# Thread: Finding F(x,y) for the x-coordinate and y-coordinate of a point on the xy plane

1. ## Finding F(x,y) for the x-coordinate and y-coordinate of a point on the xy plane

Hello all,

I have a problem that has left me a bit confused, any help or suggestions would be greatly appreciated.

Given a Unit sphere where D={(x,y,z)|x=|1|,y=|1|,0<=z<=1}. A space line projects from the north pole through the sphere and to the xy plane.

1)Find the equation of the line that goes through the general point (x,y,sqrt(1-x2-y2)).

2)Find the equation of the line that goes through the point (1/sqrt(2), 1/2, 1/2)

2)Find the function f(x,y) that gives the x-coordinate of the point on the plane.

3)Find the function g(x,y) that gives the y-coordinate of the point on the plane.

4)Look at the partial derivatives of f and g. What do they represent?

The method that I have been using is, determining the parametric equations of the line with the given general point (x,y,sqrt(1-x2-y2)) and the point where the line projects from the unit sphere N(0,0,1) "North Pole". From the determined parametric equations I assume that in order to find F(x,y) for the x-coordinate we plug "0" in for "z" as the point lies on the xy plane, then solve for "t", then using the found value for "t" I plug that in for "t" in the parametric equation for the x-coordinate and am left with F(x,y)?

Given:P(x,y,sqrt(1-x2-y2)) and N(0,0,1)

Calculated: V=<-x,-y,1-sqrt(1-x2-y2)>

Parametric Equations; X1=x-tx, Y1=y-ty, Z1=sqrt(1-x2-y2)+t(1-sqrt(1-x2-y2))

Z​1=0, t=sqrt(1-x2-y2) / (sqrt(1-x2-y2)-1) then X1=x-x*(sqrt(1-x2-y2) / (sqrt(1-x2-y2)-1)) or F(x,y)=x-x*(sqrt(1-x2-y2) / (sqrt(1-x2-y2)-1))

Is this the valid way to solve for the function F(x,y) for the x-coordinate?