The uses of homogeneous barycentric coordinates in plane Euclidean geometry

Yiu, Paul

doi:10.1080/002073900412679

Cited by 25 publications

(20 citation statements)

References 7 publications

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“…In this section we remind the definitions of barycentric coordinates, homogeneous barycentric coordinates, isotomic conjugate, and isogonal conjugate [4,10,13,15,16] . Every point in a reference triangle satisfies …”

Section: Preliminaries For Elementarymentioning

confidence: 99%

“…In particular, the use of barycentric coordinates has played an important rule in a lot of methods for the conic representation and conic approximation [1][2][3][4][5][6] . In Euclidean Plane Geometry, so many works have been done based on the use of them [1,[7][8][9][10][11] . Recently, Akopyan [12] presented the properties of the tangency of isotomically and isogonally conjugate lines of some special lines with respect to a triangle.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we show that the three intersection points correspond to the vertices of triangle are collinear. where is the origin on the plane containing the triangle and is the area of triangle [10,15] . In Eq.…”

Section: Introductionmentioning

confidence: 99%

“…The relationship between the signatures of barycentric coordinates and the position of outside of is well-known [13,17] . [10,15,16] that and has the homogeneous barycentric coordinates .…”

Section: Introductionmentioning

confidence: 99%

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Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

Seo¹,

Ahn²

2017

Journal of the Chosun Natural Science

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Section: Preliminaries For Elementarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The relationship between the signatures of barycentric coordinates and the position of outside of is well-known [13,17] . [10,15,16] that and has the homogeneous barycentric coordinates .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

Seo¹,

Ahn²

2017

Journal of the Chosun Natural Science

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“…To perform equal-mass partitioning, our approach utilizes previous work on computation using barycentric coordinates [17] and convex hulls [13].…”

Section: A Related Workmentioning

confidence: 99%

Constraint-aware coordinated construction of generic structures

Stein

Schoen

Rus

2011

2011 IEEE/RSJ International Conference on Intelligent Robots and Systems

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Abstract-This paper presents a constraint-aware decentralized approach to construction with teams of robots. We present an extension to existing work on a distributed controller for robotic construction of simple structures. Our previous work described a set of adaptive algorithms for constructing truss structures given a target geometry using continuous and graph-based equal-mass partitioning [1], [2]. Using this work as a foundation, we present an algorithm which performs construction tasks and conforms to physical constraints while considering those constraints to parallelize tasks. This is accomplished by defining a mass function which reflects the priority of part placement and prevents physically impossible states. This mass function generates a set of pointmasses in R n , and we present a novel algorithm for finding a locally optimal, equalmass, convex tessellation of such a set.

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Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

Ungar

2012

Springer Optimization and Its Applications

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In this chapter, dedicated to the 60th Anniversary of Themistocles M. Rassias, Möbius transformation and Einstein velocity addition meet in the hyperbolic geometry of Bolyai and Lobachevsky. It turns out that Möbius addition that is extracted from Möbius transformation of the complex disc and Einstein addition from his special theory of relativity are isomorphic in the sense of gyrovector spaces.(1) Möbius addition in the ball R n c forms the algebraic setting for the Cartesian-Poincaré ball model of hyperbolic geometry, and (2) Einstein addition in the ball R n c forms the algebraic setting for the Cartesian-Beltrami-Klein ball model of hyperbolic geometry, just as the common (3) vector addition in the space R n forms the algebraic setting for the standard Cartesian model of Euclidean geometry.Remarkably, Items (1)-( 3) enable Möbius addition in R n c , Einstein addition in R n c , and the standard vector addition in R n to be studied comparatively, as in [72]. for all a, b, u, v ∈ R n c . Hence, gyr[u, v] is an isometry of R n c , keeping the norm of elements of the ball R n c invariant, (34) gyr[u, v]w = w Accordingly, gyr[u, v] represents a rotation of the ball R n c about its origin for any u, v∈R n c . The invertible self-map gyr[u, v] of R n c respects Einstein addition in R n c , (35) gyr[u, v](a⊕b) = gyr[u, v]a⊕gyr[u, v]bfor all a, b, u, v∈R n c , so that gyr [u, v] is an automorphism of the Einstein groupoid (R n c , ⊕). We recall that an automorphism of a groupoid (R n c , ⊕) is a bijective self-map of the groupoid R n c that respects its binary operation, that is, it satisfies (35). Under bijection composition the automorphisms of a groupoid (R n c , ⊕) form a group known as the automorphism group, and denoted Aut(R n c , ⊕). Being special automorphisms, gyrations gyr [u, v]∈Aut(R n c , ⊕), u, v ∈R n c , are also called gyroautomorphisms, gyr being the gyroautomorphism generator called the gyrator.

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The uses of homogeneous barycentric coordinates in plane Euclidean geometry

Cited by 25 publications

References 7 publications

Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

Isotomic and Isogonal Conjugates Tangent Lines of Lines at Vertices of Triangle

Constraint-aware coordinated construction of generic structures

Möbius Transformation and Einstein Velocity Addition in the Hyperbolic Geometry of Bolyai and Lobachevsky

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