Formula derivation
The Standard equation of a line is a way to represent a 2D line with the following form: $Ax+By=C$Given two cartesian coordinates in the 2D space, one can calculate the line that intersects both points $(x_1, y_1)$ and $(x_2, y_2)$. To derivate the coefficients A, B and C of the standard equation, I will start with the equation of a straight line that intersects the point $(x_1, y_1)$ and has a slope $m$:
$y - y_1 = m(x - x_1)$ (eq. 1).
The problem with this representation is that the slope $m$ is defined as the change in 'y' with respect to the change in 'x' as $m=\frac{\Delta Y}{\Delta X}=\frac{y_2 - y_1}{x_2 - x_1}$. In the case of vertical lines $x_2 - x_1=0$ and that would undefine the slope for them.
With this context, let us find a more generic representation that can handle vertical lines. Given (eq. 1) we have the following:
$y - y_1 = mx - mx_1$(eq. 2)
$y - mx = y_1 - mx_1$(eq. 3)
The magic trick is to assign A and B as the numerator and denominator of the slope:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-A}{B}$ (eq. 4)
Then, substitute (eq. 4) in (eq. 3) and do some algebra:
$y - \left( \frac{-A}{B}\right)x = y_1 - \left( \frac{-A}{B}\right)x_1$
$\frac{A}{B}x + y = y_1 + \frac{A}{B}x_1$
$Ax + By = B(y_1 + \frac{A}{B}x_1)$
$Ax + By = By_1 + Ax_1$
Finally, if we rename the right hand side of the equation, the constant side, $C$, we'll have:
$Ax + By = C$
For the constants:
$A = y_1 - y_2$
$B = x_2 - x_1$
$C = Ax_1 + By_1$
Python 3.4 code
# Standard Form of a line Ax + By = C #------------------------------- # inputs: Two 2D points in the format: x-value, y-value # output: The standard form equation of the line # that passes through those points # by IF, 9.16.2014 #------------------------------- import unittest # GLOBAL VARIABLES #---------------------- answer = 'y' err = 0; # FUNCTIONS #---------------------- def printAnswer(): print('') if (B >= 0): print(str(round(A,2)) + "x+" + str(round(B,2)) + "y" " = " + str(round(C,2))) if (B < 0): print(str(round(A,2)) + "x" + str(round(B,2)) + "y" " = " + str(round(C,2))) print('') def parseCoords(p): global err # split by comma coordList = p.split(',') # validate for 2D coordinates if (len(coordList) != 2): err = 2 return # validate numeric input try: x = float(coordList[0]) except ValueError: x = 0 err = 1 try: y = float(coordList[1]) except ValueError: y = 0 err = 1 return x, y, err # From http://stackoverflow.com/questions/11175131/code-for-greatest-common-divisor-in-python def GCD(a, b): if b == 0: return a else: return GCD(b, a % b) def simplifyEq(a,b,c): # check if they are all integers global A, B, C if ( a%1 == 0 and b%1 == 0 and c%1 == 0): a = int(a) b = int(b) c = int(c) gcd = GCD(a,GCD(b,c)) if (gcd <= 1): return else: A = A/gcd; B = B/gcd; C = C/gcd; return # ------------------ # MAIN FUNCTION #------------------- while(answer == 'y'): # read input from user p = input("Enter 1st point in the format x1,y1: ") p = parseCoords(p) if(err == 1): print("Error 1: Values must be numbers separated by a comma") print("Format: x,y") elif(err == 2): print("Error 2: Only two values are accepted") print("Format: x,y") else: x1 = p[0] y1 = p[1] p = input("Enter 2nd point in the format x2,y2: ") p = parseCoords(p) if(err == 1): print("Error 1: Values must be numbers separated by a comma") print("Format: x,y") elif(err == 2): print("Error 2: Only two values are accepted") print("Format: x,y") else: x2 = p[0] y2 = p[1] # For the derivation go to:http://howdidisolvedit.blogspot.ca/2014/09/standard-form-of-line-given-2-points.html A = y1 - y2 B = x2 - x1 C = A*x1 + B*y1 # can the coefficients be simplified? simplifyEq(A,B,C) printAnswer() answer = input("Test again? [y/n]: ") if (answer == 'y'): err = 0 # TESTING # -------------------- class MyTest(unittest.TestCase): def test_GCD(self): self.assertEqual(GCD(144,12),12) def test_simplifyEq(self): raised = False try: simplifyEq(0,0,0) except: raised = True self.assertFalse(raised, 'Exception rised') if __name__ == '__main__': unittest.main()
Upload of python and Google App Engine
http://stdlines.appspot.com/Further info on Python + Google App Engine. Uploaded by Stefano Locati:
https://www.youtube.com/watch?v=j7OQZp2S5pY