Results

Results

Sep 13, 2025

Results and Verification

This page separates analytical results from numerical illustrations and performance measurements.

Analytical Results

The current binary radial formulation defines regular chord–tangent sectors and gives exact-arithmetic proofs of second-order error for arithmetic kernels and fourth-order error for the weighted circumference and area kernels. It also proves an area bracket for the defined regular sectors.

These statements have specific geometric hypotheses. They do not establish numerical exactness at finite resolution, hardware stability, originality, or superiority over other methods.

Numerical Checks

Numerical evaluations may check consistency with the analytical results. A publishable record must include the generating procedure, radius, sector count, precision, reference definition, raw output, and executable code.

Performance Results

The former gallery mixed example cards with timing figures and incomplete dataset records. Those figures are not retained as validated performance results. No measured efficiency advantage is established by this page.

Performance results will be added when complete reproducible evidence is available under the comparison protocol.

Parameters

Parameters

Sep 12, 2025

Parameters and Notation

This page defines the current circle-kernel conventions. Historical methods may use other conventions; their symbols must be interpreted locally.

Geometric Scale

R is the positive radius. N is the number of complete sectors across one Turn. Binary depth b gives N = 2ᵇ; the nondegenerate chord–tangent kernels require b ≥ 2.

Angular Steps

The complete sector width is 1/N Turn, its half-width is 1/(2N), and the quarter-step parameter is 1/(4N). If M counts samples in a quarter of the full Turn, state explicitly how M relates to N; do not use the same symbol for both counts.

Local Geometry

a is the chord apothem, h the half-chord, and e the half-tangent side. The geometric ratios are c = a/R, s = h/R, and q = h/a.

Implementation Parameters

Specify arithmetic format, rounding, construction procedure, stopping rule, and any stored tables. Additional parameters for adaptive or noncircular methods require their own definitions.

Full definitions and formulas.