path: root/blender/bolapeluda.py

# -*- coding: utf-8 -*-
"""BOLA PELUDA - the hairy ball theorem (Brouwer, 1912).

Each hair lies down along the tangent field v(p) = A x p, as much as the
field is strong: where |v| vanishes (the two poles of axis A) the hair has
nowhere to lie and stands up. So the swirl is not drawn by hand, it comes
from the same computation that combs the rest.
"""
import math, os, sys
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
import bpy
from mathutils import Vector, Matrix
from base import *

NAME_KEY = "bolapeluda"
R = 1.15                  # sphere radius
LARGO = 0.34              # hair length
K = 7                     # points per hair
N = 1500                  # number of hairs
THRESHOLD = 0.42             # |v| below this, the hair never lies down
CZ = 0.50                 # height of the centre in the frame

# Combings: (axis, kind). The axes lie almost in the screen plane (XZ) so the
# TWO zeros fall on the silhouette and show together.
#   rot -> field v = A x p          (combs in circles, like latitudes)
#   mer -> field v = A - (A.p) p    (combs from pole to pole, along meridians)
# Both vanish exactly where p is parallel to A: that is where the swirl is.
COMBINGS = [((0.80, -0.05, 0.60), "rot"),
            ((-0.58, -0.05, 0.82), "mer"),
            ((0.97, -0.08, -0.22), "rot"),
            ((0.12, -0.05, 0.99), "mer"),
            ((-0.86, -0.06, -0.50), "rot"),
            ((0.80, -0.05, 0.60), "rot")]


def normalize_text(v):
    n = math.sqrt(sum(c * c for c in v))
    return tuple(c / n for c in v) if n > 1e-9 else (0.0, 0.0, 1.0)


def cross_m(a, b):
    return (a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0])


def sphere_points(n):
    """Fibonacci spiral: spreads n points almost evenly over the sphere."""
    pts, ga = [], math.pi * (3 - math.sqrt(5))
    for i in range(n):
        z = 1 - 2 * (i + 0.5) / n
        r = math.sqrt(max(0.0, 1 - z * z))
        th = ga * i
        pts.append((r * math.cos(th), r * math.sin(th), z))
    return pts


def field(p, axis_obj, kind_m):
    if kind_m == "mer":
        d = sum(a * b for a, b in zip(axis_obj, p))
        return tuple(axis_obj[i] - d * p[i] for i in range(3))
    return cross_m(axis_obj, p)


def hair(p, axis_obj, combed, kind_m="rot", radio=R, largo=LARGO):
    """Returns the K points of the hair rooted at p and the value of |v| there."""
    v = field(p, axis_obj, kind_m)
    m = math.sqrt(sum(c * c for c in v))
    if m > 1e-6:
        t = tuple(c / m for c in v)
    else:                                  # at the zero there is no direction: it stands up
        t = normalize_text(cross_m(p, (0.0, 0.0, 1.0)) if abs(p[2]) < 0.9 else (1.0, 0.0, 0.0))
    b = combed * min(1.0, m / THRESHOLD)
    pos = [tuple(c * radio for c in p)]
    step = largo / (K - 1)
    for j in range(1, K):
        s = j / (K - 1.0)
        phi = b * (math.pi / 2) * s ** 0.85
        cs, sn = math.cos(phi), math.sin(phi)
        d = (p[0] * cs + t[0] * sn, p[1] * cs + t[1] * sn, p[2] * cs + t[2] * sn)
        a = pos[-1]
        pos.append((a[0] + d[0] * step, a[1] + d[1] * step, a[2] + d[2] * step))
    return pos, m


def torus_hairs(nu=60, nv=24, RT=0.90, rt=0.34, largo=0.22):
    """Donut hairs, combed by the toroidal tangent field: it never vanishes."""
    splines, radios = [], []
    for i in range(nu):
        u = 2 * math.pi * i / nu
        cu, su = math.cos(u), math.sin(u)
        for j in range(nv):
            w = 2 * math.pi * j / nv
            cw, sw = math.cos(w), math.sin(w)
            p = ((RT + rt * cw) * cu, (RT + rt * cw) * su, rt * sw)
            n = (cw * cu, cw * su, sw)                  # normal
            t = (-su, cu, 0.0)                          # toroidal tangent, |t| = 1
            pos = [p]
            step = largo / (K - 1)
            for k in range(1, K):
                s = k / (K - 1.0)
                phi = (math.pi / 2) * s ** 0.85
                cs, sn = math.cos(phi), math.sin(phi)
                d = (n[0] * cs + t[0] * sn, n[1] * cs + t[1] * sn, n[2] * cs + t[2] * sn)
                a = pos[-1]
                pos.append((a[0] + d[0] * step, a[1] + d[1] * step, a[2] + d[2] * step))
            splines.append(pos)
            radios.append([1.0 - 0.72 * (k / (K - 1.0)) for k in range(K)])
    return splines, radios


def build_scene(T):
    sc = scene_setup()
    lens, FRAME_H = 70.0, 6.0
    camera_obj((0.0, -lens / 36.0 * FRAME_H, 0.0), (0.0, 0.0, 0.0), lens=lens)
    bpy.context.scene.camera.data.sensor_fit = 'VERTICAL'
    bpy.context.scene.camera.data.sensor_height = 36.0

    light_obj("key", 'AREA', (-3.2, -4.6, 3.8), 800, "blanco", size_u=5.0, sight=(0, 0, CZ))
    light_obj("fill", 'AREA', (3.8, -3.6, -1.2), 260, "celeste", size_u=5.0, sight=(0, 0, CZ))
    light_obj("rim", 'AREA', (0.4, 4.2, 2.6), 620, "ambar", size_u=4.0, sight=(0, 0, CZ))

    rig = bpy.data.objects.new("rig", None)
    bpy.context.collection.objects.link(rig)
    rig.location = (0, 0, CZ)

    m_skin = material("piel", "azul", rough=0.55, metal=0.1)
    ball = sphere("bola", R * 0.985, m_skin, seg_m=64, rings=36)
    ball.parent = rig

    base_points = sphere_points(N)
    splines = [hair(p, COMBINGS[0][0], 0.0)[0] for p in base_points]
    radios = [[1.0 - 0.70 * (j / (K - 1.0)) for j in range(K)] for _ in base_points]
    hair_a = curve_poly("pelo_a", splines, thickness_px=0.0135, radios=radios,
                       mat=material("m_pelo", "ambar", rough=0.42, emit=0.55))
    hair_b = curve_poly("pelo_b", splines, thickness_px=0.0135, radios=radios,
                       mat=material("m_remol", "rosa", rough=0.42, emit=1.9))
    hair_a.parent = hair_b.parent = rig

    # marks on the zeros
    tick_list = []
    for k in range(2):
        g = bpy.data.objects.new(f"marca{k}", None)
        bpy.context.collection.objects.link(g)
        g.parent = rig
        ring = torus(f"anillo{k}", 0.34, 0.024,
                      material(f"mm{k}", "rosa", emit=3.2), u=48, v=10)
        ring.parent = g
        tick_list.append(g)

    # the donut, combed in one go
    sp_t, ra_t = torus_hairs()
    rig_t = bpy.data.objects.new("rig_toro", None)
    bpy.context.collection.objects.link(rig_t)
    rig_t.location = (0, 0, CZ)
    donut = torus("rosquilla", 0.90, 0.335, material("piel_t", "azul", rough=0.55))
    donut.parent = rig_t
    hair_t = curve_poly("pelo_toro", sp_t, thickness_px=0.0125, radios=ra_t,
                        mat=material("m_pelo_t", "verde", rough=0.42, emit=0.8))
    hair_t.parent = rig_t
    rig_t.rotation_mode = 'ZYX'        # first it spins about its axis, then it tilts
    rig_t.scale = (0, 0, 0)

    tag_m = {"esf": txt_m("2", size_u=0.52, color="ambar"),
           "ros": txt_m("0", size_u=0.52, color="verde"),
           "pie": txt_m("agujeros que cuentan", size_u=0.19, color="gris")}
    for o in tag_m.values():
        o.scale = (0, 0, 0)
    return dict(rig=rig, rig_t=rig_t, ball=ball, hair_a=hair_a, hair_b=hair_b,
                tick_list=tick_list, base=base_points, tag_m=tag_m, skin=m_skin, donut_obj=donut)


def main():
    T = Timeline(NAME_KEY)
    ob = build_scene(T)
    base, hair_a, hair_b = ob["base"], ob["hair_a"], ob["hair_b"]
    cache = {"clave": None}

    def comb(axis_idx, combed):
        key_name = (axis_idx, round(combed, 3))
        if cache["clave"] == key_name:
            return
        cache["clave"] = key_name
        axis_obj = normalize_text(COMBINGS[axis_idx][0])
        kind_m = COMBINGS[axis_idx][1]
        sp, ra_a, ra_b = [], [], []
        for p in base:
            pos, m = hair(p, axis_obj, combed, kind_m)
            sp.append(pos)
            tap = [1.0 - 0.70 * (j / (K - 1.0)) for j in range(K)]
            swirl = m < THRESHOLD * 0.62 and combed > 0.25
            ra_a.append([0.0 if swirl else t for t in tap])
            ra_b.append([t * 1.25 if swirl else 0.0 for t in tap])
        rebuild_curve(hair_a, sp