- #1
JyN
- 28
- 2
Why aren't there trigonometric functions for elliptic geometry? There is trigonometry for circles and hyperbolas, but why not ellipses?
Elliptic Trig: Circles, Hyperbolas & Ellipses
- Context: Graduate
- Thread starter JyN
- Start date
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- Tags
- Trigonometry
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Discussion Overview
The discussion revolves around the existence and nature of trigonometric functions in the context of elliptic geometry, particularly focusing on why there are established trigonometric functions for circles and hyperbolas but not for ellipses. Participants explore the potential for trigonometric-like relationships in non-square hyperbolas and ellipses, as well as the implications of defining angles in elliptic contexts.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions the absence of trigonometric functions for ellipses, noting the existence of such functions for circles and hyperbolas.
- Another participant suggests that trigonometry exists only for the square hyperbola and the circle, implying a limitation based on geometric properties.
- A participant raises the question of whether trigonometry-like relationships could exist for non-square hyperbolas and ellipses.
- There is speculation about the definition of an elliptic angle, considering the complexities introduced by the major and minor axes of an ellipse.
- One participant mentions the availability of various papers on elliptic trigonometry, suggesting that there is ongoing research in this area.
- A later reply introduces Jacobi elliptic functions as a potential answer to the initial question about trigonometric functions in elliptic geometry.
Areas of Agreement / Disagreement
Participants express differing views on the existence and utility of trigonometric functions for ellipses, with no consensus reached on the matter. Some acknowledge the potential for elliptic trigonometry, while others question its practicality.
Contextual Notes
The discussion highlights the complexities involved in defining angles and relationships in elliptic geometry, particularly due to the dual-axis nature of ellipses. There are unresolved questions regarding the applicability and usefulness of any proposed elliptic trigonometric functions.
- #2
tiny-tim
Science Advisor
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Hi JyN! 
ah, there is trigonometry only for the square hyperbola (ie with perpendicular asymptotes), just as there is only for the "square" ellipse (ie the circle).
JyN said:
ah, there is trigonometry only for the square hyperbola (ie with perpendicular asymptotes), just as there is only for the "square" ellipse (ie the circle).
- #3
JyN
- 28
- 2
Why can't trigonometry-like relationships exist for non-square hyperbolas and ellipses?
- #4
tiny-tim
Science Advisor
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i expect they can be, but why would anyone bother with them, when the "square" functions can easily be adapted for the purpose? 
- #5
waht
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An angle in radians is defined as the length of the arc of a circle over its radius.
A slight problem occurs if you want to extend that to an ellipse because an ellipse is defined by two variables, major axis and minor axis. It would be interesting how to define an elliptic angle, length of arc of an ellipse over major axis or minor axis or their algebraic combination? But in either case, I don't know how this could be useful other than perhaps in an elliptical coordinate system.
A slight problem occurs if you want to extend that to an ellipse because an ellipse is defined by two variables, major axis and minor axis. It would be interesting how to define an elliptic angle, length of arc of an ellipse over major axis or minor axis or their algebraic combination? But in either case, I don't know how this could be useful other than perhaps in an elliptical coordinate system.
- #6
waht
- 1,502
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If you google elliptic trig there is lots of different papers on it.
Here's one that might be of interest:
http://www.und.edu/instruct/schwalm/MAA_Presentation_10-02/handout.pdf
Here's one that might be of interest:
http://www.und.edu/instruct/schwalm/MAA_Presentation_10-02/handout.pdf
- #7
Petek
Gold Member
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