A Course in the Geometry of n Dimensions.

Pedoe, D.; Kendall, M. G.

doi:10.2307/2313174

Cited by 6 publications

(9 citation statements)

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“…This is the same as claiming that the corresponding arcs are orthogonal to the radical circle. By property 8, the arcs are orthogonal to the circle with diameter BR as they pass through inverse pairs [10,11].…”

Section: Property 10mentioning

confidence: 99%

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The Arbelos

Rangel-Mondragon¹

2014

TMJ

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This article systematically verifies a series of properties of an ancient figure called the arbelos. It includes some new discoveries and extensions contributed by the author. ‡ Introduction Motivated by the computational advantages offered by Mathematica, I decided some time ago to embark on collecting and implementing properties of the fascinating geometric figure called the arbelos. I have since been impressed by the large number of surprising discoveries and computational challenges that have sprung out of the growing literature concerning this remarkable object. I recall its resemblance to the lower part of the iconic canopied penny-farthing bicycle of the 1960s TV series The Prisoner, Punch's jester cap (of Punch and Judy fame), and a yin-yang symbol with one arc inverted; see Figure 1. There is now an online specialized catalog of Archimedean circles (circles contained in the arbelos) [1] and important applications outside the realm of mathematics and computer science [2] of arbelos-related properties.Many famous names are involved in this fascinating theme, among them Archimedes (killed by a Roman soldier in 212

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Section: Property 10mentioning

confidence: 99%

“…Let g be the length of the gap between the bases (so that the diameter of the top arc is 2 a + g + 2 b) and let v be the abscissa of the intersection of the radical axis with the x axis, assuming the origin is at the leftmost point of the arbelos [10].…”

Section: Property 28mentioning

confidence: 99%

The Arbelos

Rangel-Mondragon¹

2014

TMJ

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“…These two topologies define the projection of the node into the probe 3D space and therefore enable us to capture the many-dimensional node for this particular Hartree-Fock state. This again enables to describe the complete node using a theorem from algebraic geometry which states that any cubic surface is determined by an appropriate mapping of six points in a projective plane 17,18,19 . To use it we first need to realize the following property of the 3D projected node: The node equation above contains only a homogeneous polynomial in x, y, z which implies that in spherical coordinates the radius can be eliminated and the node is dependent only on angular variables.…”

Section: B Approximate Hartree-fock Nodes Of Thementioning

confidence: 99%

“…This enables us to project the positions of the six electrons on an arbitrary plane which does not contain the origin and the node will cut such a plane in a cubic curve. As we mentioned above, a theorem from the algebraic geometry of cubic surfaces and curves says that any cubic surface is fully described by six points in a projective plane (see 17,18,19 ). For ruled surfaces any plane not passing through the origin is a projective plane and therefore we can specify a one to one correspondence between the 6×3 = 18 dimensional space and our cubic surface in 3D.…”

Section: B Approximate Hartree-fock Nodes Of Thementioning

confidence: 99%

Approximate and exact nodes of fermionic wavefunctions: Coordinate transformations and topologies

et al. 2005

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A study of fermion nodes for spin-polarized states of a few-electron ions and molecules with s, p, d one-particle orbitals is presented. We find exact nodes for some cases of two electron atomic and molecular states and also the first exact node for the three-electron atomic system in 4 S(p 3 ) state using appropriate coordinate maps and wavefunction symmetries. We analyze the cases of nodes for larger number of electrons in the Hartree-Fock approximation and for some cases we find transformations for projecting the high-dimensional node manifolds into 3D space. The node topologies and other properties are studied using these projections. We also propose a general coordinate transformation as an extension of Feynman-Cohen backflow coordinates to both simplify the nodal description and as a new variational freedom for quantum Monte Carlo trial wavefunctions.

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“…Note that is not a closed set. However it is practical to impose impose a "collar" around the points on the image plane by enforcing the requirement that they maintain a minimum distance from one another, as well as maintaining their 6 A homography or projective transformation, Z 2 PL(1) is uniquely determined by the correspondence of three distinct points [28] [4], [7] has been modified with an "arrow" feature on the tip of the arm, which is observed by a perspective projection camera. The body frame is coincident with the tip of the arm and the z axis is pointing into the arm.…”

Section: A Example 1: Planar Rigid Body Servoingmentioning

confidence: 99%

Visual servoing via navigation functions

Cowan

Weingarten

Koditschek

2002

IEEE Trans. Robot. Automat.

182

134

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This paper presents a framework for visual servoing that guarantees convergence to a visible goal from almost every initially visible configurations while maintaining full view of all the feature points along the way. The method applies to first-and second-order fully actuated plant models. The solution entails three components: a model for the "occlusion-free" configurations; a change of coordinates from image to model coordinates; and a navigation function for the model space. We present three example applications of the framework, along with experimental validation of its practical efficacy. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.

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A Course in the Geometry of n Dimensions.

Cited by 6 publications

References 0 publications

The Arbelos

The Arbelos

Approximate and exact nodes of fermionic wavefunctions: Coordinate transformations and topologies

Visual servoing via navigation functions

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