Conic Hull & Triangulation Methods

• Conic Hull and Triangulation Methods are geometric and combinatorial techniques that decompose complex analytic and optimization problems into convex cones and simplicial pieces.
• They replace unstructured global searches with finite generators and structured triangulations, streamlining residue computations and integration processes.
• These methods are pivotal in applications such as Mellin–Barnes integrals, computer vision reconstruction, and integer optimization through dual formulations.

to=arxiv_search.search 山大发json content='{"query":"id:(Banik et al., 2023) OR id:(Banik et al., 2024) OR id:(Banik et al., 22 Dec 2025) OR id:(Gummeson et al., 2022) OR id:(Rydell et al., 2024)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}' to=arxiv_search.search ฝ่ายขายรายการjson content='{"query":"all:\"conic hull triangulation\" OR ti:\"conic hull\" OR ti:triangulation","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}' Conic hull and triangulation methods are geometric and combinatorial techniques that encode analytic, algebraic, optimization, and reconstruction problems either by cones generated from finite sets of vectors or by simplicial decompositions of associated point configurations. In one prominent formulation, a conic hull is the convex cone generated by a finite set, while a triangulation decomposes a convex hull into simplices with pairwise intersections along common faces; in another, back-projected cones and dual conics organize multiview reconstruction constraints. Across these settings, the common objective is to replace an unstructured global search by a finite family of geometrically meaningful generators, faces, cones, or simplices (Banik et al., 2023).

1. Geometric primitives and dual viewpoints

A conic hull generated by a finite set $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$ is

$$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$

A triangulation of a point configuration $P\subset\mathbb R^d$ is a decomposition of $\operatorname{conv}(P)$ into simplices with vertices in $P$, with pairwise intersections along common faces. The Mellin–Barnes literature further emphasizes regular triangulations, generated by lifting the points to one dimension higher, taking the lower convex hull, and projecting back; algorithmically, all regular triangulations are generated using TOPCOM (Banik et al., 2023).

In convex conic reformulation, the same geometric vocabulary appears at the level of cones, faces, and supporting hyperplanes. For a cone $C\subseteq V$, $\operatorname{cone}(C)=\operatorname{co}(C)$, and exact convexification is governed by the condition

$\operatorname{co}(K\cap J)=J,$

where $K$ is a nonconvex cone and $J$ is a convex cone contained in $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$0. Under this condition, optimization over $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$1 and over $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$2 on the same affine slice have the same optimal value (Arima et al., 2023).

In computer vision, the dual language is equally central. A quadric envelope $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$3 projects linearly to a dual conic $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$4 by

$$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$5

and tangency of an image line $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$6 is encoded by $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$7. This linearity makes dual conics and dual quadrics the natural objects for cylinder triangulation and conic multiview varieties (Gummeson et al., 2022).

2. Mellin–Barnes integrals: from conic hull intersections to triangulations

For higher-fold Mellin–Barnes integrals, the geometric content is encoded directly in the affine-linear arguments of gamma functions. The generic $$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$8-fold integral is written as

$$\operatorname{cone}(A)=\left\{\sum_{i=1}^m \lambda_i a_i : \lambda_i\ge 0\right\}.$$9

with $P\subset\mathbb R^d$0 and $P\subset\mathbb R^d$1 linear in $P\subset\mathbb R^d$2. A key step is passage to canonical form, where $P\subset\mathbb R^d$3 are factored out, and the remaining numerator gamma functions define a finite point configuration. If $P\subset\mathbb R^d$4, one defines points $P\subset\mathbb R^d$5 from the coefficients $P\subset\mathbb R^d$6 and adjoins $P\subset\mathbb R^d$7 unit vectors; the resulting $P\subset\mathbb R^d$8-matrix is the configuration matrix of GKZ theory. Maximal simplices in a regular triangulation correspond, via a complement map, to the maximal cones that appear in the older conic-hull intersection method, and each regular triangulation yields one linear combination of multiple series representations of the original MB integral (Banik et al., 2023).

The older conic-hull algorithm works directly with subsets of gamma-function coefficient vectors, forms the corresponding cones, computes their intersections, and interprets relevant intersections as admissible simultaneous residue choices. The triangulation method replaces this by a dual combinatorial computation on the $P\subset\mathbb R^d$9-matrix point configuration. The analytic output is unchanged: both approaches produce sums of multivariable hypergeometric series of the form

$\operatorname{conv}(P)$0

What changes is the search mechanism. This replacement is computationally decisive for high-fold integrals (Banik et al., 2024).

The performance data are explicit. For the conformal hexagon, a 9-fold MB integral with 194,160 solutions, the cone method needed about 1 min to get one solution and did not compute all solutions, whereas triangulation took 0.489 s for one solution and 40 min for all 194,160. For the conformal double box, also 9-fold, triangulation took 0.635 s for one solution and 1.8 h for all 243,186. For off-shell massless one-loop $\operatorname{conv}(P)$1-point integrals, whose MB dimensionality is $\operatorname{conv}(P)$2, the method remained tractable up to the 15-point case: the 104-fold representation produced a 32,752-term solution in 8.9 h (Banik et al., 2023).

The same framework has been extended to MB integrals with polygamma factors. The arguments of polygamma functions are treated in the same geometric way as gamma-function arguments when constructing conic hulls and triangulations, but residue extraction differs because $\operatorname{conv}(P)$3 has poles of order $\operatorname{conv}(P)$4 at non-positive integers. The resulting algorithms split according to contour type: a direct treatment for non-straight contours, and a limiting procedure based on gamma-ratio representations of polygamma functions for straight contours (Banik et al., 22 Dec 2025).

3. Dual conics, cylinders, and conic multiview triangulation

In geometric reconstruction, conic hull and triangulation methods reappear in dual form. A cylinder is modeled as a quadric whose dual representation is linear under projection, so silhouette lines impose constraints of the form

$\operatorname{conv}(P)$5

After estimating the axis direction $\operatorname{conv}(P)$6 and rotating coordinates so that the cylinder axis aligns with a coordinate axis, the problem reduces to a planar dual conic $\operatorname{conv}(P)$7 constrained to be the dual of a circle. The manifold of circular dual conics is characterized by the two polynomial constraints

$\operatorname{conv}(P)$8

This reduction turns cylinder reconstruction from image silhouettes into a conic triangulation problem from tangent lines (Gummeson et al., 2022).

The ill-posedness of unconstrained conic estimation is a central point. Linear least-squares fitting of a general dual conic to silhouette lines can return a hyperbola with almost indistinguishable reprojection error from the true circular model, especially under limited angular coverage. The proposed response is to enforce the circularity manifold algebraically. Three silhouette lines are minimally sufficient; the resulting solver yields a system of two quadratic equations in two unknowns, generically with up to 4 complex solutions, and the generated minimal solver runs in about $\operatorname{conv}(P)$9. A constrained least-squares solver that uses all tangent lines reduces to three polynomials in three variables, uses an elimination template of size $P$0, and runs in about $P$1 per instance (Gummeson et al., 2022).

A related but distinct algebraic setting is the conic multiview variety. For a camera arrangement $P$2, the conic multiview variety is the Zariski closure of the image of the degree-2 plane curve multiview map. If the arrangement contains at least two cameras with distinct centers, its dimension is $P$3. For two views in a canonical gauge, a pair of image conics $P$4 lies in the conic multiview variety if and only if

$P$5

equivalently, all $P$6 minors vanish. For any number of views, the simplest set-theoretic description holds when all camera centers are distinct, no three are collinear, and no eight lie on a conic; then the tuple of image conics arises from a single 3D conic exactly when the associated back-projected cones satisfy the corresponding inclusion and blowup conditions. The associated triangulation problem leads to a Euclidean-distance optimization on the conic multiview variety, and the paper conjectures that the two-view Euclidean distance degree is $P$7 (Rydell et al., 2024).

4. Hull-driven triangulation in computational geometry

In computational geometry, hull-based triangulation methods reverse the usual point-location viewpoint: the convex hull, rather than the current triangulation, becomes the principal dynamic object. External incremental Delaunay triangulation inserts points in sorted order so that each new point lies outside the current hull. The update then consists of finding upper and lower tangents to the hull, eroding the visible chain by in-circle tests, and reconnecting the exposed boundary to the new point. The point-sorting stage costs $P$8, the collective complexity for upper/lower tangent searches is $P$9, empirical construction time for uniformly distributed point sets is linear $C\subseteq V$0, and the overall time complexity remains $C\subseteq V$1 (Cai, 13 Mar 2025).

S-hull is another hull-first method. It constructs a radially propagating sweep-hull from a radially sorted point set, producing a non-overlapping triangulation, and then restores the Delaunay property by edge flips. The algorithm is $C\subseteq V$2, and empirical tests reported it running in approximately half the time of q-hull for 2D Delaunay triangulation on randomly generated point sets (Sinclair, 2016).

A 3D sweep-hull variant, the Newton Apple Wrapper algorithm, exploits the paraboloid lifting $C\subseteq V$3 for planar Delaunay triangulation. Points are sorted in $C\subseteq V$4, and each new point can see at least one facet touching the last point added, reducing search overhead. The algorithm is $C\subseteq V$5, slower than q-hull for the general 3D case, but significantly faster in the pathological all-on-hull regime that arises in Delaunay lifting (Sinclair, 2016).

At the combinatorial level, triangulation spaces themselves admit a hull-based description. For planar point sets in general position, the edge flip graph is $C\subseteq V$6-connected and the bistellar flip graph is $C\subseteq V$7-connected; both bounds are tight. Regular triangulations form the subfamily obtained by lifting the points to 3-space and projecting back the lower convex hull, and their flip graph is governed by the secondary polytope (Wagner et al., 2020). This suggests that triangulation methods are not merely local mesh-update procedures but are organized globally by high-dimensional convex-geometric complexes.

5. Conic hull methods in integer and nonconvex optimization

In conic integer programming, the fundamental object is

$C\subseteq V$8

with $C\subseteq V$9 a regular cone. The integer hull $\operatorname{cone}(C)=\operatorname{co}(C)$0 is described through cut-generating functions $\operatorname{cone}(C)=\operatorname{co}(C)$1 satisfying subadditivity, monotonicity with respect to the cone order, and $\operatorname{cone}(C)=\operatorname{co}(C)$2. Every such $\operatorname{cone}(C)=\operatorname{co}(C)$3 yields a valid inequality

$\operatorname{cone}(C)=\operatorname{co}(C)$4

For bounded conic sets, linear composition functions built from integer-linear-programming cut-generating functions suffice to recover the integer hull. For unbounded second-order conic sets, this reduction fails, and the paper introduces a new family of functions

$\operatorname{cone}(C)=\operatorname{co}(C)$5

which are non-decreasing with respect to the second-order cone. Under mild conditions, linear composition functions together with these $\operatorname{cone}(C)=\operatorname{co}(C)$6 are sufficient to yield the integer hull of intersections of conic sections in $\operatorname{cone}(C)=\operatorname{co}(C)$7 (Santana et al., 2016).

The canonical example is

$\operatorname{cone}(C)=\operatorname{co}(C)$8

whose integer hull is

$\operatorname{cone}(C)=\operatorname{co}(C)$9

The facet $\operatorname{co}(K\cap J)=J,$0 cannot be obtained from any finite polyhedral outer approximation of $\operatorname{co}(K\cap J)=J,$1, but it is produced directly by a suitable $\operatorname{co}(K\cap J)=J,$2. The paper’s 2D completeness theorem is therefore genuinely conic rather than polyhedral (Santana et al., 2016).

A parallel line of work studies exact convexification of nonconvex conic programs. In the framework with a nonconvex cone $\operatorname{co}(K\cap J)=J,$3, a convex cone $\operatorname{co}(K\cap J)=J,$4, and a normalizing hyperplane $\operatorname{co}(K\cap J)=J,$5, exact reformulation is equivalent to

$\operatorname{co}(K\cap J)=J,$6

The paper provides three sets of necessary-sufficient conditions for this property and applies them to a new class of quadratically constrained quadratic programs with multiple nonconvex equality and inequality constraints that can be solved exactly by semidefinite relaxation. In the matrix-lifted setting with $\operatorname{co}(K\cap J)=J,$7, if

$\operatorname{co}(K\cap J)=J,$8

then $\operatorname{co}(K\cap J)=J,$9, and for all $K$0 with finite optimal value the SDP relaxation and the original nonconvex QCQP have the same optimum (Arima et al., 2023).

6. Software, scale, and cross-domain patterns

The Mellin–Barnes program is implemented in MBConicHulls.wl. Version 1.2 adds the TriangulateMB module to the earlier conic-hull-based package, and the workflow MBRep $K$1 TriangulateMB $K$2 EvaluateSeries $K$3 SumAllSeries provides a full path from an MB representation to explicit hypergeometric series and numerical evaluation. TOPCOM is used externally for regular triangulations, and the updated package also supports MB integrals with polygamma factors (Banik et al., 2023).

In cylinder reconstruction, the minimal solver was designed explicitly for robust estimation schemes such as RANSAC, and in a water-fountain experiment with unknown correspondences the exhaustive three-line strategy reconstructed 7 cylinders with an average outlier rate of 89% (Gummeson et al., 2022). In computational geometry, hull-first Delaunay routines exploit sorted insertion, tangent searches, and local flips rather than global point-location structures (Cai, 13 Mar 2025). In optimization, both second-order conic cuts and exact convexification results depend on isolating the appropriate face or asymptotic direction of a cone rather than enumerating all constraints naively (Santana et al., 2016).

This suggests that “conic hull and triangulation methods” are best understood as a recurring duality. One side works with generators, rays, gamma-argument vectors, tangent planes, or back-projected cones; the other side works with simplices, regular triangulations, flip graphs, or facial decompositions. In Mellin–Barnes analysis, the duality is explicit: cone intersections and regular triangulations encode the same residue data (Banik et al., 2024). In vision, dual conics and back-projected cones turn reconstruction into an algebraic incidence problem (Rydell et al., 2024). In optimization and computational geometry, the same structural move replaces direct high-dimensional search by a decomposition into extremal generators or simplicial pieces.