- #1
ƒ(x)
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Are circles considered straight lines in Non-Euclidean Geometry?
Circles in Non-Euclidean Geometry
- Thread starter ƒ(x)
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In summary, the conversation discusses the concept of straight lines and circles in different types of non-Euclidean geometry, specifically in Lobachevsky space. It is mentioned that in elliptic geometry, straight lines can be closed paths and in hyperbolic geometry, all lines are unbounded. The conversation also touches on the idea of mapping Lobachevsky space where straight lines can be represented as arcs of circles.
- #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.
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HallsofIvy
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Lobachevsky space being the aforementioned "hyperbolic space". And I corrected the silly typo in my first response.
Related to Circles in Non-Euclidean Geometry
1. What is the definition of a circle in non-Euclidean geometry?
In non-Euclidean geometry, a circle is defined as a set of points in a plane that are equidistant from a fixed point called the center. However, the distance metric used in non-Euclidean geometry may differ from the traditional Euclidean distance.
2. Can circles exist in non-Euclidean geometry?
Yes, circles can exist in non-Euclidean geometry. However, they may have different properties and characteristics compared to circles in Euclidean geometry. For example, the circumference of a circle may not be constant in non-Euclidean geometry.
3. How do circles in non-Euclidean geometry differ from circles in Euclidean geometry?
In non-Euclidean geometry, circles may have different shapes and sizes compared to circles in Euclidean geometry. The distance between points on a circle may also vary depending on the type of non-Euclidean geometry being used.
4. What are some real-world applications of circles in non-Euclidean geometry?
Circles in non-Euclidean geometry have applications in fields such as physics, engineering, and computer graphics. They are used to model curved surfaces and to solve problems involving non-Euclidean spaces, such as the movement of objects in a curved universe.
5. How is the area of a circle calculated in non-Euclidean geometry?
The area of a circle in non-Euclidean geometry is calculated using the appropriate formula for the specific type of non-Euclidean geometry being used. For example, in hyperbolic geometry, the area of a circle is given by A = 2πsinh(r), where r is the radius of the circle.
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