Modifying Tilings: Surgery and Coordinate Transforms¶

Conway operators are global; sometimes you want a scalpel. This notebook collects some of the ad-hoc modification primitives that pleat.half exposes:

• finding faces / edges by predicate,
• deleting faces, edges, and (degree-2) vertices,
• subdividing edges and faces,
• transforming all positions in one go (stretch, warp).

We highlight every change by writing a color_key and showing the graph before and after.

In [1]:

import numpy as np
from pleat import (
    example_graphs,
    example_tilesets,
)
from pleat.rendering import multi_show

import numpy as np from pleat import ( example_graphs, example_tilesets, ) from pleat.rendering import multi_show

Finding by predicate¶

All collections (faces, vertices, halfedges) are plain Python sets. Filter with a list comprehension and mark hits with a colour key.

In [2]:

G = example_graphs.from_tiles(example_tilesets.t_4_6_12(), rings=2)
G.recompute_lengths_and_angles()
hexagons = [f for f in G.faces if f.order() == 6]
for f in hexagons:
    f['color_key'] = (1.0, 0.6, 0.0, 0.8)
print(f'{len(hexagons)} hexagons found')
G.show(render_faces=True, face_inset=0.05, render_vertices=False)

G = example_graphs.from_tiles(example_tilesets.t_4_6_12(), rings=2) G.recompute_lengths_and_angles() hexagons = [f for f in G.faces if f.order() == 6] for f in hexagons: f['color_key'] = (1.0, 0.6, 0.0, 0.8) print(f'{len(hexagons)} hexagons found') G.show(render_faces=True, face_inset=0.05, render_vertices=False)

6 hexagons found

Deleting faces¶

G.delete_faces(set_of_faces) removes faces and the half-edges that bordered only them. The neighbouring faces become adjacent to a new piece of border.

In [3]:

G_before = G.copy()
to_delete = [f for f in G.faces if f.order() == 6]
G.delete_faces(set(to_delete))
multi_show([G_before, G],
           titles=['before delete_faces', f'after (-{len(to_delete)} hex)'],
           render_faces=True, face_inset=0.05, render_vertices=False)

G_before = G.copy() to_delete = [f for f in G.faces if f.order() == 6] G.delete_faces(set(to_delete)) multi_show([G_before, G], titles=['before delete_faces', f'after (-{len(to_delete)} hex)'], render_faces=True, face_inset=0.05, render_vertices=False)

Deleting an edge merges two faces¶

G.delete_edge(h) removes the halfedge and its reverse, merging the two adjacent faces into one. We mark the merged face with a single colour to make the result visible.

In [4]:

G = example_graphs.from_tiles(example_tilesets.platonic(6), rings=3)
G.recompute_lengths_and_angles()
central = G.central_face()
h = next(h for h in central.halfedge_iter()
         if not h.on_border() and not h.rev.on_border())
G_before = G.copy()
h['color_key'] = h.rev['color_key'] = (0.85, 0.1, 0.1, 1.0)
h['line_width'] = h.rev['line_width'] = 5
G.delete_edge(h)
merged = next(f for f in G.faces if f.order() > 6)
merged['color_key'] = (0.6, 0.85, 0.4, 0.9)
multi_show([G_before, G],
           titles=['edge to delete (red)', 'merged face (green)'],
           render_faces=True, face_inset=0.05, render_vertices=False)

G = example_graphs.from_tiles(example_tilesets.platonic(6), rings=3) G.recompute_lengths_and_angles() central = G.central_face() h = next(h for h in central.halfedge_iter() if not h.on_border() and not h.rev.on_border()) G_before = G.copy() h['color_key'] = h.rev['color_key'] = (0.85, 0.1, 0.1, 1.0) h['line_width'] = h.rev['line_width'] = 5 G.delete_edge(h) merged = next(f for f in G.faces if f.order() > 6) merged['color_key'] = (0.6, 0.85, 0.4, 0.9) multi_show([G_before, G], titles=['edge to delete (red)', 'merged face (green)'], render_faces=True, face_inset=0.05, render_vertices=False)

Subdividing edges and faces¶

• G.subdivide_edge(h) inserts a new vertex on h (at the midpoint by default) and returns (h_new, v_new).
• G.subdivide_face(f, v1, v2) adds a new edge between two of f's vertices, splitting it. Returns (h_new, f_new).

Below we subdivide a hexagon's diagonal.

In [5]:

G = example_graphs.from_tiles(example_tilesets.platonic(6), rings=2)
G.recompute_lengths_and_angles()
f = G.central_face()
vs = list(f.vertex_iter())
G_before = G.copy()
h_new, f_new = G.subdivide_face(f, vs[0], vs[3])
f_new['color_key'] = (0.4, 0.7, 1.0, 0.9)
f['color_key'] = (1.0, 0.5, 0.1, 0.9)
multi_show([G_before, G],
           titles=['before subdivide_face', 'after — split into halves'],
           render_faces=True, face_inset=0.05, render_vertices=False)

G = example_graphs.from_tiles(example_tilesets.platonic(6), rings=2) G.recompute_lengths_and_angles() f = G.central_face() vs = list(f.vertex_iter()) G_before = G.copy() h_new, f_new = G.subdivide_face(f, vs[0], vs[3]) f_new['color_key'] = (0.4, 0.7, 1.0, 0.9) f['color_key'] = (1.0, 0.5, 0.1, 0.9) multi_show([G_before, G], titles=['before subdivide_face', 'after — split into halves'], render_faces=True, face_inset=0.05, render_vertices=False)

Coordinate transformations¶

Vertex positions live at v['pos']. Mutating them in place and then calling G.recompute_lengths_and_angles() is the standard way to bend / stretch / warp a graph.

Below we apply a horizontal shear and a sinusoidal vertical warp.

In [6]:

G = example_graphs.from_tiles(example_tilesets.platonic(4), rings=3)
G.recompute_lengths_and_angles()
G_orig = G.copy()

for v in G.vertices:
    x, y = v['pos']
    v['pos'] = np.array([x + 0.3 * y, y + 0.4 * np.sin(x)])
G.recompute_lengths_and_angles()

multi_show([G_orig, G],
           titles=['original', 'shear + sin warp'],
           face_inset=0.05, render_vertices=False)

G = example_graphs.from_tiles(example_tilesets.platonic(4), rings=3) G.recompute_lengths_and_angles() G_orig = G.copy() for v in G.vertices: x, y = v['pos'] v['pos'] = np.array([x + 0.3 * y, y + 0.4 * np.sin(x)]) G.recompute_lengths_and_angles() multi_show([G_orig, G], titles=['original', 'shear + sin warp'], face_inset=0.05, render_vertices=False)