A Symmetric GRG Kernel for Area
A Symmetric GRG Kernel for Area
The binary radial formulation combines complementary internal and external sector triangles. The area kernel uses one internal contribution and two external contributions, unlike the circumference kernel’s two-to-one chord–tangent balance.
Geometric Elements
Resolve one Turn into N equal symmetric sectors. Let a be the internal triangle’s height, h half its chord, and e half the external tangent side. Define q = h/a.
N=2^b,\quad b\ge2,\quad R>0,\qquad a^2+h^2=R^2,\quad e=R\frac ha.
$$
I_b=ah=R^2\frac{q_b}{1+q_b^2},\qquad E_b=Re=R^2q_b.
$$
Two Area Kernels
A_m=\frac N2(I_b+E_b),\qquad A_w=\frac N3(I_b+2E_b).
$$
Accuracy for the Defined Triangles
The current formulation gives an analytical verification using a local integral as an exact sector reference, not as a kernel input.
F(q)=\int_0^q\frac{dt}{1+t^2},\qquad A_T(R)=NR^2F(q_b).
$$
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$
Regular geometric bisection yields a scale bound. Together with the expansion, it proves second-order total error for the arithmetic kernel and fourth-order total error for the weighted kernel in exact arithmetic.
0\lt q_b\le\frac4N,\qquad A_m-A_T=O(N^{-2}),\qquad A_w-A_T=O(N^{-4}).
$$
The same analysis proves the following bracket for these sectors:
A_{\mathrm{in}}\le A_m\le A_T\le A_w\le A_{\mathrm{out}}.
$$
Sampling and Scope
\Delta u=\frac1{4N}.
$$
The earlier version introduced this quarter-step sampling parameter inside each module. The proof here concerns the complete sector triangles defined above. A different sample-based area construction requires its own explicit definition and proof.
Research Status
The cancellation is established for the regular triangular elements above. This does not establish original authorship of the weighting, numerical stability of an implementation, or a computational advantage. Those questions require separate literature and implementation comparisons.
Earlier revisions of this page described fourth-order behaviour as a candidate. The current analytical clarification identifies the geometric assumptions under which that behaviour follows.
