How to calculate a convex hull with SciPy

Point clouds often contain measurements that lie inside the region formed by their outermost samples. A convex hull reduces that cloud to its smallest convex boundary, which makes the perimeter and enclosed area available for later geometry work.

SciPy's scipy.spatial.ConvexHull accepts a two-dimensional NumPy array and delegates hull construction to Qhull. The returned vertices indexes preserve counterclockwise order for 2-D input, so indexing the original array produces an ordered polygon boundary.

Two interior samples are included to show that they do not become boundary vertices. For a 2-D hull, SciPy reports the perimeter through hull.area and the polygon area through hull.volume; a shoelace calculation independently cross-checks the area.

1. Create the initial convex_hull_demo.py script with SciPy imports and a two-dimensional point array.

   $ cat > convex_hull_demo.py <<'PY'
   import numpy as np
   from scipy.spatial import ConvexHull
   
   points = np.array([
       [0.0, 0.0],
       [2.0, 0.0],
       [2.0, 1.5],
       [0.0, 1.0],
       [0.8, 0.5],
       [1.2, 0.9],
   ])
   PY

   The first four points form the outer quadrilateral, while the last two points lie inside it.

2. Append the hull calculation and ordered boundary extraction to convex_hull_demo.py.

   $ cat >> convex_hull_demo.py <<'PY'
   
   hull = ConvexHull(points)
   boundary = points[hull.vertices]
   PY

3. Append the boundary coordinate and measurement report to convex_hull_demo.py.

   $ cat >> convex_hull_demo.py <<'PY'
   
   print("boundary vertex indexes:", hull.vertices.tolist())
   print("boundary coordinates:")
   for index, coordinates in zip(hull.vertices, boundary):
       print(f"  {index}: ({coordinates[0]:.1f}, {coordinates[1]:.1f})")
   print(f"perimeter: {hull.area:.3f}")
   print(f"area: {hull.volume:.3f}")
   PY

   The last two points sit inside the quadrilateral, so they should not appear in hull.vertices.

4. Append an independent shoelace-area check to convex_hull_demo.py.

   $ cat >> convex_hull_demo.py <<'PY'
   
   next_boundary = np.roll(boundary, -1, axis=0)
   shoelace_area = 0.5 * abs(
       np.sum(boundary[:, 0] * next_boundary[:, 1])
       - np.sum(boundary[:, 1] * next_boundary[:, 0])
   )
   np.testing.assert_allclose(hull.volume, shoelace_area)
   print(f"shoelace area check: {shoelace_area:.3f}")
   PY

   For 2-D input, hull.area is the perimeter and hull.volume is the polygon area. In higher dimensions, the same attributes mean surface area and volume.

5. Run convex_hull_demo.py to calculate the boundary, perimeter, and area.

   $ python3 convex_hull_demo.py
   boundary vertex indexes: [0, 1, 2, 3]
   boundary coordinates:
     0: (0.0, 0.0)
     1: (2.0, 0.0)
     2: (2.0, 1.5)
     3: (0.0, 1.0)
   perimeter: 6.562
   area: 2.500
   shoelace area check: 2.500