- #1
ƒ(x)
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Are circles considered straight lines in Non-Euclidean Geometry?
Circles in Non-Euclidean Geometry
- Context: Graduate
- Thread starter ƒ(x)
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Discussion Overview
The discussion revolves around the nature of circles and straight lines in Non-Euclidean Geometry, specifically exploring how these concepts differ in various geometrical frameworks such as elliptic and hyperbolic geometry. Participants engage in clarifying definitions and implications of these terms within the context of different geometries.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions whether circles are considered straight lines in Non-Euclidean Geometry, prompting further clarification on the specific type of geometry in question.
- Another participant notes that in spherical geometry, great circles can be defined as geodesics, which are locally distance-minimizing paths.
- In elliptic geometry, it is suggested that while geodesics may be closed paths, there exist distinct definitions of circles that differ from those of straight lines.
- A participant mentions that in hyperbolic geometry, all lines are unbounded and thus cannot be classified as circles.
- One participant shares a personal anecdote about discussing Nikolai Lobachevsky, indicating a connection to hyperbolic geometry.
- It is clarified that in Lobachevsky space (hyperbolic space), straight lines extend to infinity, and therefore are not circles, although there exists a mapping where straight lines appear as arcs of circles.
Areas of Agreement / Disagreement
Participants express differing views on the definitions and relationships between circles and straight lines in various Non-Euclidean geometries, indicating that multiple competing perspectives exist without a clear consensus.
Contextual Notes
The discussion highlights the dependence on specific definitions of "straight line" and "circle," as well as the implications of different geometrical frameworks, which remain unresolved.
- #2
yenchin
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Your question is abit vague... *which* non-Euclidean geometry? And what do you mean by "straight line"? Do you mean geodesic? Certainly on a sphere we can define a geometry where every great circle is a geodesic, which is locally distance minimising between two points.
- #3
HallsofIvy
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In elliptic geometry, "straight lines" (as yenchin said, geodesics) may be closed paths but, technically, there still exist "circles" that are quite different from those. In hyperbolic geometry, all "lines" are unbounded and so are definitely NOT "circles".
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- #4
ƒ(x)
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Well, I started wondering about this because my uncle and myself started talking about Nikolai Lobachevsky. I don't know if that will help answer the question.
- #5
tiny-tim
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ƒ(x) said:
Hi ƒ(x)!
In Lobachevsky space, all straight lines go off to infinity, so none of them are circles.
(Though there is a "map" of Lobachevsky space, in which all the straight lines are mapped as arcs of circles which meet the enclosing circle at right-angles … but they aren't circles "in" the space, only "in" the map.
)- #6
HallsofIvy
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Lobachevsky space being the aforementioned "hyperbolic space". And I corrected the silly typo in my first response.
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