# Ex 2.2 Stereographic Projection

Functional Differential Geometry

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The points on the plane can also be specified with polar coordinates $\displaystyle{(\rho, \theta)}$ and the points on the sphere are specified both by Riemann coordinates and the traditional colatitude and longitude $\displaystyle{(\phi, \lambda)}$.

(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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