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-x^{2}-y^{2})).
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-x^{2}-y^{2})) 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-x^{2}-y^{2})) and N(0,0,1)
Calculated: V=<-x,-y,1-sqrt(1-x^{2}-y^{2})>
Parametric Equations; X_{1}=x-tx, Y_{1}=y-ty, Z_{1}=sqrt(1-x^{2}-y^{2})+t(1-sqrt(1-x^{2}-y^{2}))
Z_{1}=0, t=sqrt(1-x^{2}-y^{2}) / (sqrt(1-x^{2}-y^{2})-1) then X_{1}=x-x*(sqrt(1-x^{2}-y^{2}) / (sqrt(1-x^{2}-y^{2})-1)) or F(x,y)=x-x*(sqrt(1-x^{2}-y^{2}) / (sqrt(1-x^{2}-y^{2})-1))
Is this the valid way to solve for the function F(x,y) for the x-coordinate?