Concave hull approximation from point coordinates

The deterministic result includes the ordered boundary, edge lengths, perimeter, area, and a count of edges that could not be shortened, making the approximation easy to inspect rather than presenting it as an exact geometric answer.

Choose coordinates and a meaningful threshold

Provide at least three planar points as numeric x and y coordinates, together with a positive edge-length threshold. Coordinates may represent projected map positions, drawing units, measurements, or any other two-dimensional Cartesian system. They should share one scale: do not mix longitude degrees with distances in meters and expect the threshold to retain a physical meaning. A smaller threshold asks the algorithm to follow the available points more closely, while a larger threshold preserves more of the convex outline. The threshold is a target, not a guarantee, because a sparse point set may contain no safe interior point capable of splitting a particular boundary edge. Duplicate coordinates are ignored. At least three unique, non-collinear points must remain, since a polygon cannot be formed from a line. For geographic latitude and longitude data, project the coordinates appropriately before using a threshold expressed in a distance unit. The tool performs planar geometry and does not apply a spherical or ellipsoidal Earth model. Start with a threshold near the spacing you consider a meaningful gap, then compare the returned boundary with the original points before using it downstream.

Understand how boundary refinement works

The algorithm first computes a deterministic convex hull with the monotone-chain method. It then examines boundary edges longer than the supplied threshold, longest first. For each such edge, it searches unused points that lie inside the current polygon and can replace the edge with two shorter segments without crossing another boundary segment. The candidate that minimizes the longer replacement segment is selected, with stable tie breaking based on total replacement length and original input order. After an insertion, edge selection begins again because the polygon has changed. Refinement stops when every edge is within the target or when no remaining point can safely improve any long edge. This construction deliberately favors predictable, explainable output over claiming to solve the many competing definitions of an exact concave hull. It never invents vertices, moves coordinates, uses random sampling, or calls a remote service. The returned hull is an open ordered vertex list whose final vertex connects back to the first. Edge lengths use that same cyclic order. Collinear outer points may be omitted unless one later becomes a useful refinement vertex.

Inspect limitations before relying on the polygon

Review unresolved_long_edges whenever the threshold matters operationally. A value of zero means every returned boundary edge meets the requested target; a positive value means the available point geometry could not satisfy it without a crossing, an outward move, or a replacement that failed to shorten the edge. That is useful evidence, not a hidden failure. The approximation can also produce a boundary that differs from results generated by alpha shapes, k-nearest-neighbor hulls, triangulation filters, or domain-specific shoreline methods. Compare methods when topology carries legal, scientific, or safety consequences. The reported area and perimeter belong to the returned planar polygon and use the coordinate system supplied by the caller. They are not geodesic measurements. Plot the ordered hull over the source points to check whether the chosen threshold captures the intended clusters and indentations. If the boundary remains too broad, add representative points or reduce the threshold; if it becomes overly detailed, raise the threshold. The capability accepts at most 500 input items so its repeated intersection checks remain bounded and suitable for interactive browser execution as well as deterministic API automation.