""" Written by Steven Byrnes, http://sjbyrnes.com/ Download: https://pypi.python.org/pypi/trianglesolver/ Source code repository: https://github.com/sbyrnes321/trianglesolver This little package applies the law of sines or cosines to find all the sides and angles of a triangle, if you know some of the sides and/or angles. The main function defined by this package is solve(...). Simple example:: from math import pi from trianglesolver import solve a,b,c,A,B,C = solve(b=7.6, c=8.3, A=pi/3) Following the usual convention, lower-case letters are side lengths and capital letters are angles. Corresponding letters are opposite each other, e.g. side b is opposite angle B. All angles are in radians! However, you can use the degree constant to convert:: from trianglesolver import solve, degree a,b,c,A,B,C = solve(b=7, A=5*degree, B=70*degree) print(C / degree) """ from __future__ import division, print_function from math import sin, cos, pi, sqrt, acos, asin, isclose # 5 * degree is 5 degrees in radians # X / degree is the angle X expressed in degrees degree = pi/180 # ---------- Helper functions -------------------------------------- # def aaas(D, E, F, f): """ This function solves the triangle and returns (d,e,f,D,E,F) """ d = f * sin(D) / sin(F) e = f * sin(E) / sin(F) return (d,e,f,D,E,F) def sss(d,e,f): """ This function solves the triangle and returns (d,e,f,D,E,F) """ assert d + e > f and e + f > d and f + d > e F = acos((d**2 + e**2 - f**2) / (2 * d * e)) E = acos((d**2 + f**2 - e**2) / (2 * d * f)) D = pi - F - E return (d,e,f,D,E,F) def sas(d,e,F): """ This function solves the triangle and returns (d,e,f,D,E,F) """ f = sqrt(d**2 + e**2 - 2 * d * e * cos(F)) return sss(d,e,f) def ssa(d, e, D, ssa_flag): """ This function solves the triangle and returns (d,e,f,D,E,F) See docstring for solve() for definition of ssa_flag""" assert ssa_flag in ('acute', 'obtuse', 'forbid'), 'Invalid value of ssa_flag' sinE = sin(D) * e / d # This whole part is to calculate E if isclose(sinE, 1): # Right triangle, where the solution is unique E = pi/2 else: assert sinE < 1, 'No such triangle' E_acute = asin(sinE) E_obtuse = pi - E_acute acute_is_valid = (0 < (pi - D - E_acute) < pi) obtuse_is_valid = (0 < (pi - D - E_obtuse) < pi) if ssa_flag == 'acute': assert acute_is_valid, 'No such triangle' E = E_acute elif ssa_flag == 'obtuse': assert obtuse_is_valid, 'No such triangle' E = E_obtuse else: assert ssa_flag == 'forbid' if acute_is_valid and obtuse_is_valid: raise ValueError('Two different triangles fit this description') if (not acute_is_valid) and (not obtuse_is_valid): raise ValueError('No such triangle') E = E_acute if acute_is_valid else E_obtuse # Now that we know E, the rest is straightforward F = pi - D - E e,f,d,E,F,D = aaas(E,F,D,d) return (d,e,f,D,E,F) # ---------- Main function you should use ------------------------- # def solve(a=None, b=None, c=None, A=None, B=None, C=None, ssa_flag='forbid'): """ Solve to find all the information about a triangle, given partial information. a, b, c, A, B, and C are the three sides and angles. (e.g. A is the angle opposite the side of length a.) Out of these six possibilities, you need to tell the program exactly three. Then the program will tell you all six. It returns a tuple (a, b, c, A, B, C). "ssa" is the situation when you give two sides and an angle which is not between them. This is usually not enough information to specify a unique triangle. Usually there are two possible triangles—except for a special case with right triangles where the two possible triangles are the same (the equation has a "double root"), and some cases where one of the two possible triangles has a negative angle. Therefore there is an 'ssa_flag'. You can set it to'forbid' (raise an error if the answer is not unique - the default setting), or 'acute' (where the unknown angle across from the known side is chosen to be acute) or 'obtuse' (similarly). """ if sum(x is not None for x in (a,b,c,A,B,C)) != 3: raise ValueError('Must provide exactly 3 inputs') if sum(x is None for x in (a,b,c)) == 3: raise ValueError('Must provide at least 1 side length') assert all(x > 0 for x in (a,b,c,A,B,C) if x is not None) assert all(x < pi for x in (A,B,C) if x is not None) assert ssa_flag in ('forbid', 'acute', 'obtuse') # If three sides are known... if sum(x is not None for x in (a,b,c)) == 3: a,b,c,A,B,C = sss(a,b,c) return (a,b,c,A,B,C) # If two sides and one angle are known... if sum(x is not None for x in (a,b,c)) == 2: # ssa case if all(x is not None for x in (a, A, b)): a,b,c,A,B,C = ssa(a, b, A, ssa_flag) elif all(x is not None for x in (a, A, c)): a,c,b,A,C,B = ssa(a, c, A, ssa_flag) elif all(x is not None for x in (b, B, a)): b,a,c,B,A,C = ssa(b, a, B, ssa_flag) elif all(x is not None for x in (b, B, c)): b,c,a,B,C,A = ssa(b, c, B, ssa_flag) elif all(x is not None for x in (c, C, a)): c,a,b,C,A,B = ssa(c, a, C, ssa_flag) elif all(x is not None for x in (c, C, b)): c,b,a,C,B,A = ssa(c, b, C, ssa_flag) # sas case elif all(x is not None for x in (a, b, C)): a,b,c,A,B,C = sas(a, b, C) elif all(x is not None for x in (b, c, A)): b,c,a,B,C,A = sas(b, c, A) elif all(x is not None for x in (c, a, B)): c,a,b,C,A,B = sas(c, a, B) else: raise ValueError('Oops, this code should never run') return (a,b,c,A,B,C) # If one side and two angles are known... if sum(x is not None for x in (a,b,c)) == 1: # Find the third angle... if A is None: A = pi - B - C elif B is None: B = pi - A - C else: C = pi - A - B assert A > 0 and B > 0 and C > 0 # Then solve the triangle... if c is not None: a,b,c,A,B,C = aaas(A,B,C,c) elif a is not None: b,c,a,B,C,A = aaas(B,C,A,a) else: c,a,b,C,A,B = aaas(C,A,B,b) return (a,b,c,A,B,C) raise ValueError('Oops, this code should never run') #---------------- Tests ------------------------------------------------- # def tuple_isclose(x,y): return all(isclose(x[i], y[i]) for i in range(len(x))) def test_triangle(a,b,c,A,B,C): """ Check that the triangle satisfies the law of cosines and law of sines""" assert isclose(a/sin(A), b/sin(B)) assert isclose(a/sin(A), c/sin(C)) assert isclose(c**2, a**2 + b**2 - 2 * a * b * cos(C)) assert isclose(a**2, b**2 + c**2 - 2 * b * c * cos(A)) assert isclose(b**2, c**2 + a**2 - 2 * c * a * cos(B)) def test_solver(a,b,c): """ Test that the program works, for a triangle of sides a,b,c.""" # first find the angles, testing SSS a1,b1,c1,A,B,C = solve(a=a,b=b,c=c) assert tuple_isclose((a,b,c), (a1,b1,c1)) test_triangle(a,b,c,A,B,C) tri = (a,b,c,A,B,C) # SAS tests assert tuple_isclose(tri, solve(a=a,b=b,C=C)) assert tuple_isclose(tri, solve(a=a,c=c,B=B)) assert tuple_isclose(tri, solve(b=b,c=c,A=A)) # SAA / ASA tests assert tuple_isclose(tri, solve(a=a, A=A, B=B)) assert tuple_isclose(tri, solve(a=a, A=A, C=C)) assert tuple_isclose(tri, solve(a=a, B=B, C=C)) assert tuple_isclose(tri, solve(b=b, A=A, B=B)) assert tuple_isclose(tri, solve(b=b, A=A, C=C)) assert tuple_isclose(tri, solve(b=b, B=B, C=C)) assert tuple_isclose(tri, solve(c=c, A=A, B=B)) assert tuple_isclose(tri, solve(c=c, A=A, C=C)) assert tuple_isclose(tri, solve(c=c, B=B, C=C)) Atype = 'acute' if A < pi/2 else 'obtuse' Btype = 'acute' if B < pi/2 else 'obtuse' Ctype = 'acute' if C < pi/2 else 'obtuse' # SSA tests assert tuple_isclose(tri, solve(a=a, b=b, A=A, ssa_flag=Btype)) assert tuple_isclose(tri, solve(a=a, b=b, B=B, ssa_flag=Atype)) assert tuple_isclose(tri, solve(a=a, c=c, A=A, ssa_flag=Ctype)) assert tuple_isclose(tri, solve(a=a, c=c, C=C, ssa_flag=Atype)) assert tuple_isclose(tri, solve(b=b, c=c, B=B, ssa_flag=Ctype)) assert tuple_isclose(tri, solve(b=b, c=c, C=C, ssa_flag=Btype)) def run_lots_of_tests(): for a,b,c in ((7,8,9), (5,6,10), (5,10,6), (10,5,6), (5,12,13), (12,5,13), (12,13,5)): test_solver(a,b,c) print('All tests pass!')