KT SALEEJ: Arc Calculation

Hi Friends,

Do you know!?
If you have any two dimensions of the Arc, you can calculate all the other dimensions of that Arc. I think the Arc Calculation given below would be very useful for my engineering friends to easily solve Arc.

Thank you Mr.Lee M for helping me overcome some of the obstacles I encountered when developing this.

Segment Calculation

Old post
Hi My Friends..

Do you know, if you have two dimensions of the Arc...You can calculate all other dimensions of that Arc. But this is not applicable if we only have the Arc length and Chord length of the Arc.

This is because as far as I know in Mathematics, there is no solution for finding the value of O from the form O/sinO. If any of my friends know how to find the value of O from O/sin O please let me know. As it will help me improve my Arc calculation.

I think the Arc calculation provided below will be very useful for my engineering friends to easily solve Arc.

Mobile users please use Web view for easy to use

Arc Length	Angle	Radius	Chord Length	Height

13 comments:

AnonymousApril 10, 2019 at 5:20 AM
HELPFUL
AnonymousFebruary 16, 2021 at 2:14 AM
The Arc Length and Chord length are two definite parameters which will give you infinite solutions. So it is not possible to get other parameters using Arc and Chord length. If you get the third parameter as your input then you will have a unique sollution. You have done a wonderful job. All the best.
ajwardApril 11, 2021 at 7:34 AM
It is a good attempt. Waw Nice
ajwardApril 11, 2021 at 7:34 AM
It is a good attempt. Waw Nice
UnknownApril 12, 2021 at 9:40 AM
Nice sir tomar contact karaga number please patay
Lee M.August 16, 2021 at 12:56 PM
There is a mathematical technique for finding the value of Ɵ from the form Ɵ/sinƟ. It is often referred to as numerical analysis.

https://en.wikipedia.org/wiki/Numerical_analysis

You can use numerical analysis to solve for the angle Ɵ given the arc length S, and the chord length L. Once you have Ɵ you can easily solve for the radius and height.

From a simple examination of an arc we express the relation between Ɵ, L, and S as:

f = sin(Ɵ/2)*S/Ɵ - L/2 (theta is in radians)

We know S and L the task is to find the value of Ɵ that will yield f = 0.

Assuming valid values for S and L, (the chord length cannot be greater than the arc length) we can evaluate the function f for a very small angle Ɵ1 (e.g., 0.1°) a large value Ɵ2 (360°) and an angle halfway between the two angles Ɵ3 = ( Ɵ1 + Ɵ2)/2. Note that the function is undefined when Ɵ = 0 since we cannot divide by zero and that is why the lower value is not precisely zero.

If the value of f changes sign in the interval from Ɵ1 to Ɵ3 then we know that the root of the function f is in this interval. If however the function f is positive for both f1 (the function evaluated using Ɵ1) and f3, or negative for both f1 and f3, then we look at the interval f3 to f2 where there will be a sign change.

Assuming that the the root is between Ɵ1 and Ɵ3 we can redefine Ɵ2 to be equal to Ɵ3 and define a new value for Ɵ3 = ( Ɵ1 + new Ɵ2)/2. The new Ɵ1, Ɵ3 interval is then examined for a change in sign and the process is repeated until the new value of Ɵ3 is within an acceptable level of precision to the old value of Ɵ3. This process of finding a root for an equation is called the bisection method.

The following Excel/VBA code will calculate the angle Ɵ given the arc length and chord length. I tested it with a variety of input and it usually converged to a solution with an accuracy of 0.001° within 20 iterations. There is the possibility when using numerical analysis that a satisfactory solution can not be found and that the calculation does not converge and gets stuck in an infinite loop. The program here limits the number of iterations to 50 at which point a message is displayed that a solution has not been found.

Note, the bisection method for finding a root is not the fastest numerical method for finding a root nor does it guarantee a solution but it seems to yield acceptable results for this task.

Here's the Excel/VBA code (I'd post the file and an image but there doesn't appear to be a way to do that).

Lee M.August 16, 2021 at 12:57 PM
Here's the code.

--------------------------------------
Code notes
Cell A2 contains the chord length
Cell A3 contains the arc length

The calculated value of theta is placed in cell A5.

------------------------------------------
Sub main()
' calculates the root of the function
' Sin((theta * Pi / 180# / 2#)) * S / (theta * Pi / 180#) - L / 2#
' using the bisection method and up to 50 iteration
' to a precision of 0.001 degrees
' L. Minardo 8/16/2021
'
' Get L and S
Range("A2").Select
L = ActiveCell.Value ' chord length
ActiveCell.Offset(1, 0).Select
S = ActiveCell.Value ' arc length
If L < S Then
Range("A15").Select
' set intial range from 0.001° to 360.0°
theta1 = 0.001
theta2 = 360
theta3 = (theta1 + theta2) / 2#
theta3old = theta1
f1 = F(L, S, theta1)
f2 = F(L, S, theta2)
f3 = F(L, S, theta3)
For i = 1 To 50
' if f1 anf f2 are of different signs then solution is
' between theta1 and theta2.
If f1 * f3 < 0 Then
' redefine upper angle to theta3
theta2 = theta3
f2 = f3
Else
theta1 = theta3
f1 = f3
End If
theta3old = theta3
theta3 = (theta1 + theta2) / 2#
f3 = F(L, S, theta3)
' compare last two iterations.
If Abs(theta3old - theta3) < 0.0001 Then
If i < 50 Then
msg = "Number of iterations = " & i
Else
msg = "Solution not found!"
End If
MsgBox msg
i = 100
End If
Range("A5").Select
ActiveCell.Value = theta3
Next i
Else
msg = "Error - The arc length S must be" & vbCrLf & _
" greater than the chord length L."
MsgBox msg, vbCritical, "Arc Angle Calculator"
End If

End Sub

Function F(L, S, theta)
Pi = 3.1415926535
F = Sin((theta * Pi / 180# / 2#)) * S / (theta * Pi /180#) - L/ 2#
End Function

-------------------------------------
Lee M.January 15, 2022 at 7:13 AM
Is there a way I can send you an Excel spreadsheet? It will help you understand the logic of calculating the angle and radius given the chord and arc length.
Lee M.January 16, 2022 at 7:09 AM
I've sent the files.