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SPACES OF SPECIAL QUADRILATERALS

Part of: Real and complex geometry Topological manifolds General convexity

Published online by Cambridge University Press:  07 January 2019

AHTZIRI GONZÁLEZ Open the ORCID record for AHTZIRI GONZÁLEZ [Opens in a new window]  and
JORGE L. LÓPEZ-LÓPEZ Open the ORCID record for JORGE L. LÓPEZ-LÓPEZ [Opens in a new window]
AHTZIRI GONZÁLEZ
Affiliation:
Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia, Michoacán, México email [email protected]
JORGE L. LÓPEZ-LÓPEZ*
Affiliation:
Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio Alfa, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed $n$-disc $\mathbb{D}^{n}$ is the unique topological $n$-manifold (with boundary) whose boundary and interior are homeomorphic to $\mathbb{S}^{n-1}$ and $\mathbb{R}^{n}$, respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to $\mathbb{D}^{3}$.

Keywords

similarity classshapequadrilateral

MSC classification

Primary: 57N25: Shapes
Secondary: 51M99: None of the above, but in this section 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 1 , August 2019 , pp. 155 - 167
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is partially supported by CONACYT grant FORDECYT 265667; the second author is partially supported by funding from CIC-UMSNH.

References

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