separation_axis_theorem.py

# This code performs collision testing of convex 2D polyedra by means
# of the Hyperplane separation theorem, also known as Separating axis theorem (SAT).
#
# For more information visit:
# https://en.wikipedia.org/wiki/Hyperplane_separation_theorem
#
# Copyright (C) 2016, Juan Antonio Aldea Armenteros
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.


# -*- coding: utf8 -*-

from math import sqrt


def normalize(vector):
    """
    :return: The vector scaled to a length of 1
    """
    norm = sqrt(vector[0] ** 2 + vector[1] ** 2)
    return vector[0] / norm, vector[1] / norm


def dot(vector1, vector2):
    """
    :return: The dot (or scalar) product of the two vectors
    """
    return vector1[0] * vector2[0] + vector1[1] * vector2[1]


def edge_direction(point0, point1):
    """
    :return: A vector going from point0 to point1
    """
    return point1[0] - point0[0], point1[1] - point0[1]


def orthogonal(vector):
    """
    :return: A new vector which is orthogonal to the given vector
    """
    return vector[1], -vector[0]


def vertices_to_edges(vertices):
    """
    :return: A list of the edges of the vertices as vectors
    """
    return [edge_direction(vertices[i], vertices[(i + 1) % len(vertices)]) for i in range(len(vertices))]


def project(vertices, axis):
    """
    :return: A vector showing how much of the vertices lies along the axis
    """
    dots = [dot(vertex, axis) for vertex in vertices]
    return [min(dots), max(dots)]


def overlap(projection1, projection2):
    """
    :return: Boolean indicating if the two projections overlap
    """
    return min(projection1) <= max(projection2) and min(projection2) <= max(projection1)


def separating_axis_theorem(vertices_a, vertices_b):
    edges = vertices_to_edges(vertices_a) + vertices_to_edges(vertices_b)
    axes = [normalize(orthogonal(edge)) for edge in edges]

    for axis in axes:
        projection_a = project(vertices_a, axis)
        projection_b = project(vertices_b, axis)

        overlapping = overlap(projection_a, projection_b)

        if not overlapping:
            return False

    return True


def main():
    a_vertices = [(0, 0), (70, 0), (0, 70)]
    b_vertices = [(70, 70), (150, 70), (70, 150)]
    c_vertices = [(30, 30), (150, 70), (70, 150)]

    print(separating_axis_theorem(a_vertices, b_vertices))
    print(separating_axis_theorem(a_vertices, c_vertices))
    print(separating_axis_theorem(b_vertices, c_vertices))


if __name__ == "__main__":
    main()