Question

Three people live on the unit sphere and are going to walk from North 

Pole (0,0,1) to the South Pole (0,0,−1). The first person walks along the 

arc of a great circle that lies in on the coordinate planes. The second 

person walks along an arc of a great circle that does not lie in one of the 

coordinate planes, and the third walks along a curve that spirals once 

around the sphere. Find the parametric equations that describe possible 

paths for each person.  

 

Answer 1

Given coordinates are:

$N(0,0,1) \ to S(0,0,-1)$

For the first person he can move in a circle form N to S in XZ and YZ plane (radius of the circle)

$R=1-(-1)/2=1$

$For\ \theta=0 \ to \ \pi\ or \ 0 \ to -\pi \ to \ be \ angle\ from\ z \ axis \\ In XZ \ plane\\ x=sin\theta\\ z= cos\\ y=0\\$

In YZ plane

$y=sin\theta\\ z=cos\theta\\ x=0\\$

The second person along a circle in a plane other than XZ and YZ plane other than XZ and YZ plane perpendicular to XY plane

$Take \ the \ plane \ as \ \phi \ from \ X axis \ other \ than \ multiple \ of \pi/2\\ If z=cos\theta\\ for \theta=0 \ to \ \pi \ or \ 0 \ to \ -\pi \ to \ be \ anle \ from \ z \ axis$

A

rbitrary point represented by C

The parametric equation :

$x=sin\theta cos\phi\\ y=sin\theta sin\phi\\ z=cos\theta\\$

For the third person the path is a spiral

$Take \ \theta =0 \ to \ \pi \ or \ 0 \ to \ -\pi \ to \ be \ angle \ from \ axis$

$Then \ z=cos\theta\\ Take \ \phi=c\theta \ from \ X axis \ in \ XY \ plane \ as \ the \ rotation \ of \ spiral \ about \ Z \ axis\\$

$Since it only spiral once \phi=0 to 2\pi\\ c=2$

Since to previous case arbitrary point

$x=sin\theta cos\phi=sin\theta cos2\theta\\ y=sin\theta sin2\theta\\ z=cos\theta$