# Geometric Maps

Geometric maps provide a simple, but effective way to identify points that can be considered equal up to a specified precision. A geometric map basically divides 3D space into cubes sized according to the specified resolution. Cubes can only be sized in discrete steps, each step corresponding to an order of magnitude. All points within the cube collapse to the corner closest to the origin of the coordinate system.

A geometric map uses a dictionary to map string representations of XYZ coordinates to an index in a list, a key in another dictionary, or to an actual geometric location. The keys of a dictionary form a set. Therefore the keys are unique and lookups in a dictionary are very fast.

## Comparing distances

The concept is best explained with an example. Consider a set of line segments defined by their start and end point coordinates. To determine how the lines are connected, we could compute the distance between each point and all other points, identifying a match as soon as the distance is below a tolerance value.

import compas

print(len(lines))

tol = 0.001
tol2 = tol ** 2

vertices = []
edges = []

for sp, ep in lines:
u = None
v = None

for i, xyz in enumerate(vertices):
if u is None:
# no match has been found for the start point yet
if (xyz - sp) ** 2 < tol and (xyz - sp) ** 2 < tol and (xyz - sp) ** 2 < tol:
u = i

if v is None:
# no match has been found for the end point yet
if (xyz - ep) ** 2 < tol and (xyz - ep) ** 2 < tol and (xyz - ep) ** 2 < tol:
v = i

if u is not None and v is not None:
# if both points have been found in the current list of vertices iteration can stop
break

if u is None:
# the start point is not in the list of vertices that were already identified
u = len(vertices)
vertices.append(sp)

if v is None:
# the end point is not in the list of vertices
v = len(vertices)
vertices.append(ep)

edges.append((u, v))

# verify the result
print(len(lines) == len(edges))
print(len(edges)) == len(set(edges))


Note

We are not really comparing distances between points, but rather the distance per coordinate axis. This has the advantage that when the points don’t match the calculation gets interrupted faster than when calculating the full spatial distance.

## Geometric mapping

The code above works fine and is fast enough for small sets of lines. However, computation time grows exponentially when the number of lines increases. Therefore, things slow down pretty quickly…

A better way is to use a geometric map.

import compas

print(len(lines))

tol = '3f'
vertexdict = {}
edges = []

for sp, ep in lines:
a = "{0:.{1}},{0:.{1}},{0:.{1}}".format(sp, tol)
b = "{0:.{1}},{0:.{1}},{0:.{1}}".format(ep, tol)
vertexdict[a] = sp
vertexdict[b] = ep
edges.append((a, b))

key_index = {key: index for index, key in enumerate(vertexdict)}

vertices = list(vertexdict.values())
edges[:] = [(key_index[a], key_index[b]) for a, b in edges]

# verify the result
print(len(lines) == len(edges))
print(len(edges)) == len(set(edges))


## Identifying elements

Geometric maps are used internally for many things, such as computing the connectivity between geometric objects as illustrated above. This is for example the mechanism behind Network.from_lines() or Mesh.from_polygons().

Another task geometric maps are useful for is identification of elemenets of data structures based on selected (CAD) geometry.

import compas
import compas_rhino

from compas.datastructures import Mesh

guid = compas_rhino.select_mesh()
surf = RhinoMesh.from_guid(guid)
mesh = surf.to_compas()

guids = compas_rhino.select_points()
points = compas_rhino.get_point_coordinates(guids)

gkey_vertex = {geometric_key(mesh.vertex_coordinates(vertex)): vertex for vertex in mesh.vertices()}

for point in points:
gkey = geometric_key(point)

if gkey in gkey_vertex:
vertex = gkey_vertex[gkey]

# do something to this vertex
# for example mark it as fixed
# or add a load to it based on the name of the matching point