Construction Space: It's Not Tracing

"The obvious way to grade a drawing rewards the one thing you must never reward: tracing. The whole grader exists to escape that trap."

The Trap: Grading by Overlap

The naive grader is seductive and wrong. Compare the student's lines to the photo's edges, measure how much they overlap, call that the score. But think about who wins: a perfect tracer. Someone who laid tracing paper over the reference and copied every edge scores a flawless 100 and learned absolutely nothing about construction. An overlap grader doesn't measure whether you understood the solid — it measures whether you copied its silhouette accurately. It grades the exact skill the whole engine is built to make obsolete.

The Move: Align Away the Pose

Here is the escape. Before comparing your construction to the reference, the grader runs Procrustes analysis: it finds the single best 2-D similarity transform — a translation, a uniform scale, and a rotation — that maps your construction onto the reference, applies it, and only then measures the difference. Translation removes where you drew it. Uniform scale removes how big. Rotation removes how tilted. All three are exactly the things a tracer gets right for free by copying placement — so aligning them away deletes the tracer's entire advantage before a single point is scored.

What Survives the Alignment Is Construction

After Procrustes, two constructions that are the same shape — drawn small in the corner or large in the center, upright or tilted — score identically, because position, size, and tilt were removed. What can still differ is the shape itself: the angle of the jaw relative to the brow, the proportion of the cranium to the face, where the box's far edge sits relative to the near one. That residual difference is your construction accuracy. It cannot be earned by placement, because placement no longer exists in the comparison. This is the literal, mathematical meaning of the promise from Track 2: it grades your construction, not your tracing.

The Bed of Procrustes

The name is a grim Greek myth: Procrustes made guests fit his iron bed by stretching the short ones and cutting the tall ones. The analysis borrows the image — it stretches, shifts, and rotates your drawing to best-fit the reference 'bed.' But note the crucial restraint: it uses a similarity transform only, never a shear or a free warp. It can slide, resize, and turn your construction, but it cannot bend a wrong shape into a right one. So it forgives everything that doesn't matter (placement) and forgives nothing about the shape that does. One honest footnote, answered two lessons on: similarity removes position, size, and tilt — but a deliberate stretch survives it, and that leftover is exactly what the engine will call exaggeration.

Code

Procrustes: strip placement, grade only the shape·python

import numpy as np

def construction_score(yours, reference):
    # Procrustes: best translation + uniform scale + rotation mapping yours -> ref.
    # This ALIGNS AWAY position, size, and tilt -- everything a tracer got free.
    aligned = procrustes_similarity(yours, reference)   # translate, scale, rotate only

    # What REMAINS is pure shape difference: angles and proportions, not placement.
    residual = np.linalg.norm(aligned - reference) / scale_of(reference)
    return 1.0 - residual

# A perfect tracer nails placement but may still MIS-SEE the solid.
# Procrustes deletes the placement they copied and scores only the seeing.
# (Similarity removes position/size/tilt but NOT a deliberate stretch --
#  that leftover is 'exaggeration', estimated separately two lessons on.)

Exercise

Find a comparison in your own field that's contaminated by a nuisance variable — comparing two time series with different baselines, two signals out of phase, two datasets on different scales, two shapes at different positions. Describe how you'd 'align away' the nuisance (subtract the mean, normalize the scale, register, detrend) before comparing, and what the residual would then honestly measure.

Hint

The pattern is: identify the transform group you don't care about (shifts, scalings, rotations, phase), find the best transform in that group to align the two, apply it, and measure only what's left. That residual is the real difference. Loomis's version is Procrustes over 2-D similarity; yours might be de-meaning, unit-normalizing, or time-warping — same idea, different nuisance.