Functional Differential Geometry

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The points on the plane can also be specified with polar coordinates and the points on the sphere are specified both by Riemann coordinates and the traditional colatitude and longitude .

(show-expression
 ((compose
   (chart S2-spherical)
   (point S2-Riemann)
   (chart R2-rect)
   (point R2-polar))
  (up 'rho 'theta)))

~~~

1. The code

 
(up 'rho 'theta)

represents the polar coordinates of a point.

2. The function

 
(point R2-polar)

generates an abstract point from a point in R2-polar coordinates.

3. The function

 
(chart R2-rect)

gives the rect coordinates given an abstract point on the plane R2.

(show-expression
 ((compose
   (chart R2-rect)
   (point R2-polar))
  (up 'rho 'theta)))

4.

The procedure (point S2-Riemann) gives the point on the sphere given rectangular coordinates on the plane.

In other words, the function

 
(point S2-Riemann)

generates an abstract point-on-the-sphere (S2) from a point-on-the-plane (R2) in rect coordinates. In other words,

 
S2-Riemann

means

 
S2-rect

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5.

Perform an analogous computation to get the polar coordinates of the point on the plane corresponding to a point on the sphere given by its colatitude and longitude.

(show-expression
 ((compose
   (chart R2-polar)
   (point R2-rect)
   (chart S2-Riemann)
   (point S2-spherical))
  (up 'phi 'lambda)))

— Me@2023-04-22 10:42:50 PM

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2023.04.25 Tuesday (c) All rights reserved by ACHK