What are the mathematical concepts used to describe angles in three dimensions?

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Discussion Overview

The discussion revolves around the mathematical concepts used to describe angles in three dimensions, contrasting them with two-dimensional representations. Participants explore the differences in angular representation and accuracy in calculations, particularly in the context of physics and mathematics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants note that angles in three dimensions can be understood by relating them to a plane oriented in those dimensions, similar to the x-y plane.
  • One participant describes the use of polar coordinates in two dimensions with theta and r, and introduces phi as an additional angle in three dimensions, referred to as the "vertical angle."
  • Concerns are raised about the accuracy of mathematical representations, particularly the loss of precision in calculations involving the inverse cosine function (acos) for angles near pi.
  • Another participant expresses surprise at the simplicity of the concepts, indicating a lack of exposure to three-dimensional angle calculations in their current coursework.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and familiarity with the topic, with some questioning the foundational knowledge of others. There is no clear consensus on the best approach to describing angles in three dimensions, and concerns about accuracy in mathematical representations are noted but not resolved.

Contextual Notes

Participants mention limitations in their current studies, such as not having covered three-dimensional angles in their coursework, which may affect their understanding and contributions to the discussion.

Nano-Passion
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We usually describe angles in two dimensions (x and y plane). What is the mathematics behind three dimensions?
 
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Nano-Passion said:
We usually describe angles in two dimensions (x and y plane). What is the mathematics behind three dimensions?
I don't get it. You're an undergrad in physics and you're asking this question? Why don't you already know?

It's really no different. When you talk about the angle between things in three dimensions, you can think of them as being on some plane oriented in those three dimensions and then think about the angle in relation to that plane, just as you would with the familiar x-y plane...
 
Actually, I'd say we usually describe angles in two-dimensions with theta and r. That is, an angle from the designated "zero line", and a distance from the origin. In three dimensions, we simply add another angle, usually phi, that I call the "vertical angle", or the angle between the line and the x-y plane.
 
Jocko Homo said:
I don't get it. You're an undergrad in physics and you're asking this question? Why don't you already know?

It's really no different. When you talk about the angle between things in three dimensions, you can think of them as being on some plane oriented in those three dimensions and then think about the angle in relation to that plane, just as you would with the familiar x-y plane...

I'm in Calculus based Classical Mechanics at the moment. We haven't gone into figuring out angles in three-dimensions.

Char. Limit said:
Actually, I'd say we usually describe angles in two-dimensions with theta and r. That is, an angle from the designated "zero line", and a distance from the origin. In three dimensions, we simply add another angle, usually phi, that I call the "vertical angle", or the angle between the line and the x-y plane.

I'm surprised its that simple. I've read around and seen that sometimes the math isn't incredibly accurate (such as near pi with using cos).

As you see, there is a very distinct loss of accuracy in 'acos' for angles
near pi. Some seven entire decimal places have been lost - that is,
errors are several million times as large as normal. On the other hand,
the angle near pi/2 yields the customary 1 in 2^52 accuracy.
 

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