The Universe of Conics

Glaeser, Georg; Stachel, Hellmuth; Odehnal, Boris

doi:10.1007/978-3-662-45450-3

Cited by 60 publications

(20 citation statements)

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“…XII, XVI, and XVIII]. Many other theorems about conics can be found in [13]. For any billiard in the grey conic, there is an associated incircular net.…”

Section: Preliminariesmentioning

confidence: 99%

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Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

Bobenko

Fairley

2021

Discrete Comput Geom

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In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

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“…XII, XVI, and XVIII]. Many other theorems about conics can be found in [13]. For any billiard in the grey conic, there is an associated incircular net.…”

Section: Preliminariesmentioning

confidence: 99%

“…The notation p i,i+1 will be used to denote a point that is contained in the line (v i , v i+1 ). Proposition 2.1 is a degenerate case of Brianchon's theorem [2,13].…”

Section: Preliminariesmentioning

confidence: 99%

Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

Bobenko

Fairley

2021

Discrete Comput Geom

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“…As a result, Q1Q2 is a ruling of the developable surface connecting C1 and C2. The developable surfaces connecting two cospherical circles are quadratic cones (Glaeser, Stachel, and Odehnal 2016). Consequently, the transversal edge Q1Q2 is a ruling of one of these quadratic cones (in blue).…”

Section: Proofmentioning

confidence: 99%

Surfaces with planar curvature lines: Discretization, generation and application to the rationalization of curved architectural envelopes

Tellier

Douthe

Hauswirth

et al. 2019

Automation in Construction

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Motivated by architectural applications, we propose a method to generate circular and conical meshes with planar curvature lines in both directions. The method is based on the discretization of the Gauss map of surfaces with planar curvatures lines. It allows generation of meshes in real-time via two planar guide curves. The resulting meshes can be used as a geometric base to build gridshells with flat panels, torsion free-nodes, node offset and planar arches. A particular technological application is for gridshells built with curved members: those can be built with planar piecewise-circular beams, and identical nodes if beams have circular cross-section.

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“…Figure 4. As on the sphere points can be replaced by their antipodes, it can easily be seen that every spherical conic can be interpreted as a spherical ellipse (e.g., [35] (Section 10.1)). The left and the right figure show the same symmetric rolling motion.…”

Section: Planar/spherical Symmetric Rollings With Circular Pathsmentioning

confidence: 99%

Hexapods with Plane-Symmetric Self-Motions

Nawratil

2018

Robotics

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A hexapod is a parallel manipulator where the platform is linked with the base by six legs, which are anchored via spherical joints. In general, such a mechanical device is rigid for fixed leg lengths, but, under particular conditions, it can perform a so-called self-motion. In this paper, we determine all hexapods possessing self-motions of a special type. The motions under consideration are so-called plane-symmetric ones, which are the straight forward spatial counterpart of planar/spherical symmetric rollings. The full classification of hexapods with plane-symmetric self-motions is achieved by formulating the problem in terms of algebraic geometry by means of Study parameters. It turns out that besides the planar/spherical symmetric rollings with circular paths and two trivial cases (butterfly self-motion and two-dimensional spherical self-motion), only one further solution exists, which is the so-called Duporcq hexapod. This manipulator, which is studied in detail in the last part of the paper, may be of interest for the design of deployable structures due to its kinematotropic behavior and total flat branching singularities.

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The Universe of Conics

Cited by 60 publications

References 0 publications

Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

Surfaces with planar curvature lines: Discretization, generation and application to the rationalization of curved architectural envelopes

Hexapods with Plane-Symmetric Self-Motions

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