Computing and inspheres and circumspheres

Under different conditions, various types of spheres containing or contained within a shape can be useful. For example, for spheres contained in a shape we may be interested in the largest sphere contained by a shape, or the largest concentric sphere contained within the shape. For a polyhedron, we may instead want to find the sphere that touches all the faces, if it exists.

import fresnel
import numpy as np
from matplotlib import pyplot as plt

import coxeter

def plot_polyhedron_with_sphere(shape, insphere=True):
    """Image a polyhedron along with a sphere contained within it."""
    device = fresnel.Device()
    scene = fresnel.Scene(device)

    transparent_material = fresnel.material.Material(
        color=fresnel.color.linear([1, 1, 1]),
        spec_trans=0.95,
        roughness=0.2,
        primitive_color_mix=0.0,
    )
    colored_material = fresnel.material.Material(
        color=fresnel.color.linear([0.9, 0.714, 0.169]),
        roughness=0.8,
    )

    # First make the shape and set up its properties.
    primitive = fresnel.geometry.ConvexPolyhedron(
        scene,
        fresnel.util.convex_polyhedron_from_vertices(shape.vertices),
        N=1,
        outline_width=0.01,
        material=transparent_material if insphere else colored_material,
    )
    primitive.color_by_face = 0.0

    # Now draw the insphere within the shape.
    sphere = fresnel.geometry.Sphere(
        scene,
        N=1,
        material=colored_material if insphere else transparent_material,
    )

    # Make the sphere a little bit smaller than it really is,
    # otherwise you get artifacts near the intersection of the
    # polyhedron and the insphere.
    sphere.radius[:] = [
        shape.insphere.radius * 0.99 if insphere else shape.circumsphere.radius * 1.01
    ]

    scene.camera = fresnel.camera.Orthographic.fit(scene, view="front")
    tracer = fresnel.tracer.Path(device=device, w=300, h=300)
    return tracer.sample(scene, samples=24, light_samples=40)

The Platonic solids are a canonical set of shapes we can use for our analysis. Conveniently, they can easily be generated using coxeter. A good example is the dodecahedron.

dodecahedron = coxeter.families.PlatonicFamily.get_shape("Dodecahedron")

We can query different types of spheres from this shape now:

# The sphere tangent to all the faces of the polyhedron.
print(dodecahedron.insphere)

# The largest concentric sphere contained in the shape.
print(dodecahedron.maximal_centered_bounded_sphere)

coxeter.shapes.Sphere(radius=0.5648000780281438, center=[-5.551115123125784e-17, -2.220446049250313e-16, -3.23771046624367e-16])
coxeter.shapes.Sphere(radius=0.564800078028144, center=[-1.7347234759768062e-17, -1.1564823173178707e-17, 1.619075244245019e-17])

Let’s visualize what these shapes look like:

plot_polyhedron_with_sphere(dodecahedron)

If we instead want to look at spheres containing a shape, we can get those as well.

# The sphere tangent to all the faces of the polyhedron.
print(dodecahedron.circumsphere)

# The largest concentric sphere contained in the shape.
print(dodecahedron.minimal_centered_bounding_sphere)

coxeter.shapes.Sphere(radius=0.7107492598629351, center=[0.0, -2.7755575615628914e-17, -1.1102230246251565e-16])
coxeter.shapes.Sphere(radius=0.7107492598629351, center=[-1.7347234759768062e-17, -1.1564823173178707e-17, 1.619075244245019e-17])

plot_polyhedron_with_sphere(dodecahedron, False)