designspace deskew

a.py

#%%
import collections
import itertools

import fontTools.designspaceLib as designspaceLib
import matplotlib.pyplot as plt
import numpy as np
import scipy.spatial


def designspace_vertices(sources):
    source_locations = [
        tuple(x for x in source.location.values()) for source in sources
    ]
    n_axes = len(source_locations[0])
    vertices = sorted(scipy.spatial.ConvexHull(source_locations).vertices)

    expected_vertices = 2 ** n_axes
    if len(vertices) != expected_vertices:
        raise ValueError(
            f"Designspace has {n_axes} axes. Expected {expected_vertices} vertices, "
            f"got {vertices}, cannot rectangularize."
        )

    return [source_locations[i] for i in vertices]


def project_coordinates(coords, projection_matrix):
    coords_np = np.array([x for x in coords] + [1])
    coords_projected_homogenous_full = projection_matrix.dot(coords_np)
    coords_projected_homogenous = coords_projected_homogenous_full[:-1]
    coords_projected = (
        coords_projected_homogenous / coords_projected_homogenous_full[-1]
    )
    return coords_projected.round()


#%%
d = designspaceLib.DesignSpaceDocument.fromfile("Encode-Sans-skewed.designspace")

#%%
plt.xlabel(d.axes[0].name)
plt.ylabel(d.axes[1].name)
for source in d.sources:
    plt.plot(*source.location.values(), "sy")
for instance in d.instances:
    plt.plot(*instance.location.values(), ".k")
plt.show()

#%%
# https://math.stackexchange.com/questions/296794/finding-the-transform-matrix-from-4-projected-points-with-javascript/339033

vertices_orig = sorted(designspace_vertices(d.sources))
axis_bounds = [
    (axis.map_forward(axis.minimum), axis.map_forward(axis.maximum)) for axis in d.axes
]
vertices_new = sorted(itertools.product(*axis_bounds))

coord_orig = [(*t, 1) for t in vertices_orig]
coord_orig_a = coord_orig[:-1]
coord_orig_b = coord_orig[-1]
coord_orig_a_m = np.array(coord_orig_a).transpose()
coord_orig_b_m = np.array(coord_orig_b).transpose()
A = coord_orig_a_m * np.linalg.solve(coord_orig_a_m, coord_orig_b_m)

coord_new = [(*t, 1) for t in vertices_new]
coord_new_a = coord_new[:-1]
coord_new_b = coord_new[-1]
coord_new_a_m = np.array(coord_new_a).transpose()
coord_new_b_m = np.array(coord_new_b).transpose()
B = coord_new_a_m * np.linalg.solve(coord_new_a_m, coord_new_b_m)

H = B.dot(np.linalg.inv(A))

#%%
plt.xlabel(d.axes[0].name)
plt.ylabel(d.axes[1].name)
for source in d.sources:
    coords = project_coordinates(source.location.values(), H)
    plt.plot(*coords, "sy")
for instance in d.instances:
    coords = project_coordinates(instance.location.values(), H)
    plt.plot(*coords, ".k")
plt.show()

Encode-Sans-skewed.designspace

<?xml version='1.0' encoding='UTF-8'?>
<designspace format="4.0">
  <axes>
    <axis tag="wght" name="Weight" minimum="100" maximum="900" default="100">
      <map input="100" output="34"/>
      <map input="200" output="45"/>
      <map input="300" output="62"/>
      <map input="400" output="84"/>
      <map input="500" output="111"/>
      <map input="600" output="141"/>
      <map input="700" output="173"/>
      <map input="800" output="204"/>
      <map input="900" output="232"/>
    </axis>
    <axis tag="wdth" name="Width" minimum="75" maximum="125" default="75">
      <map input="75" output="0"/>
      <map input="87.5" output="250"/>
      <map input="100" output="500"/>
      <map input="112.5" output="750"/>
      <map input="125" output="1000"/>
    </axis>
  </axes>
  <sources>
    <source filename="EncodeSans-ThinCondensed.ufo">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </source>
    <source filename="EncodeSans-CondensedBold.ufo">
      <location>
        <dimension name="Weight" xvalue="193"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </source>
    <source filename="EncodeSans-ThinExpanded.ufo">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </source>
    <source filename="EncodeSans-BoldExpanded.ufo">
      <location>
        <dimension name="Weight" xvalue="232"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </source>
  </sources>
  <instances>
    <instance stylename="Thin">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="ExtraLight">
      <location>
        <dimension name="Weight" xvalue="44"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Light">
      <location>
        <dimension name="Weight" xvalue="58"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Regular">
      <location>
        <dimension name="Weight" xvalue="76"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Medium">
      <location>
        <dimension name="Weight" xvalue="97"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="SemiBold">
      <location>
        <dimension name="Weight" xvalue="121"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Bold">
      <location>
        <dimension name="Weight" xvalue="146"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="ExtraBold">
      <location>
        <dimension name="Weight" xvalue="171"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Black">
      <location>
        <dimension name="Weight" xvalue="193"/>
        <dimension name="Width" xvalue="0"/>
      </location>
    </instance>
    <instance stylename="Thin">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="ExtraLight">
      <location>
        <dimension name="Weight" xvalue="44"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Light">
      <location>
        <dimension name="Weight" xvalue="59"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Regular">
      <location>
        <dimension name="Weight" xvalue="78"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Medium">
      <location>
        <dimension name="Weight" xvalue="101"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="SemiBold">
      <location>
        <dimension name="Weight" xvalue="126"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Bold">
      <location>
        <dimension name="Weight" xvalue="153"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="ExtraBold">
      <location>
        <dimension name="Weight" xvalue="180"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Black">
      <location>
        <dimension name="Weight" xvalue="203"/>
        <dimension name="Width" xvalue="250"/>
      </location>
    </instance>
    <instance stylename="Thin">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="ExtraLight">
      <location>
        <dimension name="Weight" xvalue="44"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Light">
      <location>
        <dimension name="Weight" xvalue="60"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Regular">
      <location>
        <dimension name="Weight" xvalue="80"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Medium">
      <location>
        <dimension name="Weight" xvalue="104"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="SemiBold">
      <location>
        <dimension name="Weight" xvalue="131"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Bold">
      <location>
        <dimension name="Weight" xvalue="160"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="ExtraBold">
      <location>
        <dimension name="Weight" xvalue="188"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Black">
      <location>
        <dimension name="Weight" xvalue="213"/>
        <dimension name="Width" xvalue="500"/>
      </location>
    </instance>
    <instance stylename="Thin">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="ExtraLight">
      <location>
        <dimension name="Weight" xvalue="45"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Light">
      <location>
        <dimension name="Weight" xvalue="61"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Regular">
      <location>
        <dimension name="Weight" xvalue="82"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Medium">
      <location>
        <dimension name="Weight" xvalue="108"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="SemiBold">
      <location>
        <dimension name="Weight" xvalue="136"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Bold">
      <location>
        <dimension name="Weight" xvalue="167"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="ExtraBold">
      <location>
        <dimension name="Weight" xvalue="196"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Black">
      <location>
        <dimension name="Weight" xvalue="222"/>
        <dimension name="Width" xvalue="750"/>
      </location>
    </instance>
    <instance stylename="Thin">
      <location>
        <dimension name="Weight" xvalue="34"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="ExtraLight">
      <location>
        <dimension name="Weight" xvalue="45"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="Light">
      <location>
        <dimension name="Weight" xvalue="62"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="Regular">
      <location>
        <dimension name="Weight" xvalue="84"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="Medium">
      <location>
        <dimension name="Weight" xvalue="111"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="SemiBold">
      <location>
        <dimension name="Weight" xvalue="141"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="Bold">
      <location>
        <dimension name="Weight" xvalue="173"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="ExtraBold">
      <location>
        <dimension name="Weight" xvalue="204"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
    <instance stylename="Black">
      <location>
        <dimension name="Weight" xvalue="232"/>
        <dimension name="Width" xvalue="1000"/>
      </location>
    </instance>
  </instances>
</designspace>