

Title  Determine where two circles intersect in Visual Basic .NET 
Description  This example shows how to determine where two circles intersect in Visual Basic .NET. 
Keywords  circles,intersect,intersect two circles,intersections,find circle intersections, VB.NET, Visual Basic .NET 
Categories  Graphics, Algorithms 


If you don't like math, skip to the code below.
Consider the figure on the right showing two circles with radii r_{0} and r_{1}. The points p_{0}, p_{1}, p_{2}, and p_{3} have coordinates (x_{0}, y_{0}) and so forth.
Let d = the distance between the circles' centers so . Solving for a gives .Now there are three cases:
 If d > r_{0} + r_{1}: The circles are too far apart to intersect.
 If d < r_{0}  r_{1}: One circle is inside the other so there is no intersection.
 If d = 0 and r_{0} = r_{1}: The circles are the same.
 If d = r_{0} + r_{1}: The circles touch at a single point.
 Otherwise: The circles touch at two points.
The Pythagorean theorem gives:
So:
Substituting and multiplying this out gives:
The b^{2} terms on each side cancel out. You can then solve for b to get:
Similarly:
All of these values are known so you can solve for a and b. All that remains is using those distances to find the points p_{3}.
If a line points in direction , then two perpendicular lines point in the directions <dy, dx> and <dy, dx>. Scaling the result gives the following coordinates for the points p_{3}:
Be careful to notice the ± and ∓ symbols.
Click and drag to create two circles on the example program. The following code shows the FindCircleCircleIntersections method that the program uses to find the intersections.


' Find the points where the two circles intersect.
Private Function FindCircleCircleIntersections( _
ByVal cx0 As Single, ByVal cy0 As Single, ByVal radius0 As _
Single, _
ByVal cx1 As Single, ByVal cy1 As Single, ByVal radius1 As _
Single, _
ByRef intersection1 As PointF, ByRef intersection2 As _
PointF) As Integer
' Find the distance between the centers.
Dim dx As Single = cx0  cx1
Dim dy As Single = cy0  cy1
Dim dist As Double = Math.Sqrt(dx * dx + dy * dy)
' See how many solutions there are.
If (dist > radius0 + radius1) Then
' No solutions, the circles are too far apart.
intersection1 = New PointF(Single.NaN, Single.NaN)
intersection2 = New PointF(Single.NaN, Single.NaN)
Return 0
ElseIf (dist < Math.Abs(radius0  radius1)) Then
' No solutions, one circle contains the other.
intersection1 = New PointF(Single.NaN, Single.NaN)
intersection2 = New PointF(Single.NaN, Single.NaN)
Return 0
ElseIf ((dist = 0) AndAlso (radius0 = radius1)) Then
' No solutions, the circles coincide.
intersection1 = New PointF(Single.NaN, Single.NaN)
intersection2 = New PointF(Single.NaN, Single.NaN)
Return 0
Else
' Find a and h.
Dim a As Double = (radius0 * radius0  _
radius1 * radius1 + dist * dist) / (2 * _
dist)
Dim h As Double = Math.Sqrt(radius0 * radius0  a * _
a)
' Find P2.
Dim cx2 As Double = cx0 + a * (cx1  cx0) / dist
Dim cy2 As Double = cy0 + a * (cy1  cy0) / dist
' Get the points P3.
intersection1 = New PointF( _
CSng(cx2 + h * (cy1  cy0) / dist), _
CSng(cy2  h * (cx1  cx0) / dist))
intersection2 = New PointF( _
CSng(cx2  h * (cy1  cy0) / dist), _
CSng(cy2 + h * (cx1  cx0) / dist))
' See if we have 1 or 2 solutions.
If (dist = radius0 + radius1) Then Return 1
Return 2
End If
End Function





