VectorPaint Reuleaux triangles: distinguish constant width from a circle through three equal-radius arcs

English translation of the Chinese original, translated on 2026-11-10. Original publication dates, attribution, code and version notes are retained.

A constant-width shape need not be circular. Construct a Reuleaux triangle by distinguishing three centres, three minor arcs and width, rather than merely rounding an ordinary triangle. VectorPaint supplies manual paths and references; calculations remain external. An approximate image is not a precision component or evidence about drilling, axles or fabrication.

Establish an equilateral reference of side 300

Set A(400, 500), B(700, 500), C(550, 500−150√3), approximately y 240.192. All three theoretical sides are 300; the displayed rounded coordinate does not provide infinite precision. Preserve pale sides and a coordinate list before arcs. Keep titles outside and text centres distinct from vertices. If controls permit only reference-based placement, label the drawing illustrative and retain theoretical coordinates separately; zoom does not repair unequal sides.

Place all three centres at original vertices

Construct radius 300 circles centred on A, B, C, each passing through the other two vertices. Verify radius rather than mistaking 300 for diameter, retaining pale full-circle checks. The University of Florida Reuleaux note supplies this equilateral/equal-circle construction; mechanical applications are not adopted. The central intersection, not the union’s outer envelope, is the target. Without a confirmed intersection control, construct the three boundaries manually rather than assuming a hidden command.

Take only the sixty-degree minor arc for each edge

Take the 60-degree B–C minor arc from A’s circle, C–A from B’s and A–B from C’s, forming three outward-bulging arcs. Exclude 300-degree major arcs and original straight sides. Joins are original vertices; inspect closure and overlap closely. Bézier approximations remain approximate, not exact circular arcs merely because handles look smooth. Inspect the unfilled three-part outline before colouring, without hiding gaps under heavier strokes.

Measure width between parallel supports

Theoretical width is the distance between parallel supporting lines in any direction, not an equal centre-to-boundary radius. Horizontal extent is 300; vertical extent from C to the lowest opposite arc is also 300. The original triangle height, about 259.808, is not the new vertical width. Place pale support pairs tangent to the outside, not through the interior. Two inspected directions do not prove every direction. The theoretical construction and approximate artwork have different evidential limits; visual checks are not comprehensive measurement certification.

Do not infer rotation about a fixed centre

For an optional reoriented copy retain shape scale and reconstruct parallel support references. Constant width does not guarantee that rotation about an arbitrary fixed centre remains fitted within a fixed square; position may also change. No motion constraint or mechanism simulation is performed. Without exact references compare outline/labels only, not invented dynamic results. Keep text readable; reversed labels after a flip are not a mathematical requirement. Three arcs have distinct centres, unlike one circle, which pale centre markers can explain.

Test the conditions with an unequal-radius copy

In a separate wrong copy change only A’s circle radius to 280, retaining 300 for the others. B/C no longer lie on that circle, so the same joins and constant-width 300 caption cannot remain. Inspect full-circle differences and identify a construction-condition error, not colour. Preserve the correct version and do not move B/C to accommodate the small circle while claiming an unchanged equilateral reference. No full width calculation for the wrong shape is needed; the original conclusion no longer follows from these conditions.

Separate construction and presentation in output

Save the editable source and inspect an actual PNG for joins, outward arcs and both width labels. Actively hide unwanted circles, straight references and supports; ordinary references still export. Check closure on light/dark backgrounds instead of concealing cracks with white patches. Isolate arcs in a copy for another format’s differences rather than overwriting the source. A simple final still needs records identifying each arc’s parent circle, not an unexplained rounded icon.

Troubleshoot vertices, radii and minor arcs in order

For a petal-like outline inspect major-arc selection; for displaced ends inspect vertices/radii; for width mismatch inspect nonuniform scaling and support positions. Do not change numerical labels to fit the mistaken drawing. Retain equilateral reference, three circles, minor arcs, two-direction checks and counterexample, distinguishing theory from approximation. This is one traceable construction, not all-angle numerical proof, automatic precision geometry or fabrication assurance.