Sphere intersection and spherical cap volumes (#5579)

* sphere intersection + spherical cap volume formulas

* reformatted

* Update volume.py

Co-authored-by: Christian Clauss <[email protected]>

Author: matteomessmer

maths/volume.py

@@ -20,6 +20,56 @@ def vol_cube(side_length: int | float) -> float:
     return pow(side_length, 3)
 
 
+def vol_spherical_cap(height: float, radius: float) -> float:
+    """
+    Calculate the Volume of the spherical cap.
+    :return 1/3 pi * height ^ 2 * (3 * radius - height)
+
+    >>> vol_spherical_cap(1, 2)
+    5.235987755982988
+    """
+    return 1 / 3 * pi * pow(height, 2) * (3 * radius - height)
+
+
+def vol_spheres_intersect(
+    radius_1: float, radius_2: float, centers_distance: float
+) -> float:
+    """
+    Calculate the volume of the intersection of two spheres.
+
+    The intersection is composed by two spherical caps and therefore its volume is the
+    sum of the volumes of the spherical caps.  First it calculates the heights (h1, h2)
+    of the the spherical caps, then the two volumes and it returns the sum.
+    The height formulas are
+    h1 = (radius_1 - radius_2 + centers_distance)
+       * (radius_1 + radius_2 - centers_distance)
+       / (2 * centers_distance)
+    h2 = (radius_2 - radius_1 + centers_distance)
+       * (radius_2 + radius_1 - centers_distance)
+       / (2 * centers_distance)
+    if centers_distance is 0 then it returns the volume of the smallers sphere
+    :return vol_spherical_cap(h1, radius_2) + vol_spherical_cap(h2, radius_1)
+
+    >>> vol_spheres_intersect(2, 2, 1)
+    21.205750411731103
+    """
+    if centers_distance == 0:
+        return vol_sphere(min(radius_1, radius_2))
+
+    h1 = (
+        (radius_1 - radius_2 + centers_distance)
+        * (radius_1 + radius_2 - centers_distance)
+        / (2 * centers_distance)
+    )
+    h2 = (
+        (radius_2 - radius_1 + centers_distance)
+        * (radius_2 + radius_1 - centers_distance)
+        / (2 * centers_distance)
+    )
+
+    return vol_spherical_cap(h1, radius_2) + vol_spherical_cap(h2, radius_1)
+
+
 def vol_cuboid(width: float, height: float, length: float) -> float:
     """
     Calculate the Volume of a Cuboid.
@@ -127,6 +177,8 @@ def main():
     print("Pyramid: " + str(vol_pyramid(2, 2)))  # ~= 1.33
     print("Sphere: " + str(vol_sphere(2)))  # ~= 33.5
     print("Circular Cylinder: " + str(vol_circular_cylinder(2, 2)))  # ~= 25.1
+    print("Spherical cap: " + str(vol_spherical_cap(1, 2)))  # ~= 5.24
+    print("Spheres intersetion: " + str(vol_spheres_intersect(2, 2, 1)))  # ~= 21.21
 
 
 if __name__ == "__main__":