# Bézier Curves in Bing Silverlight Maps

Connecting geographical locations on Bing Maps with Bézier curves.

## Introduction

Bing Maps Silverlight Control library has a MapPolyline class for showing connected points on the map. I wanted my points to be smoothly connected but there wasn’t out-of-the-box support so I developed a custom control deriving from MapShapeBase class.

## Background

Every developer who messed with Expression Blend or Gimp long enough knows by experimentation how a Bézier curve behaves. Basically a cubic Bézier curve has an initial point (P1), two control points (B1, B2) and a final point (P2).

Cubic Bézier Curve

The formula that defines a cubic Bézier curve is:

$P(t) = (1-t)^3P_1 + 3(1-t)^2tB_1 + 3(1-t)t^2B_2 + t^3P_2$

Cubic Bézier Curve

where t is in the interval [0,1].

Terms multiplying P1, B1, B2, P2 are called the basis functions for the cubic Bézier. Our points determine how much of these basis functions does the curve contains.

Cubic Bézier Basis Functions

The thing is we need to calculate coordinates of the control points such that our points of interest are on the curve.

Polyline, Bézier, Catmull-Rom

So how can we calculate these control points? After researching(read googling) I have landed on cardinal splines and Catmull-Rom splines. It appears that every control point of a Catmull-Rom spline is on the curve and it is also a Bézier curve which means we can use it as PathGeometry with a Silverlight Path object.

### Calculating Control Points

If we rewrite formulas from the cardinal splines page as the following, we can easily calculate control points.

$\\P'_0 = \frac{P_1 - P_0}{a}\\\\P'_i = \frac{P_{i+1} - P_{i-1}}{a} , i \in [1, n-1]\\\\P'_n = \frac{P_n - P_{n-1}}{a}$

Point Derivative

Control Points:

$\\B1_i = P_i + \frac{P'_i}{3}\\\\B2_i = P_{i+1} - \frac{P'_{i+1}}{3}$

Control Points

Fitted Bézier with Control Points Visible

## Using the code

The method that calculates the Bézier Points is as the following. GetB1 and GetB2 are straight forward implementations of the aforementioned formulas.

private PointCollection GetBezierPoints(PointCollection pts, double tension)
{
PointCollection ret = new PointCollection();

for (int i = 0; i < pts.Count; i++)
{
// for first point append as is.
if (i == 0)
{
continue;
}

// for each point except first and last get B1, B2. next point.
// Last point do not have a next point.
}

return ret;
}

To use the PointCollection returned from GetBezierPoints method in Silverlight, we need to build a Path with BezierSegments in it.

private PathFigure GetBezierSegments(PointCollection pts, double tension)
{
PathFigure ret = new PathFigure();
ret.Segments.Clear();
ret.IsClosed = false;

var bzPoints = GetBezierPoints(_projectedPoints, tension);

// First point is the starting point.
ret.StartPoint = bzPoints[0];

for (int i = 1; i < bzPoints.Count; i += 3)
{
{
Point1 = bzPoints[i],       // B1 control point.
Point2 = bzPoints[i + 1],   // B2 control point.
Point3 = bzPoints[i + 2]    // P2 start / end point.
});
}

return ret;
}

And use this PathFigure in the MapBezier as:

// Create a new PathGeometry.
var pGeo = new PathGeometry();

// Add the Bezier PathFigure to the PathGeometry.

// Set data of the Path to the created PathGeometry.
((Path)EncapsulatedShape).Data = pGeo;

You can use the MapBezier class just like MapPolyline and MapPolygon classes in your silverlight XAML file. See the attached sample for an example silverlight application.

<m:Map x:Name="myMap" CredentialsProvider="***">
<m:MapLayer x:Name="layerDurak" >
<local:MapBezier Tension="2" x:Name="plDurakGidis" Stroke="Orange" StrokeThickness="3" Opacity=".6" Locations="{Binding MyLocations, Mode=OneWay}" />
</m:MapLayer>
</m:Map>

Note: Bing Maps Silverlight SDK is needed for compiling the sample application.

## Points of Interest

It was fun messing with the Bing Maps Silverlight Control toolkit and I hope MapBezier is what you are looking for.

If you liked this, vote the article at http://www.codeproject.com/KB/silverlight/MapBezier.aspx