectedCoordinates);
}

## Have a Good Aspect

In this article, we’re dealing with two rectangles: the “viewport,” a projected area to be painted, and the `Form`

itself, where everything will be displayed. If the shape of the viewport differs greatly from the shape of the `Form`

, however, distortion can occur (see the “Before” picture below). To fix this problem, we must make the shape of the viewport match the shape of the `Form`

. This is done by adjusting the “aspect ratio” of the viewport.

Figure 1.5: The state of Nebraska is drawn with no correction (left), then with aspect ratio correction (right) to preserve its shape.

Aspect ratio is calculated by dividing the width of a rectangle by its height. For example, if the width of a rectangle were ten pixels, and its height were twenty pixels, then the aspect ratio would be 0.5. To adjust the aspect ratio of the viewport, its aspect ratio is compared to the aspect ratio of the rectangular `Form`

itself. If the viewport’s aspect ratio is greater than the `Form`

’s, the viewport’s height is increased. Otherwise, the viewport’s width is increased. The resulting code looks like this:

float pixelAspectRatio = (float)this.Width / this.Height;
float projectedAspectRatio = viewport.Width / viewport.Height;
RectangleF adjustedViewport = viewport;
if (pixelAspectRatio > projectedAspectRatio)
{
adjustedViewport.Inflate(
(pixelAspectRatio * adjustedViewport.Height - adjustedViewport.Width)
/ 2,
0);
}
else if (pixelAspectRatio < projectedAspectRatio)
{
adjustedViewport.Inflate(
0,
(adjustedViewport.Width / pixelAspectRatio - adjustedViewport.Height)
/ 2);
}

… with the aspect ratio adjusted, all geographic objects painted will now preserve their shape even as the `Form`

’s shape changes.

## Navigating a Map

Now that we have the ability to paint a portion of a map, the final step in this example is to implement some form of navigation. Panning a map means shifting the viewport without changing its size. Zooming, however, is somewhat counter-intuitive: to zoom a map *in*, you must make the projected viewport *smaller*. A smaller viewport means a greater scale factor is applied.

If we’re using the `PointF`

class to represent projected coordinates, we can use the `RectangleF`

class to represent the projected viewport. Zooming becomes a matter of calling the `Inflate`

method to either shrink or grow the projected viewport to zoom in or out, respectively. Another important thing to mention here is the concept of "zooming by percentage." Zooming should always be done using a percentage of the current viewport. Otherwise, zooming will appear to have an exaggerated effect the more you zoom in, and closer to no effect as you zoom out:

public void ZoomIn()
{
float zoomWidthAmount = -viewport.Width * 0.10f;
float zoomHeightAmount = -viewport.Height * 0.10f;
viewport.Inflate(zoomWidthAmount, zoomHeightAmount);
Invalidate();
}
public void ZoomOut()
{
float zoomWidthAmount = viewport.Width * 0.10f;
float zoomHeightAmount = viewport.Height * 0.10f;
viewport.Inflate(zoomWidthAmount, zoomHeightAmount);
Invalidate();
}

Developers may recognize how easily these methods can be plugged into the `MouseWheel`

event of a `Form`

. Panning methods are just as straightforward, but involve use of the `Offset`

method:

public void PanUp()
{
float zoomHeightAmount = -viewport.Height * 0.10f;
viewport.Offset(0.0f, zoomHeightAmount);
}
public void PanDown()
{
float zoomHeightAmount = viewport.Height * 0.10f;
viewport.Offset(0.0f, zoomHeightAmount);
}

... again, you may have already recognized how to plug these methods into the `KeyDown`

event. With these methods implemented, you can now explore your map at any zoom level. If you are familiar with parts one and two of this article, you are well on your way to developing a commercial application which can plot your GPS location, along with all kinds of geographic features. Whether your intent is to draw points, lines, or polygons, the approach is the same.

## Play it Backwards

At this point, we’ve successfully drawn a map and provided a way to pan and zoom. But, it would be helpful to be able to see where the mouse is pointing, but in terms of geographic or projected coordinates, not pixel coordinates. So, we’ll add some code into the `MouseMove`

event of the `Form`

, which will show the mouse’s location in all three coordinate systems.

Conversion starts with pixel coordinates, which are then converted to projected coordinates using the inverse of the `Matrix`

we set up earlier in this article. Finally, the `Deproject`

method of our projection is used to convert the projected coordinate back into its geographic equivalent. The code will look like this:

protected override void OnMouseMove(MouseEventArgs e)
{
Matrix reverseTransform = transform.Clone();
reverseTransform.Invert();
Point[] projectedCoordinate = new Point[] { e.Location };
reverseTransform.TransformPoints(projectedCoordinate);
PointF geographicCoordinate = plateCaree.Deproject(projectedCoordinate[0]);
Console.WriteLine("Pixel: " + e.Location.ToString());
Console.WriteLine("Projected: "
+ projectedCoordinate[0].ToString();
Console.WriteLine("Geographic: "
+ geographicCoordinate.ToString();
}

… with this code, you can now freely convert between all three coordinate systems, in both directions.

## Conclusion

The task of displaying geographic data on-screen involves conversion of the data to two other coordinate systems. Map projections are used to flatten 3D coordinates into 2D coordinates, and then matrix math is used to actually paint geographic data in a meaningful way. Panning and zooming a map involves changing the location and size of the viewport, and navigation is typically tied into keyboard and mouse events. The aspect ratio of the viewport is adjusted to match the aspect ratio of the `Form`

to prevent distortion. Finally, an inverse of the `Matrix`

is used to convert coordinates from pixel to projected, and the `Deproject`

method of the projection converts the projected coordinate back into its geographic equivalent.

There are many topics which we have yet to cover in order to develop a commercial-quality mapping application. Topics such as geographic data sources, spatial indexing, vector normalization, and paint optimization could easily take up several articles. Many commercial components for .NET exist which address these topics. However, this article can at least help you to gain a solid understanding of how to display geographic data in your own .NET applications.

*Please indicate your interest in part four by rating this article.*

Phil has a highly technical mind as well as having adepts in both art and music. His twenty-five years of computer experience brings knowledge to the table for any enterprise endeavor. But his skills in problem solving provide practical solutions to a myriad of problems, both technical and administrative. He also has a propensity for math and science that suit him well for tasks associated with geospacial information systems.

As a .Net architect and developer, Phil has six years under his belt in C# development, on top of twelve years of Visual Basic, starting with version four. He has an extensive background in business enterprise development, which includes SQL server database design and development. Now Phil prefers to exercise his talents in fields of science and mathematics.