Formula for getting Middle point of an ARC

[engowen]

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?

[Guv]

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?

[Good Dreams]

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

He wants to know the middle point of the arc.

[Guv]

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.

[Guv]

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).

[engowen]

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.

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

BTW, thanks for ur response, GUV!