Chests & hearts

The green region is the convex hull of X(1), X(10), the centroid, and samples of the power centers M_p, where 4 − 2√2 ≤ p ≤ 4 + 2√2. M_p minimizes the sum of the p-th powers of the distances to the three vertices. Papers III and IV prove attractivity for this family. The app retains whichever sampled centers are hull vertices for the current triangle. Hover over a center to identify it.

This is a numerical approximation of a proved inner family, not the exact maximal Heart. The power interval is the sharp universal guarantee for the power family, and the X(1)–X(10) segment is sharp on its centroidal ray. These facts do not assert that the displayed polygon is the global Heart boundary. The earlier 50-point ETC base came from a scan with subsequently identified evaluation problems, so it does not enter this inner family.

Noninteger power centers need not be globally analytic at every input configuration. The displayed family concerns attractive centers with the regularity and extension conventions of Paper III. Computation uses a normalized convex solver; unresolved samples are omitted and counted.

This browser app has no server, account, tracking, or external JavaScript dependency. Geometry stays on your device. The JavaScript source runs on GitHub Pages. A dependency-free Julia geometry module implements the same constructions for scientific use. Methods and usage · Regression tests · MIT license.

Percentages compare each checked method on its own with the hull, then show the intersection of checked regions. The four-point network uses parallelogram chests at its intermediate nodes. Stability assumes absorption across numbers of inputs. It is optional and has a separate mathematical hypothesis.

Read the four Attractive Centers papers. The papers remain PDF downloads.