## #1,147 – Converting from Polar Coordinates to Cartesian Coordinates

You can represent a two dimensional point as either a cartesian coordinate (X,Y) or a polar coordinate (r,theta).  Conversions between these two coordinate systems are shown below.

From polar coordinates to cartesian: To convert from cartesian coordinates to polar, the following formula works, as long as the X value is positive. If X is 0 or negative, then the calculation for theta becomes: ## #1,146 – Polar Coordinate System

In most cases when you’re working with graphical objects, you use a cartesian coordinate system, where each point is represented as an X and a Y value, indicating the point’s distance from an origin in two different dimensions.

You can also express points in a two-dimensional system using a polar coordinate system.  Each point in a polar coordinate system is represented with two values:

• A radius value, describing how far the point is from an origin  (range is any non-negative number)
• An angular coordinate, describing how many degrees around the circle the point is located, typically from a horizontal ray extending to the right of the origin  (range typically [0, 360) degrees or [0, 2*pi) radians)

Below is a picture showing two sample points expressed in polar coordinates.

• (2.0, 60) – Radius = 2, Angle = 60 degrees (counterclockwise) from horizontal
• (1.0, 180) – Radius = 1, Angle = 180 degrees (counterclockwise) from horizontal ## #236 – Drawing an Arbitrary Geometry into a DrawingVisual

You can render content in a DrawingVisual object by using a DrawingContext to draw objects that you want rendered.  Use the DrawingContext‘s DrawGeometry method to draw an arbitrary geometry.

Below is an example of using DrawGeometry, where we render our object in the DrawingVisual constructor.

```        public MyDrawingVisual()
{
// First define the geometric shape
StreamGeometry geom = new StreamGeometry();
using (StreamGeometryContext gc = geom.Open())
{
// Start new object, filled=true, closed=true
gc.BeginFigure(new Point(150.0, 150.0), true, true);

// isStroked=true, isSmoothJoin=true
gc.LineTo(new Point(210.0, 105.0), true, true);
gc.LineTo(new Point(270.0, 150.0), true, true);
gc.LineTo(new Point(270.0, 45.0), true, true);
gc.LineTo(new Point(210.0, 90.0), true, true);
gc.LineTo(new Point(150.0, 45.0), true, true);
}

using (DrawingContext dc = RenderOpen())
{
// Draw our geometry
dc.DrawGeometry(Brushes.DarkRed, new Pen(Brushes.Blue, 2), geom);
}
}
```

We create a new StreamGeometry object, which is what we’ll render into the DrawingContext using the DrawGeometry method.  To create the actual object, we use the StreamGeometryContext of the StreamGeometry and call methods like BeginFigure and LineTo. 