Formula for getting Middle point of an ARC

Hi,

I know the radius, start point, end point and the origin. By applying distance formula,

d = sqr((x1 - x2)^2 + (y1 - y2)^2)


I end up with 2 equations.

1.) distance between middle point and origin = radius,

2.) distance between start point and middle point = distance between end point and middle point


Therefore,

(x1,y1) = middle point

x2 = oArcGeom.Origin.X
y2 = oArcGeom.Origin.Y
x3 = oArcGeom.Start.X
y3 = oArcGeom.Start.Y
x4 = oArcGeom.End.X
y4 = oArcGeom.End.Y


1.) distance between middle point and origin = radius:

x1 ^ 2 - 2 * (x1 * x2) + x2 ^ 2 = oArcGeom.Radius - y1 ^ 2 + 2 * (y1 * y2) - y2 ^ 2



2.) distance between start point and middle point = distance between end point and middle point:

2 * (x1 * x4) - 2 * (x1 * x3) + x3 ^ 2 - x4 ^ 2 = 2 * (y1 * y3) - 2 * (y1 * y4) + y4 ^ 2 - y3 ^ 2


But I don't know how to simplify more, and VB cannot accept the above formula.

Can any one help me?

I am confused. I am not sure what variables are known and which are unknown.

What are you trying to determine or draw or whatever?

[quote]
I know the radius, start point, end point and the origin.
[/b]

He wants to know the middle point of the arc.

It looks like you are trying to use simultaneous equations to solve this problem when trig is probably necessary.

My approach might be a long way around to get where you want to go, but I think it will work. It involves converting to and from polar coordinates.

I really do not want to go through the work of solving this, but can provide an algorithmic approach, which should be programmable.

I hope i understand the notation you are using. If not, let me know and I will try to post a corrected description of the following.

If circle is not centered at origin, do a coordinate transform which puts it there. This is accomplished by subtracting circle center coordinates from end point coordinates. In you original notation, this looks like

• Newx3 = x3 - x2
• Newy3 = y3 - y2
• Newx4 = x4 - x2
• Newy4 = y4 - y2

If circle is centered on the origin, use (x3, y3) & (x4, y4) as is.
Now determine angles for polar instead of Cartesian coordinates.

• AngleStart = Atn(Newy3 / Newx3)
• AngleEnd = Atn(Newy4 / Newx4)
• AngleMid = (AngleStart + AngleEnd) / 2
• Midx = R * cos(AngleMid)
• Midy = R * sin(AngleMid)

Now transform back to original coordinate system.

• x1 = Midx + x2
• y1 = Midy + y2

The Atn Function probably cannot be used directly. I suggest writing a function which accepts XY-Coordinates as arguments and returns an angle from zero to 2*Pi (0 to 360 degrees). The function would use Atn internally. If you use the Atn function directly, there is a problem with an X-coordinate of zero being used a s a denominator. There is also a problem with points in the second and third quadrants (90 to 270 degrees) The Atn function always returns a value in the first or fourth quadrant (+90 to - 90 degrees.).

I posted a usable function somewhere. If you search the forum for ArkTangent you should find it.

Excuse the spelling. In previous post I mentioned an ArkTangent Function which has been posted elsewhere.

I did a search and discovered it has been posted at least twice. Once with quadrant logic to return an angle between 0 and 2Pi (0 and 360 degrees), and once with no quadrant logic, returning an angle between zero and Pi/2 (0 and 90 degrees).

i compare the value with the one i should get, it is different. anyway, I wroted a rather dinosour procedure for it, and it work.

Code:

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Function ReturnARCMiddleXY(startX As Double, startY As Double, endX As Double, endY As Double, originX As Double, _
originY As Double, Radius As Double) As Double()

Dim distanceX As Double
Dim distanceY As Double
Dim distanceT As Double
Dim PointXY(1) As Double
Dim PointMidStartEnd(1) As Double

    PointMidStartEnd(0) = (startX + endX) / 2
    PointMidStartEnd(1) = (startY + endY) / 2

    If startX = endX Then
        PointXY(1) = PointMidStartEnd(1)
        If startX > originX Then
            PointXY(0) = originX + Radius
        Else
            PointXY(0) = originX - Radius
        End If
    ElseIf startY = endY Then
        PointXY(0) = PointMidStartEnd(0)
        If startY > originY Then
            PointXY(1) = originY + Radius
        Else
            PointXY(1) = originY - Radius
        End If
    'right side of origin
    ElseIf PointMidStartEnd(0) > originX Then
        distanceT = Sqr((PointMidStartEnd(0) - originX) ^ 2 + (PointMidStartEnd(1) - originY) ^ 2)
        distanceX = Radius * (PointMidStartEnd(0) - originX) / distanceT
        distanceY = Sqr(Radius ^ 2 - distanceX ^ 2)
        PointXY(0) = originX + distanceX
        If PointMidStartEnd(1) > originY Then
            PointXY(1) = originY + distanceY
        Else
            PointXY(1) = originY - distanceY
        End If
    'left side of origin
    ElseIf PointMidStartEnd(0) < originX Then
        distanceT = Sqr((originX - PointMidStartEnd(0)) ^ 2 + (PointMidStartEnd(1) - originY) ^ 2)
        distanceX = Radius * (originX - PointMidStartEnd(0)) / distanceT
        distanceY = Sqr(Radius ^ 2 - distanceX ^ 2)
        PointXY(0) = originX - distanceX
        If PointMidStartEnd(1) > originY Then
            PointXY(1) = originY + distanceY
        Else
            PointXY(1) = originY - distanceY
        End If
    End If
   
    ReturnARCMiddleXY = PointXY()
End Function

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BTW, thanks for ur response, GUV!