4d Cartesian to Polar Transform

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SUMMARY

The discussion focuses on deriving the Cartesian to polar transform for \mathbb{R}^4, building on established transformations for \mathbb{R}^2 and \mathbb{R}^3. The transformation equations provided include w = r sin(θ) sin(ψ) cos(φ), x = r sin(θ) sin(ψ) sin(φ), y = r sin(θ) cos(ψ), and z = r cos(θ). The challenge lies in visualizing angles in higher dimensions, which complicates the understanding of the transformation. Participants are encouraged to share insights or corrections to the proposed equations.

PREREQUISITES
  • Understanding of Cartesian and polar coordinates in \mathbb{R}^2 and \mathbb{R}^3
  • Familiarity with trigonometric functions and their applications in geometry
  • Basic knowledge of high-dimensional mathematics
  • Ability to visualize geometric transformations in multiple dimensions
NEXT STEPS
  • Research the mathematical foundations of high-dimensional geometry
  • Explore the concept of spherical coordinates in \mathbb{R}^4
  • Study the inverse transformations for Cartesian to polar coordinates
  • Learn about applications of polar transformations in data analysis and machine learning
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Mathematicians, physicists, and computer scientists interested in high-dimensional data analysis and geometric transformations.

chasehusky
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Howdy everyone,

I'm on a quest for something that is proving a bit elusive at the moment: a Cartesian to polar transform (along with its inverse) for \mathbb{R}^4. I'm well aware of how to derive the transform for both \mathbb{R}^2 and \mathbb{R}^3, as it is just a matter of looking at the angles made, with respect to the origin and appropriate coordinate axes, for the vector in question; e.g., for the \mathbb{R}^3 case: x = r \sin(\theta)\cos(\psi), y = r \sin(\theta)\sin(\psi), z = r \cos(\theta). Unfortunately, as with all high-dimensional spaces, visualizing these angles becomes much trickier. If anyone can help me with this, I'd greatly appreciate it.
 
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Well, after toying around a bit, it appears that the conversion would go something like: w = r\sin(\theta)\sin(\psi)\cos(\phi), x = r\sin(\theta)\sin(\psi)\sin(\phi), y = r\sin(\theta)\cos(\psi), z = r\cos(\theta).
 

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