diff --git a/maths/points_are_collinear_3d.py b/maths/points_are_collinear_3d.py new file mode 100644 index 000000000..3bc0b3b9e --- /dev/null +++ b/maths/points_are_collinear_3d.py @@ -0,0 +1,126 @@ +""" +Check if three points are collinear in 3D. + +In short, the idea is that we are able to create a triangle using three points, +and the area of that triangle can determine if the three points are collinear or not. + + +First, we create two vectors with the same initial point from the three points, +then we will calculate the cross-product of them. + +The length of the cross vector is numerically equal to the area of a parallelogram. + +Finally, the area of the triangle is equal to half of the area of the parallelogram. + +Since we are only differentiating between zero and anything else, +we can get rid of the square root when calculating the length of the vector, +and also the division by two at the end. + +From a second perspective, if the two vectors are parallel and overlapping, +we can't get a nonzero perpendicular vector, +since there will be an infinite number of orthogonal vectors. + +To simplify the solution we will not calculate the length, +but we will decide directly from the vector whether it is equal to (0, 0, 0) or not. + + +Read More: + https://math.stackexchange.com/a/1951650 +""" + +Vector3d = tuple[float, float, float] +Point3d = tuple[float, float, float] + + +def create_vector(end_point1: Point3d, end_point2: Point3d) -> Vector3d: + """ + Pass two points to get the vector from them in the form (x, y, z). + + >>> create_vector((0, 0, 0), (1, 1, 1)) + (1, 1, 1) + >>> create_vector((45, 70, 24), (47, 32, 1)) + (2, -38, -23) + >>> create_vector((-14, -1, -8), (-7, 6, 4)) + (7, 7, 12) + """ + x = end_point2[0] - end_point1[0] + y = end_point2[1] - end_point1[1] + z = end_point2[2] - end_point1[2] + return (x, y, z) + + +def get_3d_vectors_cross(ab: Vector3d, ac: Vector3d) -> Vector3d: + """ + Get the cross of the two vectors AB and AC. + + I used determinant of 2x2 to get the determinant of the 3x3 matrix in the process. + + Read More: + https://en.wikipedia.org/wiki/Cross_product + https://en.wikipedia.org/wiki/Determinant + + >>> get_3d_vectors_cross((3, 4, 7), (4, 9, 2)) + (-55, 22, 11) + >>> get_3d_vectors_cross((1, 1, 1), (1, 1, 1)) + (0, 0, 0) + >>> get_3d_vectors_cross((-4, 3, 0), (3, -9, -12)) + (-36, -48, 27) + >>> get_3d_vectors_cross((17.67, 4.7, 6.78), (-9.5, 4.78, -19.33)) + (-123.2594, 277.15110000000004, 129.11260000000001) + """ + x = ab[1] * ac[2] - ab[2] * ac[1] # *i + y = (ab[0] * ac[2] - ab[2] * ac[0]) * -1 # *j + z = ab[0] * ac[1] - ab[1] * ac[0] # *k + return (x, y, z) + + +def is_zero_vector(vector: Vector3d, accuracy: int) -> bool: + """ + Check if vector is equal to (0, 0, 0) of not. + + Sine the algorithm is very accurate, we will never get a zero vector, + so we need to round the vector axis, + because we want a result that is either True or False. + In other applications, we can return a float that represents the collinearity ratio. + + >>> is_zero_vector((0, 0, 0), accuracy=10) + True + >>> is_zero_vector((15, 74, 32), accuracy=10) + False + >>> is_zero_vector((-15, -74, -32), accuracy=10) + False + """ + return tuple(round(x, accuracy) for x in vector) == (0, 0, 0) + + +def are_collinear(a: Point3d, b: Point3d, c: Point3d, accuracy: int = 10) -> bool: + """ + Check if three points are collinear or not. + + 1- Create tow vectors AB and AC. + 2- Get the cross vector of the tow vectors. + 3- Calcolate the length of the cross vector. + 4- If the length is zero then the points are collinear, else they are not. + + The use of the accuracy parameter is explained in is_zero_vector docstring. + + >>> are_collinear((4.802293498137402, 3.536233125455244, 0), + ... (-2.186788107953106, -9.24561398001649, 7.141509524846482), + ... (1.530169574640268, -2.447927606600034, 3.343487096469054)) + True + >>> are_collinear((-6, -2, 6), + ... (6.200213806439997, -4.930157614926678, -4.482371908289856), + ... (-4.085171149525941, -2.459889509029438, 4.354787180795383)) + True + >>> are_collinear((2.399001826862445, -2.452009976680793, 4.464656666157666), + ... (-3.682816335934376, 5.753788986533145, 9.490993909044244), + ... (1.962903518985307, 3.741415730125627, 7)) + False + >>> are_collinear((1.875375340689544, -7.268426006071538, 7.358196269835993), + ... (-3.546599383667157, -4.630005261513976, 3.208784032924246), + ... (-2.564606140206386, 3.937845170672183, 7)) + False + """ + ab = create_vector(a, b) + ac = create_vector(a, c) + return is_zero_vector(get_3d_vectors_cross(ab, ac), accuracy)