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

.

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

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