StereographicProjection - Maple Help

geom3d

 StereographicProjection
 find the stereographic projection of a point

 Calling Sequence StereographicProjection(P, P1, s)

Parameters

 P - the name of the point to be created P1 - a point s - a sphere

Description

 • Let S and N be the south pole and the north pole of the sphere s, respectively. If P1 is a point on s, then the computed point P is the stereographic projection of P1 on s to the tangent plane sp at S, i.e., P is the intersection of the line l, which passes through N and P, and sp. If P1 is a point on the tangent plane sp, then the computed point P is a point on the sphere s such that P1 is the stereographic projection of P on s to the tangent plane sp.
 • For a detailed description of the object created P, use the routine detail (i.e., detail(P))
 • The command with(geom3d,StereographicProjection) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Define the point P(4/3,4/3,4/3) on the sphere s with center at (0,0,2) and radius 2

 > $\mathrm{sphere}\left(s,\left[\mathrm{point}\left(o,0,0,2\right),2\right]\right),\mathrm{point}\left(P,\frac{4}{3},\frac{4}{3},\frac{4}{3}\right)$
 ${s}{,}{P}$ (1)

Find the stereographic projection P1 of P

 > $\mathrm{StereographicProjection}\left(\mathrm{P1},P,s\right)$
 ${\mathrm{P1}}$ (2)

Find the stereographic projection P2 of P1

 > $\mathrm{StereographicProjection}\left(\mathrm{P2},\mathrm{P1},s\right)$
 ${\mathrm{P2}}$ (3)

The points P and P2 should have the same coordinates

 > $\mathrm{coordinates}\left(P\right)=\mathrm{coordinates}\left(\mathrm{P2}\right)$
 $\left[\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}\right]{=}\left[\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}\right]$ (4)