Bundle Adjustment (BA) is a cornerstone of 3D computer vision and has benefited from decades of advances in sparse optimization and numerical methods. It was originally developed for jointly optimizing camera intrinsics, poses and sparse 3D points. While extensions incorporate lines and other primitives, integrating richer geometric structures such as parallelism, coplanarity, or wireframes often introduces significantly increased computational cost and reduced numerical stability. In this paper, we propose a unified framework that extends bundle adjustment to jointly optimize geometric features and higher-order relations. We first introduce a taxonomy that distinguishes scalable geometric features with direct 2D measurements (e.g., points and lines), from groups encoding higher-order relations (e.g., coplanarity, parallelism, etc.), where we show that groups can be modeled as camera-like entities within the bundle adjustment framework. Building on this formulation, we propose that both group constraints and cross-feature relations (i.e., point-line associations) can be expressed through 2D reprojection measurements. By formulating group-induced and cross-feature reprojection errors, we preserve the sparsity structure of classical point-based BA under Schur elimination, while avoiding direct 3D regularization that degrades the conditioning and stability. Experiments on both real-world and synthetic datasets demonstrate runtime performance comparable to classical point-only bundle adjustment, while producing significantly richer 3D structures and improved geometric accuracy.
Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.
Converting hand-drawn sketches into structured 3D geometries remains challenging due to the difficulty of representing non-Euclidean surfaces and maintaining topological consistency. Existing generative models such as GANs, NeRFs, and diffusion architectures often fail to produce editable manifolds directly usable in downstream design workflows. We present Sketch2MinSurf, a hybrid vision-language and geometric optimization framework that integrates vision-language guidance with minimal-surface theory to generate smooth and editable 3D surfaces from hand-drawn sketches. The core of our approach is a spatial-topological encoding that represents geometry as tuples of node coordinates and real/virtual edge skeletons, enabling stable topological control during generation. We further introduce the Sketch2MinSurf Structural Loss (S2MS-Loss), a reward-modulated objective that jointly constrains geometric reconstruction and topological coherence. On a test set of 100 sketches, Sketch2MinSurf achieves a topological similarity score of 0.844, outperforming existing sketch-to-shape baselines. The generated manifolds are directly editable and free from non-manifold artifacts. A public art installation at a university showcases the method's potential for human-intent-driven 3D form generation. The dataset and code are available at https://anonymous.4open.science/r/Sketch2MinSurf/.