A drawing experiment, examined through geometry

How round is round?

Draw-a-perfect-circle games turn a familiar shape into a numerical challenge. Yet a high score can accompany a visibly imperfect loop. That mismatch raises a geometric question: what should closeness to a circle actually mean?

This lab compares average error, worst-case deviation, compactness, local curvature, and harmonic structure. A stronger criterion should resist size tricks, expose local defects, and distinguish a complete loop from a well-fitted arc.

Inspect the counterexamples

Specimen

Near circle

Ready to inspect

Complete the last gap

Joins the endpoints without reshaping the stroke. A straight seam can leave a small corner.

What does gap completion change?

Straight: the shortest join adds no inferred shape. Its corners can affect curvature, especially across a wider gap.

Tangent-matched: a short cubic follows estimated endpoint directions. This softens the join, but does not match curvature or guarantee a better score. Unstable or crossing curves fall back to straight.

Only a nearly complete loop inside the release zone is joined. A short overshoot past the start can be trimmed; longer tails remain part of the stroke. The added seam is teal, discarded tail amber. Scores describe the completed contour; exports retain the captured stroke and all completion details.

02 Measurements → result

A score with an explicit criterion

Analysis scale 1.3% circumference

Declared aggregation

One result, five measurements

Calculating…

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Scoring criterion

Active score · S

How a measurement becomes a quality

Anchors and weights are declared design choices. The measurements overlap; this score is not fitted to human ratings.

The criterion determines how normalized measurements contribute to the result. A fixed 1.3% smoothing scale keeps drawings comparable under the same criterion.

Measurement evidence — Checks sampling, drawing size, closure, and smoothing sensitivity.

The scale slider still changes the diagnostic plots below; the overall score always uses 240 equal-distance samples at the declared 1.3% scale.

03 Intermediate geometry

From gesture to measurement

Equal-arclength sampling separates boundary geometry from drawing speed.

Raw input— points

Arc-length sample240 points

Analysis contourscale 1.3%

04 Signals around the loop

The structure behind the error

Radial error is indexed by angle, curvature by boundary position, and harmonics by spatial frequency.

Radial signature(r − R) / R

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Angle 0° points right from the fitted center.

Curvature signalnormalized κR; ideal = 1

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Boundary 0% is the first equal-distance sample along the stroke.

Harmonic fingerprintradial error by lobe count

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Mode 1 mostly reflects centering; 2, 3, and 4 read as oval, triangular, and squarish.

05 The caliper paradox Theorem-backed

Same width. Different roundness.

A noncircular constant-width family isolates what a caliper measurement cannot distinguish.

Shape deformation a = 0.080 circle stops at 0.120; singular boundary is a = ⅛ Roll between fixed calipers 0° scrub the angle; the contact midpoint moves but the gap does not

Exact construction

h(θ) = R[1 + a cos(3θ)]

The three-lobed term changes the boundary, but cancels when opposite support distances are added.

Exact gap variation: 0
Existing circle score: —
Local bend ratio: —
Rim length / area: —

Directional width tracew(θ) / mean(w) − 1

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The theorem is exact; this trace exposes the tiny residual introduced by representing a smooth curve with finitely many points.

Why the paradox workssupport-function spectrum; exact for this constructed family

odd n = 3 cancels

Opposite directions multiply mode n by (−1)ⁿ. Adding them doubles even modes and erases odd modes; n = 1 is translation, while n = 3 changes this shape without changing its width. The two DC bars are normalized independently: support has coefficient R and width has 2R.

The support-function theorem: established Sampled drawing diagnosis: experimental

What directional width leaves unresolved

Directional width sees only the convex hull, so outlines that share a hull have identical width traces and inward dent depth beyond that hull is not measured. A single outward speck can also control a support line. The slider stops below a = ⅛, where the minimum radius of curvature reaches zero and numerical curvature becomes singular.

Research basis: Hynd, support functions and constant width · Resnikoff, Fourier structure

06 The tangent walk Theorem-backed

Same total turning. Different local geometry.

A regular simple closed loop has one full tangent rotation. A circle distributes that turning uniformly along arclength.

Walk around the current analysis contour 0% The arrow is the local tangent; the filled path tracks measured arclength, not sample count. Osculating-circle microscope Experimental · blue local circle · yellow center trail · near-flat centers may leave the stage.

Cumulative tangent turnmeasured versus a circle's uniform 360° ramp

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Above the dashed ramp: more turning has accumulated than on a circle at the same relative arclength. The shared endpoint alone does not distinguish roundness.

Simple-loop theorem: established Sampled local diagnosis: exploratory

The total is exact; the route is diagnostic

Open strokes and self-crossings do not receive the simple-loop guarantee. Discrete tangent turns depend on resampling and smoothing, and a cusp or near-180° sample jump can make angle unwrapping unstable. Reversing stroke direction flips the raw sign, so the plot normalizes orientation while reporting the original direction.

Research basis: Hopf's tangent-turning theorem (1935) · Whitney on rotation number (1937) · Ghys, Tabachnikov & Timorin on osculating circles

Curvature as an evolution law Numerical telescope

The approach to a round point

Each point follows its curvature vector. Convex bulges retreat, concave dents can fill outward, and tighter bends move faster.

Curve-shortening time 72% both views recentered · physical scale versus equal-area magnification

The universal area clockmeasured polygon area versus the continuum law

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The dashed diagonal is exact for the smooth continuum flow. The orange path is this finite polygon solver; the distance between them is shown, not hidden.

Round-point theorem: established Polygon animation: experimental

Continuum prediction and numerical resolution

The theorem belongs to smooth embedded curves. This preview first withholds contours that its 96-point grid cannot resolve faithfully. Every step must decrease area and stay within five percentage points of the universal clock; crossings are checked every tenth step and before every displayed frame. If a guard fails, the telescope stops early. The equal-area enlargement is display-only. Nothing here changes the roundness score.

Research basis: Gage & Hamilton (1986) · Grayson (1987)

07 Scale sensitivity Experimental

What survives when the smoothing scale changes?

Feature persistence across scale is a separate diagnostic; it does not enter the composite score.

Multiscale curvature mapκR − 1 across boundary position and scale

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lower signed κR near κR = 1 higher signed κR κ = 0 crossing

Fine scales sit at the bottom; each pixel preserves the strongest signed deviation in its bin. Boundary 0% is tied to the stroke start.

Bending excessB(σ) = L∫κ²ds / 4π² − 1

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Zero is the continuous-theory circle limit. Negative values are shown as numerical undershoot, not “better than a circle.”

Theory: established Roundness interpretation: experimental

Scale-space theory and this implementation

Curvature scale-space tracks how curvature features change under path-based Gaussian smoothing. This lab adds a nonstandard deviation heatmap and bending-energy view so convex shapes are not blank. Neither is an accepted metrology parameter, probability, or uncertainty interval.

Numerical limits and interpretation

Fine scales amplify pointer sampling and numerical differentiation.
Feature locations depend on the stroke start, and smoothing can merge or shift narrow dents.
Classical zero-crossing scale-space can be blank for both a circle and a noncircular convex shape.
Discrete curvature and bending estimates may rebound across scale because of aliasing.

Research basis: Mokhtarian & Mackworth (1986) · multiscale theory (1989)

Interpreting the result

A score depends on what it rewards.

Small average error, a tight tolerance band, efficient use of area, and consistent local bending are different requirements. I kept the measurements separate so a high aggregate score can be traced back to its assumptions. Counterexamples and controlled defects test those assumptions.

Least-squares circle

Typical radial error relative to a fitted reference.

average

Minimum-zone annulus

Metrology-style worst case; sensitive to the single deepest dent.

worst case

Isoperimetric compactness

Center-free and scale-free, but rough edges inflate perimeter.

global shape

Curvature consistency

Local bending relative to a circle; sensitive to sampling and smoothing.

local bend

Radial harmonics

Explains the wobble: mode 2 is oval, 3 is triangular, 4 is squarish.

diagnosis

Implementation and evidence

Explicit geometry makes the assumptions testable.

The geometry pipeline is deterministic: resampling, smoothing, fitting, and signal analysis have inspectable intermediate results. Known circles, transformed copies, local defects, and constant-width counterexamples distinguish numerical behavior from choices in the scoring policy.

Agreement with human preference remains a separate validation problem. The current calibration expresses a set of geometric priorities; it was not fitted to human ratings. That boundary matters when a plausible number conflicts with the visible shape—the observation that started this experiment.

Question → criterion → measurements → result → validation.

Measurement background: NIST · roundness measurement · Chernov & Lesort · circle fitting · OECD/JRC · composite indicators