Describe an algorithm to get Bezier curves from a set of points. This curve will pass through all the points and will have a smooth factor. In the case of the article, this is implemented in WPF, but also I've updated the article with some implementations made in JavaScript, which can be used in Web or Mobile apps based on JavaScript.

## 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 of 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 to the original antigrain site being down (I can find that Sourceforge is still supporting this library and we can find the original article over here!), 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 form one straight line along with the vertex between them.

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 with the line relative 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 is the class that makes the calculation of the spline segments, based on the algorithm, exposed above. This class is named `InterpolationUtils`

, it has a `static `

method (named `InterpolatePointWithBezierCurves`

) that returns a list of `BezierCurveSegment`

, that will be the solution to our problem.

The class `BezierCurveSegment `

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 `InterpolatePointWithBezierCurves `

method receives a second parameter, a boolean variable named `isClosedCurve`

, which determines if the algorithm will return an open or closed curve. Since we take four points (x1 = *the current point*, x2 = *the next point*, but also are required two more points for creating the three edges. x0 = *the current's previous point* and x3 = *the next's next point*), the x0 and x3 points selection will be like this:

- If it is a closed curve if x1 is the first point, then x0 is going to be the latest point (in this implementation, it is the latest but one because the latest point is the same that the first one), and if x2 is the latest point, then x3 is going to be the first point (in a similar way, in this implementation, is going to be the second point).
- If it is an open curve, then x0 = x1 and x3 = x2 for the previous cases.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Windows;
namespace BezierCurveSample.View.Utils
{
public class InterpolationUtils
{
public class BezierCurveSegment
{
public Point StartPoint { get; set; }
public Point EndPoint { get; set; }
public Point FirstControlPoint { get; set; }
public Point SecondControlPoint { get; set; }
}
public static List<BezierCurveSegment>
InterpolatePointWithBezierCurves(List<Point> points, bool isClosedCurve)
{
if (points.Count < 3)
return null;
var toRet = new List<BezierCurveSegment>();
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 BezierCurveSegment
{
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 another 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 bezierSegments = InterpolationUtils.InterpolatePointWithBezierCurves
(points, IsClosedCurve);
if (bezierSegments == null || bezierSegments.Count < 1)
{
foreach (var point in points.GetRange(1, points.Count - 1))
{
var myLineSegment = new LineSegment { Point = point };
myPathSegmentCollection.Add(myLineSegment);
}
}
else
{
foreach (var bezierCurveSegment in bezierSegments)
{
var segment = new BezierSegment
{
Point1 = bezierCurveSegment.FirstControlPoint,
Point2 = bezierCurveSegment.SecondControlPoint,
Point3 = bezerCurveSegment.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 to 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

## History

- 6
^{th} May 2014: Initial version - 4
^{th} July 2023: Update the links and references. Added references to the original article, and also to some other implementations.