You might put your curve into a PathGeometry (or StreamGeometry) in the Data property of a Path control.
<Canvas>
<Path Stroke="Black" StrokeThickness="3">
<Path.Data>
<PathGeometry>
<PathFigure x:Name="figure" IsClosed="False">
<PathFigure.Segments>
<PolyLineSegment x:Name="polyline"/>
</PathFigure.Segments>
</PathFigure>
</PathGeometry>
</Path.Data>
</Path>
</Canvas>
Update the PathGeometry in code-behind:
figure.StartPoint = ...;
polyline.Points.RemoveAt(...);
polyline.Points.Add(...);