Circle boundaries and zero-radius degeneracy

Mathematical statement

In the Euclidean plane, C(O,r)={P:d(P,O)=r} for r≥0. For r>0 this is the ordinary circle boundary; for r=0 it is the singleton {O}. Compare distance with r to classify a point as strictly inside, on the boundary, or outside. The diameter length is 2r.

Representative formula (exact TeX)

C(O,r)

Assumptions

• Coordinate exercises use the same equal-unit orthonormal frame as the distance entry, with integer center and point coordinates and a nonnegative integer radius.
• Negative radii are outside this pulse domain. At r=0 there are no strictly inside points, and the center lies on the boundary.
• Axial points O±(r,0) and O±(0,r) lie on the boundary. At r=0 all four representations give the same point.

Symbol scope

• C — circle boundary point set
• O — center point
• r — nonnegative radius

Intuition, definition, and misconceptions

With a nonnegative radius, compare the squared distance to r² exactly. Points strictly inside the disk are not members of the circle boundary.

At radius zero, only the center is on the boundary and every other point is outside. The diameter length is zero; this is an explicit degenerate case rather than an ordinary positive-radius circle.

The stated axial construction changes exactly one coordinate by ±r and leaves the other unchanged. Use the declared coordinates rather than infer incidence from a sketch.

Common errors include identifying the boundary with the filled disk, overlooking the zero-radius case, or admitting a negative radius into this domain.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Euclidean length — Radius and diameter use length units.
• Points, lines, and planes — A circle is a point set in the plane.
• Euclidean distance in an equal-unit orthonormal frame — Integer-coordinate decisions compare squared distances.

Verification boundary

Exact coordinate comparisons can be recomputed; no new formal Evidence relating sphere and ball APIs is attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind EuclideanSpace and distinguish Metric.sphere from Metric.ball, retaining separate positive-radius and zero-radius cases.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.