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I'm trying to write a program on CNC. Basically I have circular arc starting x, y , radius and finishing x, y also I know the direction of the arc clockwise or cc. So I need to find out the value of y on the arc at the specific x position. What is the best way to do that? I found similar problem on this website here. But i not sure how to get angle a.

sample

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aidas
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2 Answers2

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equation of a circle is x^2 + y^2 = r^2

in your case, we know x_random and R

substituting in knows we get,

x_random ^ 2 + y_random ^ 2 = R ^ 2

and solving for y_random get get

y_random = sqrt( R ^ 2 - x_random ^ 2 )

Now we have y_random

Edit: this will only work if your arc is a circular arc and not an elliptical arc

to adapt this answer to an ellipse, you'll need to use this equation, instead of the equation of a circle

( x ^ 2 / a ^ 2 ) + ( y ^ 2 / b ^ 2 ) = 1, where a is the radius along the x axis and b is the radius along y axis

Simple script to read data from a file called data.txt and compute a series of y_random values and write them to a file called out.txt

import math                                                                 

def fromFile():                                                             
    fileIn = open('data.txt', 'r')                                          
    output = ''                                                             
    for line in fileIn:                                                     
        data = line.split()                                                 
        # line of data should be in the following format                    
        # x h k r                                                           
        x = float(data[0])                                                  
        h = float(data[1])                                                  
        k = float(data[2])                                                  
        r = float(data[3])                                                  
        y = math.sqrt(r**2 - (x-h)**2)+k                                    
        if ('\n' in line):                                                  
            output += line[:-1] + ' | y = ' + str(y) + '\n'                 
        else:                                                               
            output += line + ' | y = ' + str(y)                             
    print(output)                                                           
    fileOut = open('out.txt', 'w')                                          
    fileOut.write(output)                                                   
    fileIn.close()                                                          
    fileOut.close()                                                         

if __name__ == '__main__':                                                  
    fromFile()

data.txt should be formatted as such

x0 h0 k0 r0
x1 h1 k1 r1
x2 h2 k2 r2
... for as many lines as required
francium
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  • i thought equation for circle is (x – h)^2 + (y – k)^2 = r^2. Am i wrong? h, k centre of a circle. – aidas Mar 24 '16 at 23:33
  • @aidas h and k are used to translate the circle along the x and y axis. If you don't translate the circle, h=0, k=0, then you get x^2 + y^2 = r^2 – francium Mar 25 '16 at 00:26
  • but in my case h and k isn't 0. i could offset origin to zero with every arc in the program i'm making. but i'm hoping that there's cleaner method. – aidas Mar 25 '16 at 00:47
  • @aidas well for every arc the calculation is still the same, only difference is the numbers you plug in. You could easily write a simple program (python for example) that will accept your numbers and quickly compute the result. – francium Mar 25 '16 at 00:56
  • @fracium could you show me how its done? python is fine. just centre of circle is unknown. i found equation for it [here](http://math.stackexchange.com/questions/27535/how-to-find-center-of-an-arc-given-start-point-end-point-radius-and-arc-direc). its just my math skills to solve this one isn't good enough. – aidas Mar 25 '16 at 07:17
  • @aidas, I updated the answer to include a python script – francium Mar 25 '16 at 17:23
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At first you have to find circle equation. Let's start point Pst = (xs,ys), end point Pend = (xend,yend)

For simplicity shift all coordinates by (-xs, -ys), so start point becomes coordinate origin.

New Pend' = (xend-xs,yend-ys) = (xe, ye), new 'random point' coordinate is xr' = xrandom - xs, unknown circle center is (xc, yc)

xc^2 + yc^2 = R^2    {1}
(xc - xe)^2 + (yc-ye)^2 = R^2  {2}  //open the brackets
xc^2 - 2*xc*xe + xe^2 + yc^2 - 2*yc*ye + ye^2 = R^2    {2'}
subtract {2'} from {1}
2*xc*xe - xe^2  + 2*yc*ye - ye^2 = 0    {3}
yc =  (xe^2 + ye^2 - 2*xc*xe) / (2*ye) {4}
substitute {4} in {1}
xc^2 + (xe^2 + ye^2 - 2*xc*xe)^2 / (4*ye^2) = R^2 {5}
solve quadratic equation {5} for xc, choose right root (corresponding to arc direction), find yc

having center coordinates (xc, yc), write
yr' = yc +- Sqrt(R^2 -(xc-xr')^2) //choose right sign if root exists
and finally exclude coordinate shift
yrandom = yr' + ys
MBo
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