312 lines
13 KiB
Python
312 lines
13 KiB
Python
"""
|
||
Butterfly Catastrophe Surface — ASCII STL Generator
|
||
|
||
Potential: V(x) = x^6 + a*x^4 + c*x^2 + d*x (a = -3 fixed, b = 0)
|
||
Equilibrium: dV/dx = 6x^5 - 12x^3 + 2c*x + d = 0
|
||
|
||
Control parameters (print base): c (horizontal), d (depth)
|
||
State variable (print height): x
|
||
|
||
With a = -3, the bifurcation set in the (c, d) plane forms the characteristic
|
||
butterfly shape: a self-intersecting loop passing through (c=3, d=0), enclosing
|
||
a region with 5 equilibria ("butterfly pocket"), surrounded by a 3-root region
|
||
with outer fold wings, and a single-root region outside. This is structurally
|
||
different from the cusp catastrophe and cannot be seen with c = d = 0 fixed.
|
||
"""
|
||
|
||
import numpy as np
|
||
import os
|
||
|
||
A_FIXED = -3.0 # butterfly unfolding parameter (must be negative)
|
||
|
||
# ── Tuning ──────────────────────────────────────────────────────────────────
|
||
GRID = 200 # control-space resolution — increase to 100–120 for print
|
||
C_RANGE = (-2.0, 7.0) # range of control parameter c
|
||
D_RANGE = (-6.0, 6.0) # range of control parameter d
|
||
X_RANGE = 2.5 # half-width of root search window
|
||
MAX_MATCH_DZ = 0.8 # max z-gap for inter-row branch matching
|
||
|
||
# ── 1. Root finding ──────────────────────────────────────────────────────────
|
||
|
||
def dV(x, c, d):
|
||
return 6*x**5 + 4*A_FIXED*x**3 + 2*c*x + d
|
||
|
||
def d2V(x, c, d):
|
||
return 30*x**4 + 12*A_FIXED*x**2 + 2*c
|
||
|
||
def find_roots(c, d, n_starts=80):
|
||
"""Return sorted real roots of dV/dx = 0 for the given (c, d)."""
|
||
xs = np.linspace(-X_RANGE, X_RANGE, n_starts)
|
||
roots = []
|
||
for x0 in xs:
|
||
x = float(x0)
|
||
for _ in range(200):
|
||
fx = dV(x, c, d)
|
||
if abs(fx) < 1e-12:
|
||
break
|
||
dfx = d2V(x, c, d)
|
||
if abs(dfx) < 1e-14:
|
||
break
|
||
step = fx / dfx
|
||
x -= step
|
||
if abs(x) > 2.0 * X_RANGE: # diverged — abandon
|
||
break
|
||
if abs(step) < 1e-10:
|
||
break
|
||
if abs(dV(x, c, d)) < 1e-7 and abs(x) <= X_RANGE + 0.15:
|
||
if not any(abs(x - r) < 1e-4 for r in roots):
|
||
roots.append(x)
|
||
return sorted(roots)
|
||
|
||
# ── 2. Branch tracking ───────────────────────────────────────────────────────
|
||
|
||
def track_branches(roots_along_axis):
|
||
"""
|
||
Track branches along one axis (fixed d, varying c) using greedy
|
||
nearest-neighbour matching. Returns a list of tracks; each track is a
|
||
list of length GRID where entry i is the root value at column i, or None
|
||
when the branch does not exist there.
|
||
|
||
Sorting-index matching (branch 0 always = branch 0) breaks at folds
|
||
because two adjacent branches coalesce, shifting every higher index by
|
||
one. Nearest-neighbour tracking follows each physical sheet through the
|
||
fold correctly: the two merging branches each get None past the fold, and
|
||
the surviving sheet keeps its track unbroken.
|
||
"""
|
||
n = len(roots_along_axis)
|
||
if n == 0:
|
||
return []
|
||
|
||
tracks = [[r] for r in roots_along_axis[0]]
|
||
|
||
for i in range(1, n):
|
||
curr = roots_along_axis[i]
|
||
|
||
prev_live = [(ti, t[-1]) for ti, t in enumerate(tracks)
|
||
if t[-1] is not None]
|
||
prev_matched, curr_matched = set(), set()
|
||
matches = {} # track_idx → curr_root_idx
|
||
|
||
cands = sorted(
|
||
[(abs(pv - curr[ci]), ti, ci)
|
||
for ti, pv in prev_live
|
||
for ci in range(len(curr))],
|
||
key=lambda x: x[0]
|
||
)
|
||
for _, ti, ci in cands:
|
||
if ti not in prev_matched and ci not in curr_matched:
|
||
matches[ti] = ci
|
||
prev_matched.add(ti)
|
||
curr_matched.add(ci)
|
||
|
||
for ti, t in enumerate(tracks):
|
||
t.append(curr[matches[ti]] if ti in matches else None)
|
||
|
||
# Branches that appear for the first time at this column
|
||
for ci in range(len(curr)):
|
||
if ci not in curr_matched:
|
||
tracks.append([None] * i + [curr[ci]])
|
||
|
||
return tracks
|
||
|
||
# ── 3. Build mesh ────────────────────────────────────────────────────────────
|
||
|
||
def _emit_quad(triangles, p00, p10, p11, p01):
|
||
triangles.append((p00, p10, p11))
|
||
triangles.append((p00, p11, p01))
|
||
|
||
|
||
def _fold_terminations(tracks, axis_vals):
|
||
"""
|
||
Scan tracks along one axis and return a list of fold-termination events.
|
||
Each event is (axis_val, xa, xb) where axis_val is the last valid position,
|
||
and xa < xb are the two branch values that die together at a fold.
|
||
|
||
Branches come in pairs at fold lines (two coalesce), so we pair adjacent
|
||
sorted dying values. Events are indexed by the axis position of the last
|
||
valid step.
|
||
"""
|
||
events = []
|
||
n = len(axis_vals) - 1 # number of steps
|
||
for step in range(n):
|
||
dying = sorted(t[step] for t in tracks
|
||
if t[step] is not None and t[step + 1] is None)
|
||
for k in range(0, len(dying) - 1, 2):
|
||
events.append((axis_vals[step], dying[k], dying[k + 1]))
|
||
return events
|
||
|
||
|
||
def build_mesh():
|
||
c_vals = np.linspace(*C_RANGE, GRID)
|
||
d_vals = np.linspace(*D_RANGE, GRID)
|
||
|
||
print(f' Computing roots on {GRID}×{GRID} grid…')
|
||
# roots_grid[i][j] = sorted roots at (c_vals[i], d_vals[j])
|
||
roots_grid = [[find_roots(c, d) for d in d_vals] for c in c_vals]
|
||
|
||
# Track branches along rows (fixed d, varying c) and columns (fixed c, varying d).
|
||
print(' Tracking branches…')
|
||
row_tracks = [track_branches([roots_grid[i][j] for i in range(GRID)])
|
||
for j in range(GRID)]
|
||
col_tracks = [track_branches([roots_grid[i][j] for j in range(GRID)])
|
||
for i in range(GRID)]
|
||
|
||
triangles = []
|
||
|
||
# ── Main surface quads ────────────────────────────────────────────────────
|
||
for j in range(GRID - 1):
|
||
d0, d1 = d_vals[j], d_vals[j + 1]
|
||
tracks_j = row_tracks[j]
|
||
tracks_j1 = row_tracks[j + 1]
|
||
|
||
for i in range(GRID - 1):
|
||
c0, c1 = c_vals[i], c_vals[i + 1]
|
||
|
||
segs_j = [(t[i], t[i + 1], k) for k, t in enumerate(tracks_j)
|
||
if t[i] is not None and t[i + 1] is not None]
|
||
segs_j1 = [(t[i], t[i + 1], k) for k, t in enumerate(tracks_j1)
|
||
if t[i] is not None and t[i + 1] is not None]
|
||
|
||
if not segs_j or not segs_j1:
|
||
continue
|
||
|
||
j1_used = set()
|
||
for x00, x10, _ in sorted(segs_j, key=lambda s: (s[0] + s[1]) / 2):
|
||
best_k1 = best_x01 = best_x11 = None
|
||
best_dist = MAX_MATCH_DZ
|
||
for x01, x11, k1 in segs_j1:
|
||
if k1 in j1_used:
|
||
continue
|
||
dist = max(abs(x00 - x01), abs(x10 - x11))
|
||
if dist < best_dist:
|
||
best_dist, best_k1 = dist, k1
|
||
best_x01, best_x11 = x01, x11
|
||
if best_k1 is None:
|
||
continue
|
||
j1_used.add(best_k1)
|
||
_emit_quad(triangles,
|
||
(c0, d0, x00), (c1, d0, x10),
|
||
(c1, d1, best_x11), (c0, d1, best_x01))
|
||
|
||
# ── C-direction fold caps ─────────────────────────────────────────────────
|
||
# The fold line runs at an angle through the (c, d) grid, so the column
|
||
# where two branches die can differ by ±1 between adjacent rows.
|
||
# We collect all fold terminations per row, then match them across the
|
||
# row pair regardless of exact column, connecting the dying edges with
|
||
# (possibly trapezoidal) quads.
|
||
c_terms = [_fold_terminations(row_tracks[j], c_vals) for j in range(GRID)]
|
||
|
||
for j in range(GRID - 1):
|
||
d0, d1 = d_vals[j], d_vals[j + 1]
|
||
terms_j = c_terms[j]
|
||
terms_j1 = c_terms[j + 1]
|
||
|
||
j1_used = set()
|
||
for c_j, xa, xb in terms_j:
|
||
mid = (xa + xb) / 2
|
||
best_k = None
|
||
best_dist = 1.0 # max allowed x-midpoint distance between matched pairs
|
||
for k1, (c_j1, xa1, xb1) in enumerate(terms_j1):
|
||
if k1 in j1_used:
|
||
continue
|
||
dist = abs(mid - (xa1 + xb1) / 2)
|
||
if dist < best_dist:
|
||
best_dist, best_k = dist, k1
|
||
if best_k is None:
|
||
continue
|
||
j1_used.add(best_k)
|
||
c_j1, xa1, xb1 = terms_j1[best_k]
|
||
# Cap quad: lies at the fold edge, spanning d0→d1 between the two
|
||
# dying branches. c may differ slightly between the two rows if
|
||
# the fold line is diagonal.
|
||
_emit_quad(triangles,
|
||
(c_j, d0, xa), (c_j, d0, xb),
|
||
(c_j1, d1, xb1), (c_j1, d1, xa1))
|
||
|
||
# ── D-direction fold caps ─────────────────────────────────────────────────
|
||
d_terms = [_fold_terminations(col_tracks[i], d_vals) for i in range(GRID)]
|
||
|
||
for i in range(GRID - 1):
|
||
c0, c1 = c_vals[i], c_vals[i + 1]
|
||
terms_i = d_terms[i]
|
||
terms_i1 = d_terms[i + 1]
|
||
|
||
i1_used = set()
|
||
for d_i, xa, xb in terms_i:
|
||
mid = (xa + xb) / 2
|
||
best_k = None
|
||
best_dist = 1.0
|
||
for k1, (d_i1, xa1, xb1) in enumerate(terms_i1):
|
||
if k1 in i1_used:
|
||
continue
|
||
dist = abs(mid - (xa1 + xb1) / 2)
|
||
if dist < best_dist:
|
||
best_dist, best_k = dist, k1
|
||
if best_k is None:
|
||
continue
|
||
i1_used.add(best_k)
|
||
d_i1, xa1, xb1 = terms_i1[best_k]
|
||
_emit_quad(triangles,
|
||
(c0, d_i, xa), (c1, d_i1, xa1),
|
||
(c1, d_i1, xb1), (c0, d_i, xb))
|
||
|
||
return triangles
|
||
|
||
# ── 4. Flat base ─────────────────────────────────────────────────────────────
|
||
|
||
def add_base(triangles, z_base=-2.8):
|
||
c0, c1 = C_RANGE
|
||
d0, d1 = D_RANGE
|
||
zb, zt = z_base, z_base + 0.15
|
||
|
||
triangles += [
|
||
((c0,d0,zt),(c1,d0,zt),(c1,d1,zt)),
|
||
((c0,d0,zt),(c1,d1,zt),(c0,d1,zt)),
|
||
((c0,d0,zb),(c1,d1,zb),(c1,d0,zb)),
|
||
((c0,d0,zb),(c0,d1,zb),(c1,d1,zb)),
|
||
]
|
||
for (x0,y0),(x1,y1) in [
|
||
((c0,d0),(c1,d0)), ((c1,d0),(c1,d1)),
|
||
((c1,d1),(c0,d1)), ((c0,d1),(c0,d0)),
|
||
]:
|
||
triangles += [
|
||
((x0,y0,zb),(x1,y1,zb),(x1,y1,zt)),
|
||
((x0,y0,zb),(x1,y1,zt),(x0,y0,zt)),
|
||
]
|
||
return triangles
|
||
|
||
# ── 5. ASCII STL output ───────────────────────────────────────────────────────
|
||
|
||
def normal(v0, v1, v2):
|
||
a = np.subtract(v1, v0)
|
||
b = np.subtract(v2, v0)
|
||
n = np.cross(a, b)
|
||
length = np.linalg.norm(n)
|
||
return n / length if length > 1e-14 else np.array([0.0, 0.0, 1.0])
|
||
|
||
def write_ascii_stl(triangles, filename):
|
||
with open(filename, 'w') as f:
|
||
f.write('solid butterfly_catastrophe\n')
|
||
for tri in triangles:
|
||
v0, v1, v2 = [np.array(v, dtype=float) for v in tri]
|
||
nx, ny, nz = normal(v0, v1, v2)
|
||
f.write(f' facet normal {nx:.6e} {ny:.6e} {nz:.6e}\n')
|
||
f.write(' outer loop\n')
|
||
for v in (v0, v1, v2):
|
||
f.write(f' vertex {v[0]:.6e} {v[1]:.6e} {v[2]:.6e}\n')
|
||
f.write(' endloop\n')
|
||
f.write(' endfacet\n')
|
||
f.write('endsolid butterfly_catastrophe\n')
|
||
size_kb = os.path.getsize(filename) / 1024
|
||
print(f' Triangles : {len(triangles):,}')
|
||
print(f' File size : {size_kb:.0f} KB → {filename}')
|
||
|
||
# ── 6. Main ───────────────────────────────────────────────────────────────────
|
||
|
||
if __name__ == '__main__':
|
||
print(f'Building butterfly catastrophe mesh (grid={GRID})…')
|
||
tris = build_mesh()
|
||
tris = add_base(tris)
|
||
print('Writing ASCII STL…')
|
||
write_ascii_stl(tris, 'butterfly_catastrophe.stl')
|
||
print('Done.')
|