NDLI: Discrete geometric structures for architecture

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Optimal partition trees

Discrete geometric structures for architecture

Better bounds on the union complexity of locally fat objects

Adding one edge to planar graphs makes crossing number hard

Visualizing 3D Euler spirals

A geometric approach to collective motion

Output-sensitive algorithm for the edge-width of an embedded graph

A traveller's problem

The complexity of the normal surface solution space

Klee's measure problem on fat boxes in time $∂(n^{(d+2)/3})$

The tidy set: a minimal simplicial set for computing homology of clique complexes

On degrees in random triangulations of point sets

Manifold reconstruction using tangential Delaunay complexes

Improving the first selection lemma in $R^{3}$

Applications of geometry and topology for quadrilateral meshprocessing and verifiable visualization

Computing planar Voronoi diagrams in double precision: a further example of degree-driven algorithm design

Incidences in three dimensions and distinct distances in the plane

Consistent digital line segments

On the complexity of sets of free lines and line segments among balls in three dimensions

Planar visibility: testing and counting

Shape approximation using k-order alpha-hulls

Kinetic stable Delaunay graphs

Finding shortest non-trivial cycles in directed graphs on surfaces

A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane

Distance k-sectors exist

An improved algorithm for computing the volume of the union of cubes

A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes

Acute triangulations of polyhedra and the Euclidean space

Optimal reconstruction might be hard

Points with large quadrant-depth

3D Euler spirals for 3D curve completion

Tangencies between families of disjoint regions in the plane

The geometry of flocking

Lines avoiding balls in three dimensions revisited

The 2-center problem in three dimensions

Constructing Reeb graphs using cylinder maps

A kinetic triangulation scheme for moving points in the plane

Approximating loops in a shortest homology basis from point data

New constructions of SSPDs and their applications

Zone diagrams in Euclidean spaces and in other normed spaces

Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements

Topological inference via meshing

Dynamic well-spaced point sets

Reconstructing shapes with guarantees by unions of convex sets

Approximating the Fréchet distance for realistic curves in near linear time

I/O-efficient computation of water flow across a terrain

Steinitz theorems for orthogonal polyhedra

Tight lower bounds for halfspace range searching

Linkless and flat embeddings in 3-space and the unknot problem

Convexification of polygons by length preserving transformations

A dynamic data structure for approximate range searching

Geometric tomography with topological guarantees

Computing similarity between piecewise-linear functions

Geometric methods for multi-agent collision avoidance

Discrete geometric structures for architecture

Content Provider	ACM Digital Library
Author	Pottmann, Helmut
Abstract	The emergence of freeform structures in contemporary architecture raises numerous challenging research problems, most of which are related to the actual fabrication and are a rich source of research topics in geometry and geometric computing. The talk will provide an overview of recent progress in this field, with a particular focus on discrete geometric structures. Most of these result from practical requirements on segmenting a freeform shape into planar panels and on the physical realization of supporting beams and nodes. A study of quadrilateral meshes with planar faces reveals beautiful relations to discrete differential geometry. In particular, we discuss meshes which discretize the network of principal curvature lines. Conical meshes are among these meshes; they possess conical offset meshes at a constant face/face distance, which in turn leads to a supporting beam layout with so-called torsion free nodes. This work can be generalized to a variety of multilayer structures and laid the ground for an adapted curvature theory for these meshes. There are also efforts on segmenting surfaces into planar hexagonal panels. Though these are less constrained than planar quadrilateral panels, this problem is still waiting for an elegant solution. Inspired by freeform designs in architecture which involve circles and spheres, we present a new kind of triangle mesh whose faces' in-circles form a packing, i.e., the in-circles of two triangles with a common edge have the same contact point on that edge. These "circle packing (CP) meshes" exhibit an aesthetic balance of shape and size of their faces. They are closely tied to sphere packings on surfaces and to various remarkable structures and patterns which are of interest in art, architecture, and design. CP meshes constitute a new link between architectural freeform design and computational conformal geometry. Recently, certain timber structures motivated us to study discrete patterns of geodesics on surfaces. This topic is closely related to the classical geometry of webs. There are numerous open problems in the geometry of webs, in particular when one asks for webs formed by special types of curves, as preferred in applications such as architecture.
Starting Page	38
Ending Page	38
Page Count	1
File Format	PDF
ISBN	9781450300162
DOI	10.1145/1810959.1810966
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2010-06-13
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Architecture Discrete differential geometry Geometry of webs
Content Type	Text
Resource Type	Article

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