lin-0212

17.3 Infinitesimal relational descent

Writing \(K=\left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right]\), the tangent incidence equation is

\[ \delta g_{iv} = R_iKz_i(v)\, \delta \alpha _i + R_i\delta z_i(v) + \delta t_i - \delta y_v. \]

Stacking these rows gives a sparse Jacobian \(Dg\). The computational repair solves

\[ Dg\, \delta =-g \]

in a damped metric. This selects a tangent direction; it does not define the semantic meaning of compatibility.

Private boundaries are activated before they are crossed. For \(s_{ie}=\lVert z_i(p)-z_i(q)\rVert ^2\),

\[ \dot s_{ie} = 2(z_i(p)-z_i(q))^\top (\delta z_i(p)-\delta z_i(q)). \]

If the proposed direction crosses an edge band or anchor-area floor, its normal covector enters the active solve. Backtracking then evaluates the exact nonlinear declarations.

RADAR relational descent

  1. Construct typed face and join-witness foundries

  2. Compute centered local sections and deterministic orientations

  3. Build private fuzzy bands and signed-area anchors

  4. Initialize the global section from the apex; align its faces

  5. While an improving admissible step exists:

  6. Linearize incidence and gauge equalities

  7. Solve the damped sparse tangent system

  8. Activate threatened edge and area faces; re-solve

  9. Clip units separately and backtrack against the exact audit

  10. Retract centering and the global \(SE(2)\) gauge exactly

  11. Return the realization, rejected steps, and terminal obstruction