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On equidistant lines of given line configurations

  • Takayasu Kuwata , Hiroshi Maehara und Horst Martini EMAIL logo
Veröffentlicht/Copyright: 14. August 2019
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Georgian Mathematical Journal
Aus der Zeitschrift Georgian Mathematical Journal Band 26 Heft 4

Abstract

The equidistant set of a collection F of lines in 3-space is the set of those points whose distances to the lines in F are all equal. We present many examples and results related to the lines possibly contained in the equidistant set of F. In particular, we determine the possible numbers of lines in the equidistant set of a collection of n lines for every n>0. For example, if n=3, then the possible number of such lines is either 4 or 2 or 1 or 0. In a natural way, our results are connected with properties of special types of (ruled) surfaces. For example, we obtain also results on the number of lines in the intersection of quadratic surfaces.

Keywords: Doubly ruled surface; equidistant set; equidistant line; hyperboloid of one sheet; hyperbolic paraboloid; quadratic surfaces
MSC 2010: 51N20; 52C99; 51F99; 51M04

Dedicated to Professor Alexander Kharazishvili on his 70th birthday

A Appendix

Let us present here a proof of Theorem 1.1. First, we prove a lemma.

Lemma A.1.

Let g,ξ be a pair of distinct lines, and let P∈ξ,Q∈g (possibly P=Q) be points such that |P⁢Q| attains the minimum distance between g and ξ. Let X∈ξ and Y,Z∈g be points such that Y⁢X⊥ξ and X⁢Z⊥g. Then

(A.1)|X⁢Y|2=|P⁢Q|2+|P⁢X|2⁢(|X⁢Z|2-|P⁢Q|2|P⁢X|2-|X⁢Z|2+|P⁢Q|2),
(A.2)|X⁢Z|2=|P⁢Q|2+|P⁢X|2⁢(|X⁢Y|2-|P⁢Q|2|P⁢X|2+|X⁢Y|2-|P⁢Q|2).

Proof.

Let us consider the case P≠Q. In this case, PQ is perpendicular to both g and ξ. Let g′ be the line passing through P and parallel to g, and let Y′,Z′∈g′ be such that Q⁢P⁢∥Y⁢Y′∥⁢Z⁢Z′; see Figure 7. Then Y⁢Y′ and Z⁢Z′ are both perpendicular to the plane P⁢X⁢Y′.

Figure 7

Lemma A.1.

Let φ=∠⁢X⁢P⁢Y′. Then we have

|P⁢X|2⁢tan2⁡φ+|P⁢Q|2=|X⁢Y|2,
|P⁢X|2⁢sin2⁡φ+|P⁢Q|2=|X⁢Z|2.

From these, we have (A.1) and (A.2). ∎

Proof of Theorem 1.1.

Let P∈ξ, Q1∈g1 be the points such that |P⁢Q1| attains the minimum distance between ξ and g1. Let us consider the case that P≠Q1. (The proof of the case P=Q1 is easier.) Let HP be the plane that perpendicularly intersects ξ at P.

First, suppose that all lines in F lie on a surface of revolution with axis ξ. For i=2,3,…,n, let Qi be the intersection point of gi and HP. Then |P⁢Q1|=…=|P⁢Qn|. For X∈ξ, let HX denote the plane that perpendicularly intersects ξ at X, and let {Yi}=HX∩gi. Since g1,…,gn lie on the surface of revolution with axis ξ, we have |X⁢Y1|=…=|X⁢Yn|. Hence, by (A.2), we have d⁢(X,g1)=…=d⁢(X,gn), which implies that ξ is an equidistant line of F.

Next, suppose that ξ is an equidistant line of F. Let ℋ be the surface obtained by rotating g1 around ξ. This time, define Qi∈gi as the point such that |P⁢Qi|=|P⁢Q1|. Since |P⁢Q1| is the minimum distance between ξ and g1, |P⁢Qi| is also the minimum distance between ξ and gi for i=2,…,n. For X∈ξ, define (this time) Yi∈gi as a point such that X⁢Yi⊥ξ is satisfied. Since d⁢(X,g1)=…=d⁢(X,gn) and |P⁢Q1|=…=|P⁢Qn|, it follows by (A.1) of Lemma A.1 that |X⁢Y1|=…=|X⁢Yn|. Hence Y1,…,Yn lie on ℋ. Now, moving X on ξ, we can see that the lines g1,…,gn all lie on ℋ. Thus, all lines in F lie on a surface of revolution with axis ξ. ∎

Acknowledgements

The authors are grateful to the reviewer for careful reading of the manuscript and helpful advises.

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Received: 2018-08-18
Revised: 2019-02-10
Accepted: 2019-03-12
Published Online: 2019-08-14
Published in Print: 2019-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On absolutely Baire nonmeasurable functions
  3. Existence of special rainbow triangles in weak geometries
  4. Images of Bernstein sets via continuous functions
  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
  10. On the Gitik–Shelah theorem
  11. On equidistant lines of given line configurations
  12. On convergence of sequences of functions possessing closed graphs
  13. On combinatorial and set-theoretical aspects of some finite and infinite point sets
  14. On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
  15. Automorphism groups of mono-unary algebras and CH
  16. On topologies related to the extension of the Lebesgue measure
  17. An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
  18. Local affine selections of convex set-valued functions
  19. On orbits without the Baire property
  20. Mann iteration process for monotone nonexpansive mappings with a graph
  21. Mixing coded systems
  22. Methods of comparison of families of real functions in porosity terms
https://doi.org/10.1515/gmj-2019-2037
Schlagwörter für diesen Artikel
Doubly ruled surface; equidistant set; equidistant line; hyperboloid of one sheet; hyperbolic paraboloid; quadratic surfaces

Artikel in diesem Heft

  1. Frontmatter
  2. On absolutely Baire nonmeasurable functions
  3. Existence of special rainbow triangles in weak geometries
  4. Images of Bernstein sets via continuous functions
  5. Square-integrable representations and multipliers
  6. Semigroups of pathological sets
  7. Points of density and ideals of subsets of ℕ
  8. On Kharazishvili type measures in infinite-dimensional Polish vector spaces
  9. Families of symmetric Cantor sets from the category and measure viewpoints
  10. On the Gitik–Shelah theorem
  11. On equidistant lines of given line configurations
  12. On convergence of sequences of functions possessing closed graphs
  13. On combinatorial and set-theoretical aspects of some finite and infinite point sets
  14. On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
  15. Automorphism groups of mono-unary algebras and CH
  16. On topologies related to the extension of the Lebesgue measure
  17. An abstract formulation of a theorem of Sierpiński on the nonmeasurable sum of two measure zero sets
  18. Local affine selections of convex set-valued functions
  19. On orbits without the Baire property
  20. Mann iteration process for monotone nonexpansive mappings with a graph
  21. Mixing coded systems
  22. Methods of comparison of families of real functions in porosity terms
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