Arc lengths are harder to compare than straight lengths

What it is

Arc-length comparison under radius-angle confounding starts from the fact that arc length depends jointly on radius and central angle: in radians, s = rθ. The title summarizes a common design risk in radial charts, not a psychophysical law that curvature itself must impair length judgment; some experiments find similar length functions for isolated straight and curved lines. Difficulty chiefly arises when radius, angle, starting point, orientation, or other geometric cues vary together.

Why it happens

Straight segments are easily arranged on a shared axis and origin, whereas arcs in radial charts are often assigned different radii, starts, or orientations. On one circle, arc length is proportional to angle. Across radii, equal angles have different arc lengths, and equal arc lengths can result from different combinations of r and θ. Ring thickness, chord length, area, and outward position add competing cues. The mechanism is cue conflict and poor alignment, not simply that “bending a length makes it harder.”

Studying it

Encode the same values as common-baseline straight segments, isolated equal-length curves, equal-radius arcs, varying-radius arcs, and conditions where angle and radius both vary. Test arc-length ordering, ratio estimation, angle estimation, and radius judgment separately to distinguish a pure curvature effect from layout confounding and identify which cue participants use. Manipulate size, stroke thickness, curvature, spacing, numeric labels, interaction, and practice systematically. Report task-specific error and strategies rather than generalizing from one radial chart to all curves.

Where it stops holding

When arcs share a radius, are large enough, and support a coarse part-to-whole task, angle and the full circle provide useful references. Progress rings and gauges can show one state using familiar endpoints but are poor for precise comparison across many values. When geographic routes, trajectories, or curves are themselves the subject, straightening them for easier magnitude comparison may destroy the phenomenon being studied.

Applying it

• Prefer straight bars or dots for precise multi-value comparison. If using arcs, state whether data map to angle, radius, arc length, or area; do not let several geometric properties vary accidentally.
• Keep radius, thickness, and starting direction consistent for arcs meant to be compared. If meaning requires varying radius, add direct labels and a second comparable view.
• Test labels and selection targets at minimum size, high zoom, and on touch. Interaction should reveal value, unit, and the geometric property used.
• Give screen readers an ordered value list, important differences, and part-to-whole relation rather than requiring users to derive values from geometric descriptions such as “long inner arc.”

Related

• Same group: U1.03.1 Length judgment requires a common starting point · U1.03.2 Angle judgments are markedly less accurate than length under equal conditions · U1.03.3 Angles near horizontal or vertical are judged more accurately than oblique ones · U1.03.4 Length can only encode non-negative quantities; negatives need position
• Adjacent: U3.08.1 Polar coordinates map one variable to angle, the other to radius · U3.08.5 Magnitude comparison under polar coordinates is harder than in Cartesian
• Search terms: arc length · radius-angle confound · radial encoding