Be sure to use rotation rather than mirroring, since the latter would cause inconsistent triangle winding. To finish making the actual sphere, make 7 copies of the patch at various orientations. The following Python snippet generates the vertices for a single patch.ĭef compute_geodesic(point_a, point_b, num_segments): angle_between_endpoints = math.acos(np.dot(point_a, point_b)) rotation_axis = np.linalg.norm(np.cross(point_a, point_b)) point_list = if num_segments = 0: return point_list dtheta = angle_between_endpoints / num_segments for point_index in range( 1, num_segments): theta = point_index * dtheta q = quaternion.create_from_axis_rotation(rotation_axis, theta) point_list.append(quaternion.apply_to_vector(q, point_a)) point_list.append(point_b) return point_list This is shown in blue in the above image. To leverage parametric evaluation instead of subdivision, simply march along the sequence of geodesic lines that stretch between two patch boundaries. To make an octasphere, you can tessellate one-eighth of the shape (i.e. the mesh patch that arises from a single triangle in the octahedron), then make 7 copies at various rotations. If this is a problem, you might want to consider using octahedral mapping instead of lat-long mapping. Note however that octaspheres suffer from undersampling near the poles when the geometric LOD is low. Octaspheres can be parameterized with plain old lat-long mapping whereas icospheres force you to use an alternative texture mapping scheme. For example, it can be generalized to include rounded cuboids like the ones below. Moreover its 8-way symmetry is useful in a number of ways. However it’s easy to directly generate an octasphere to any level of detail parametrically, without actual subdivision. This is subdivided from a mere 8-sided polyhedron, and is arguably less visually pleasing than the icosphere. In this post I’m championing the octasphere (right). While icospheres are slightly more complex to create than UV spheres, they do not suffer from irregular faces. The icosphere is generated by starting with a regular icosahedron (the 20-sided Platonic solid) and applying multiple subdivisions. The UV sphere is a parametric surface, typically generated with a double-nested loop over θ and φ. Two popular ways of tessellating spheres are the UV sphere (left) and icosphere (middle). When choosing from the Platonic solids, careful you must be.
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