3D representations of large-scale and urban scenes are crucial across various industries, including autonomous driving, planning, natural resource supervision many more. Large-scale industrial reconstructions inherently complex multifaceted. Many existing surveys primarily focus on academic progressions often neglect the intricate diverse needs industry. This survey aims to bridge this gap by providing a comprehensive analysis reconstruction methods, with requirements such as scalability...

Geometrically continuous splines are piecewise polynomial functions defined on a collection of patches which stitched together through transition maps. They called <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript r"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">G^{r}</mml:annotation>...

Designing mechanical devices, called linkages, that draw a given plane curve has been topic interested engineers and mathematicians for hundreds of years, recently also computer scientists. Already in 1876, Kempe proposed procedure solving the problem full generality, but his constructions tend to be extremely complicated. We provide novel algorithm produces much simpler works only parametric curves. Our approach is transform into factorization task over some noncommutative algebra. show how...

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution such problem can be expressed as convex combination certain interpolants when sought in any finite-dimensional vector space; secondly, provide general strategy iteratively update weights according approximation error and apply it spline fitting problem. In experiments, we numerical examples case polynomials splines spaces. Subsequently,...

Powersum varieties, also called varieties of sums powers, have provided examples interesting relations between since their first appearance in the 19th century.One most useful tools to study them is apolarity, a notion originally related action differential operators on polynomial ring.In this work, we make explicit how one can see apolarity terms Cox ring variety.In way, powersum are special case apolar schemes; explicitly describe such two toric surfaces, when particularly well-behaved.

An irrational toric variety X is an analytic subset of the simplex associated to a finite configuration real vectors.The positive torus acts on by translation, and we consider limits sequences these translations.Our main result identifies all possible Hausdorff translations as degenerations using elementary methods geometry secondary fan vector configuration.This generalizes work García-Puente et al., who used algebraic Kapranov, Sturmfels, Zelevinsky, when vectors were integral.

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called star. Vertex stars are natural generalization planar triangulations, and understanding splines on crucial step to analyzing trivariate splines. It particularly difficult compute the dimension in which completely surrounded by tetrahedra---we call these closed stars. formula due Alfeld, Neamtu, Schumaker gives $C^r$ degree least $3r+2$. We show that this lower bound $(3r+2)/2$. Our proof uses apolarity so-called...

In this paper we study the dimension of splines mixed smoothness on axis-aligned T-meshes. This is setting when different orders are required across edges mesh. Given a spline space whose independent its T-mesh's geometric embedding, present constructive and sufficient conditions that ensure subset mesh can be reduced while maintaining stability dimension. The have simple interpretation. Examples presented to show applicability results both hierarchical non-hierarchical For hierarchal...

Abstract Splines are piecewise polynomial functions which continuously differentiable to some order r . For a fixed integer d the space of splines degree at most is finite dimensional vector space, and largely open problem in numerical analysis determine its dimension. While considerable attention has been given this bivariate setting, literature on trivariate less conclusive. In particular, dimension generic not known even large when $$r&gt;1$$ <mml:math...

We consider spaces of multivariate splines defined on a particular type simplicial partitions that we call (generalized) oranges. Such are composed finite number maximal faces with exactly one shared medial face. reduce the problem finding dimension oranges to computing dimensions simpler, lower-dimensional projected use both algebraic and Bernstein-Bézier tools.

We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution such problem can be expressed as convex combination certain interpolants when sought in any finite-dimensional vector space; secondly, provide general strategy iteratively update weights according approximation error and apply it spline fitting problem. In experiments, we numerical examples case polynomials splines spaces. Subsequently,...

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, computer aided geometric design. In this paper we address various challenges arising study splines with enhanced mixed (super-)smoothness conditions at vertices across interior faces partition. Such supersmoothness can be imposed but also appear...