Sample on GitHub

## Introduction

Interpolating points sometimes is hard mathematical work, even more if the points are ordered. The solution is to create a function using the points, and using an extra parameter `t`

that represents the time dimension. This often is called a parametric representation of the curve. This article shows a simple way of interpolating a set points using Bezier curves in WPF.

## Background

The idea of this solution comes after asking this question in Stack Overflow. The accepted answer makes references to a simple and interesting method proposed by Maxim Shemanarev, where the control points are calculated from the original points (called anchor points).

Here we create a WPF `UserControl `

that draws the curve from any collection of points. This control can be used with the pattern MVVM. If any point's coordinate changes, the curve also will change automatically. For instance, it can be used for a draw application, where you can drag & drop the points for changing the drawing, or curve.

## The Algorithm Behind

Due the original antigrain site is down, I'm going to explain what is the algorithm proposed by Maxim Shemanarev.

A Bezier curve has two anchor points (begin and end) and two control ones (CP) that determine its shape. Our anchor points are given, they are pair of vertices of the polygon.
The question is, how to calculate the control points. It is obvious that the control points of two adjacent edges
plus the vertex between them should form one straight line.

The solution found is a very simple method that does not require any
complicated math. First, we take the polygon and calculate the
middle points A_{i} of its edges.

Here we have line segments C_{i} that connect two points A_{i} of the
adjacent segments. Then, we should calculate points B_{i} as shown in
this picture.

The third step is final. We simply move the line segments C_{i} in
such a way that their points B_{i} coincide with the respective vertices.
That's it, we calculated the control points for our Bezier curve and
the result looks good.

One little improvement. Since we have a straight line that
determines the place of our control points, we can move them
as we want, changing the shape of the resulting curve.
I used a simple coefficient K that moves the points along
the line relatively to the initial distance between vertices
and control points. The closer the control points to the vertices
are, the sharper figure will be obtained.

The method works quite well with self-intersecting polygons.
The examples below show that the result is pretty interesting.

## The Class for Calculation

Below it is exposed the class that makes the calculation of the spline segments, based in the algorithm, exposed above. This class is named `InterpolationUtils `

, it has a `static `

method (named `InterpolatePointWithBeizerCurves`

) that returns a list of `BeizerCurveSegment`

, that will be the solution of our problem.

The class `<code>BeizerCurveSegment `

has the four properties that define a spline segment: `StartPoint`

, `EndPoint`

, `FirstControlPoint`

, and the `SecondControlPoint`

.

As the above algorithm is originally implemented for closed curves, and it is desired that it can be applied for open curves too, a little change is needed. For this reason, the `InterpolatePointWithBeizerCurves `

method receive as second parameter, a boolean variable named `isClosedCurve`

, that determines if the algorithm will return an open or closed curve. The change consists in: if `isClosedCurve==true`

, then for building the first segment, the first point will be used two times, and the second point, and for the last segment will be used the last but one point, and the last point two times.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Windows;
namespace BezierCurveSample.View.Utils
{
public class InterpolationUtils
{
public class BeizerCurveSegment
{
public Point StartPoint { get; set; }
public Point EndPoint { get; set; }
public Point FirstControlPoint { get; set; }
public Point SecondControlPoint { get; set; }
}
public static List<BeizerCurveSegment> InterpolatePointWithBeizerCurves(List<Point> points, bool isClosedCurve)
{
if (points.Count < 3)
return null;
var toRet = new List<BeizerCurveSegment>();
if (isClosedCurve)
points.Add(points.First());
for (int i = 0; i < points.Count - 1; i++) {
double x1 = points[i].X;
double y1 = points[i].Y;
double x2 = points[i + 1].X;
double y2 = points[i + 1].Y;
double x0;
double y0;
if (i == 0) {
if (isClosedCurve)
{
var previousPoint = points[points.Count - 2]; x0 = previousPoint.X;
y0 = previousPoint.Y;
}
else {
var previousPoint = points[i]; x0 = previousPoint.X;
y0 = previousPoint.Y;
}
}
else
{
x0 = points[i - 1].X; y0 = points[i - 1].Y;
}
double x3, y3;
if (i == points.Count - 2) {
if (isClosedCurve)
{
var nextPoint = points[1]; x3 = nextPoint.X;
y3 = nextPoint.Y;
}
else {
var nextPoint = points[i + 1]; x3 = nextPoint.X;
y3 = nextPoint.Y;
}
}
else
{
x3 = points[i + 2].X; y3 = points[i + 2].Y;
}
double xc1 = (x0 + x1) / 2.0;
double yc1 = (y0 + y1) / 2.0;
double xc2 = (x1 + x2) / 2.0;
double yc2 = (y1 + y2) / 2.0;
double xc3 = (x2 + x3) / 2.0;
double yc3 = (y2 + y3) / 2.0;
double len1 = Math.Sqrt((x1 - x0) * (x1 - x0) + (y1 - y0) * (y1 - y0));
double len2 = Math.Sqrt((x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1));
double len3 = Math.Sqrt((x3 - x2) * (x3 - x2) + (y3 - y2) * (y3 - y2));
double k1 = len1 / (len1 + len2);
double k2 = len2 / (len2 + len3);
double xm1 = xc1 + (xc2 - xc1) * k1;
double ym1 = yc1 + (yc2 - yc1) * k1;
double xm2 = xc2 + (xc3 - xc2) * k2;
double ym2 = yc2 + (yc3 - yc2) * k2;
const double smoothValue = 0.8;
double ctrl1_x = xm1 + (xc2 - xm1) * smoothValue + x1 - xm1;
double ctrl1_y = ym1 + (yc2 - ym1) * smoothValue + y1 - ym1;
double ctrl2_x = xm2 + (xc2 - xm2) * smoothValue + x2 - xm2;
double ctrl2_y = ym2 + (yc2 - ym2) * smoothValue + y2 - ym2;
toRet.Add(new BeizerCurveSegment
{
StartPoint = new Point(x1, y1),
EndPoint = new Point(x2, y2),
FirstControlPoint = i == 0 && !isClosedCurve ? new Point(x1, y1) : new Point(ctrl1_x, ctrl1_y),
SecondControlPoint = i == points.Count - 2 && !isClosedCurve ? new Point(x2, y2) : new Point(ctrl2_x, ctrl2_y)
});
}
return toRet;
}
}
}

## The User Control

The user control that we propose is very simple to use, and it works with the MVVM pattern.

The `LandMarkControl `

has only two dependency properties, one for the points, and other for the color of the curve. The most important property is the `Points `

attached property. It is of `IEnumerable `

type, and it assumes that each item, has an `X`

and `Y`

properties.

In case the collection of points implements the `INotifyCollectionChanged `

interface, the control will register to the `CollectionChanged `

event, and if each point implements the `INotifyPropertyChanged `

interface, the control also will register to the `PropertyChanged `

event. In this way, every time any point is added or removed, or any point's coordinates changed, the control will be refreshed.

This is the complete user control code behind:

using System.Collections;
using System.Collections.Generic;
using System.Collections.Specialized;
using System.ComponentModel;
using System.Linq;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Media;
using BezierCurveSample.View.Utils;
namespace BezierCurveSample.View
{
public partial class LandmarkControl : UserControl
{
#region Points
public IEnumerable Points
{
get { return (IEnumerable)GetValue(PointsProperty); }
set { SetValue(PointsProperty, value); }
}
public static readonly DependencyProperty PointsProperty =
DependencyProperty.Register("Points", typeof(IEnumerable),
typeof(LandmarkControl), new PropertyMetadata(null, PropertyChangedCallback));
private static void PropertyChangedCallback(DependencyObject dependencyObject,
DependencyPropertyChangedEventArgs dependencyPropertyChangedEventArgs)
{
var landmarkControl = dependencyObject as LandmarkControl;
if (landmarkControl == null)
return;
if (dependencyPropertyChangedEventArgs.NewValue is INotifyCollectionChanged)
{
(dependencyPropertyChangedEventArgs.NewValue as
INotifyCollectionChanged).CollectionChanged += landmarkControl.OnPointCollectionChanged;
landmarkControl.RegisterCollectionItemPropertyChanged
(dependencyPropertyChangedEventArgs.NewValue as IEnumerable);
}
if (dependencyPropertyChangedEventArgs.OldValue is INotifyCollectionChanged)
{
(dependencyPropertyChangedEventArgs.OldValue as
INotifyCollectionChanged).CollectionChanged -= landmarkControl.OnPointCollectionChanged;
landmarkControl.UnRegisterCollectionItemPropertyChanged
(dependencyPropertyChangedEventArgs.OldValue as IEnumerable);
}
if (dependencyPropertyChangedEventArgs.NewValue != null)
landmarkControl.SetPathData();
}
#endregion
#region PathColor
public Brush PathColor
{
get { return (Brush)GetValue(PathColorProperty); }
set { SetValue(PathColorProperty, value); }
}
public static readonly DependencyProperty PathColorProperty =
DependencyProperty.Register("PathColor", typeof(Brush), typeof(LandmarkControl),
new PropertyMetadata(Brushes.Black));
#endregion
#region IsClosedCurve
public static readonly DependencyProperty IsClosedCurveProperty =
DependencyProperty.Register("IsClosedCurve", typeof (bool), typeof (LandmarkControl),
new PropertyMetadata(default(bool), OnIsClosedCurveChanged));
private static void OnIsClosedCurveChanged(DependencyObject dependencyObject, DependencyPropertyChangedEventArgs dependencyPropertyChangedEventArgs)
{
var landmarkControl = dependencyObject as LandmarkControl;
if (landmarkControl == null)
return;
landmarkControl.SetPathData();
}
public bool IsClosedCurve
{
get { return (bool) GetValue(IsClosedCurveProperty); }
set { SetValue(IsClosedCurveProperty, value); }
}
#endregion
public LandmarkControl()
{
InitializeComponent();
}
void SetPathData()
{
if (Points == null) return;
var points = new List<Point>();
foreach (var point in Points)
{
var pointProperties = point.GetType().GetProperties();
if (pointProperties.All(p => p.Name != "X") ||
pointProperties.All(p => p.Name != "Y"))
continue;
var x = (float)point.GetType().GetProperty("X").GetValue(point, new object[] { });
var y = (float)point.GetType().GetProperty("Y").GetValue(point, new object[] { });
points.Add(new Point(x, y));
}
if (points.Count <= 1)
return;
var myPathFigure = new PathFigure { StartPoint = points.FirstOrDefault() };
var myPathSegmentCollection = new PathSegmentCollection();
var beizerSegments = InterpolationUtils.InterpolatePointWithBeizerCurves(points, IsClosedCurve);
if (beizerSegments == null || beizerSegments.Count < 1)
{
foreach (var point in points.GetRange(1, points.Count - 1))
{
var myLineSegment = new LineSegment { Point = point };
myPathSegmentCollection.Add(myLineSegment);
}
}
else
{
foreach (var beizerCurveSegment in beizerSegments)
{
var segment = new BezierSegment
{
Point1 = beizerCurveSegment.FirstControlPoint,
Point2 = beizerCurveSegment.SecondControlPoint,
Point3 = beizerCurveSegment.EndPoint
};
myPathSegmentCollection.Add(segment);
}
}
myPathFigure.Segments = myPathSegmentCollection;
var myPathFigureCollection = new PathFigureCollection {myPathFigure} ;
var myPathGeometry = new PathGeometry { Figures = myPathFigureCollection };
path.Data = myPathGeometry;
}
private void RegisterCollectionItemPropertyChanged(IEnumerable collection)
{
if (collection == null)
return;
foreach (INotifyPropertyChanged point in collection)
point.PropertyChanged += OnPointPropertyChanged;
}
private void UnRegisterCollectionItemPropertyChanged(IEnumerable collection)
{
if (collection == null)
return;
foreach (INotifyPropertyChanged point in collection)
point.PropertyChanged -= OnPointPropertyChanged;
}
private void OnPointCollectionChanged(object sender, NotifyCollectionChangedEventArgs e)
{
RegisterCollectionItemPropertyChanged(e.NewItems);
UnRegisterCollectionItemPropertyChanged(e.OldItems);
SetPathData();
}
private void OnPointPropertyChanged(object sender, PropertyChangedEventArgs e)
{
if (e.PropertyName == "X" || e.PropertyName == "Y")
SetPathData();
}
}
}

And this is the XAML code:

<UserControl x:Class="BezierCurveSample.View.LandmarkControl"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
mc:Ignorable="d"
x:Name="UserControl"
d:DesignHeight="300" d:DesignWidth="300">
<Path x:Name="path" Stroke="{Binding PathColor, ElementName=UserControl}" StrokeThickness="1"/>
</UserControl>

## Examples of Usage

Using the control for creating the data template for the `LandMarkViewModel`

:

<DataTemplate DataType="{x:Type ViewModel:LandmarkViewModel}">
<PointInterpolation.View:LandmarkControl x:Name="control"
Points="{Binding LandmarkPoints}" Visibility="{Binding IsVisible,
Converter={StaticResource BoolToVisibilityConverter}}" ToolTip="{Binding Label}"/>
<DataTemplate.Triggers>
<DataTrigger Binding="{Binding IsSelected}" Value="True">
<Setter Property="PathColor" TargetName="control" Value="Red"/>
</DataTrigger>
</DataTemplate.Triggers>
</DataTemplate>

Now everywhere a `LandMarkViewModel`

is displayed, this data template will show the item as a `LandMarkControl`

. It needs be rendered on a `Canvas`

:

<ListBox x:Name="landMarks" ItemsSource="{Binding Landmarks}">
<ListBox.Template>
<ControlTemplate>
<Canvas IsItemsHost="True"/>
</ControlTemplate>
</ListBox.Template>
</ListBox>

This is a final image example:

## References