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.

Complexity

Complexity

Sep 12, 2025

Computational Complexity

Complexity must identify the task, algorithm, arithmetic precision, and output representation. This page distinguishes costs rather than assigning a single complexity to all GRG methods.

Scalar Kernel Evaluation

Given local geometric ratios, the current circle kernels require a fixed number of arithmetic operations. Obtaining the ratios has its own cost. A binary-bisection evaluation performs a number of refinement stages proportional to the binary depth; square-root cost depends on precision and implementation.

Explicit Geometry

Generating and storing all sector nodes has a cost proportional to the output count. Streaming may reduce storage, but does not remove output work.

Raster and Volume Tasks

Boundary drawing, disk filling, surface generation, and voxel filling have different output sizes. Their cost statements must specify whether every output element is generated.

Comparisons

Report setup, precomputation, precision, memory, and runtime at equal achieved accuracy under the benchmark protocol. Earlier repeated cost models have been consolidated here.

Methods

Methods

Sep 12, 2025

Methods and Their Scope

Current Binary Radial Kernels

The current formulation defines regular equal sectors, local chord–tangent lengths, triangular areas, and weighted kernels. Its proofs concern those definitions.

Earlier Methods

Algorithm Specifications

A runnable method must identify initial values, refinement or direct evaluation, arithmetic format, outputs, and stopping conditions. A geometry identity alone is not a runtime benchmark.

Extensions

Adaptive sampling, general curves, multiple seeds, and three-dimensional constructions require their own definitions and validation. No error order or performance advantage is transferred automatically from the regular-circle proof.