7-spheres-3d
Python scripts to generate 3D models of some complex curves
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""" Python script to generate 3D models of some surfaces using the methods of [1] [1]: Hanson, Andrew J. "A Construction for Computer Visualization of Certain Complex Curves." (1994). https://homes.luddy.indiana.edu/hansona/papers/CP2-94.pdf """ import numpy as np import numpy.linalg as la from typing import List, Tuple, Iterator, Callable Vector3 = np.ndarray[Tuple[3], np.dtype[np.float64]] Patch = np.ndarray[Tuple[3, int, int], np.dtype[np.float64]] ComplexGrid = np.ndarray[Tuple[int, int], np.dtype[np.complex128]] Projection = Callable[[ComplexGrid, ComplexGrid], Patch] class Model: def __init__(self, p: Projection): self.patches: List[Patch] = [] self.projection = p def append_patch(self, z1: ComplexGrid, z2: ComplexGrid): """ Appends a 2D array of points in C^2 to the model and updates the bounding box """ self.patches.append(self.projection(z1, z2)) def vertices(self) -> Iterator[Vector3]: """ Iterate through the vertices of the mesh """ return (x[:, j, i] for x in self.patches for i in range(x.shape[2]) for j in range(x.shape[1])) def write_obj(self, path: str): """ Exports the model to the an OBJ file at `path` """ with open(path, "w") as f: vs = set() # vertices eff_vis = list() # effective indixes f.write("# vertices\n") vi = 0 # OBJ indices start at 1 for x1, x2, x3 in self.vertices(): v = f"{x1:.3f} {x2:.3f} {x3:.3f}" eff_vis.append(vi) if v not in vs: vs.add(v) vi += 1 f.write(f"v {v}\n") f.write("\n# faces\n") v_i = 0 fs = set() # faces for patch in self.patches: _, cols, rows = patch.shape for theta_i in range(rows - 1): for xi_i in range(cols - 1): v1 = eff_vis[v_i + xi_i + theta_i * cols] v2 = eff_vis[v_i + xi_i + (theta_i + 1) * cols] v3 = eff_vis[v_i + (xi_i + 1) + theta_i * cols] v4 = eff_vis[v_i + (xi_i + 1) + (theta_i + 1) * cols] if (v1, v2, v3) not in fs: fs.add((v1, v2, v3)) f.write(f"f {v1} {v2} {v3}\n") if (v2, v4, v3) not in fs: fs.add((v1, v2, v3)) f.write(f"f {v2} {v4} {v3}\n") v_i += rows * cols def normalized_p(z1: ComplexGrid, z2: ComplexGrid) -> Patch: "Projection R^4 -> S^3 + inclusion R^3 -> S^3" assert z1.shape == z2.shape z = np.stack((z1, z2), axis=0) w = z/la.norm(z, axis=0) v = np.vstack([w.real, w.imag]) x = v[:3, :, :]/(1 - v[3, :, :]) return x def riemman_sphere_p(z1: ComplexGrid, z2: ComplexGrid) -> Patch: "Riemman-sphere transform + projection on the first 3 coordinates" assert z1.shape == z2.shape norm_sqrd = z1.real ** 2 + z1.imag ** 2 + z2.real ** 2 + z2.imag ** 2 u0 = 2 * (norm_sqrd/4) u1 = z1.real u2 = z1.imag return np.vstack([u0, u1, u2])/(1 + norm_sqrd/4) def s(k: int, n: int) -> np.complex128: return np.exp(2 * np.pi * 1j * k / n) EPS = 0.1 XI_MAX = 15 def fermat_surface(n: int, eps: float = EPS, xi_max: float = XI_MAX) -> Model: """ Returns a model of the Fermat surface z1^n + z2^n = 1 """ if n <= 0: raise ValueError(f"Parameter \"n\" should be positive: n = {n}") m = Model(normalized_p) theta, xi = np.meshgrid(np.arange(0, np.pi / 2 + eps, eps), np.arange(-xi_max, xi_max + eps, eps)) for k1 in range(n): for k2 in range(n): u1 = (np.exp(xi + 1j * theta)+np.exp(-xi - 1j * theta))/2 z1 = s(k1, n) * (u1**(2/n)) u2 = (np.exp(xi + 1j * theta)-np.exp(-xi - 1j * theta))/2j z2 = s(k2, n) * (u2**(2/n)) z = np.stack((z1, z2), axis=0) m.append_patch(z) return m def brieskorn_surface(z3: complex, n: int, patches: Iterator[Tuple[int, int]] = None, eps: float = EPS, xi_max: float = XI_MAX) -> Model: """ Returns the model of the Brieskorn surface z1^(6 n - 1) + z2^3 + z3^2 = 0 """ if not (1 <= n <= 11): raise ValueError(f"\"n\" should be in between 1 and 11: n = {n}") if abs(z3) > 1: raise ValueError(f"\"z3\" should have norm smaller than 1: z3 = {z3}") k0 = 6*n - 1 if patches is None: patches = ((k1, k2) for k1 in range(3) for k2 in range(k0)) m = Model(normalized_p) theta, xi = np.meshgrid(np.arange(0, np.pi / 2 + eps, eps), np.arange(-xi_max, xi_max + eps, eps)) for k1, k2 in patches: if not (0 <= k1 < 3) or not (0 <= k2 < k0): continue u1 = (np.exp(xi + 1j * theta)-np.exp(-xi - 1j * theta))/2j z1 = -(z3**(2/k0)) * s(k2, k0) * (u2**(2/k0)) u2 = (np.exp(xi + 1j * theta)+np.exp(-xi - 1j * theta))/2 z2 = -(z3**(2/3)) * s(k1, 3) * (u1**(2/3)) m.append_patch(z2, z1) return m