Hi.
I know (from many books) that normal to an ellipsoid of equation (x^2+y^2)/a^2+(z^2)/b^2=1 at any point will passes the minor axis of that ellipsoid. i just want to know how to prove it?
All Answers are Appreciated /////
Hi.
I know (from many books) that normal to an ellipsoid of equation (x^2+y^2)/a^2+(z^2)/b^2=1 at any point will passes the minor axis of that ellipsoid. i just want to know how to prove it?
All Answers are Appreciated /////
This question is beyond what I would consider to be "Geometry and Trig." To find the equation of the normal to a surface, I would find the gradient, which involves partial derivatives.
Surface: [tex] x^2/a^2 + y^2/a^2 + z^2/b^2 - 1 = 0 [/tex]
Normal vector: [tex]\vec{n} = (2x/a^2)\ \hat{x} + (2y/a^2)\ \hat{y} + (2z/b^2)\ \hat{z} [/tex]
Let [tex](X, Y, Z)[/tex] be a point on the ellipsoid. The equation of the normal line through that point is
........[tex] (2X/a^2)\ (x - X) + (2Y/a^2)\ (y - Y) + (2Z/b^2)\ (z - Z) = 0 [/tex]
........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - \left[ X^2/a^2 + Y^2/a^2 + Z^2/b^2\right] = 0 [/tex]
........[tex] (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - 1 = 0 [/tex]
Can you show that that line intersects the z-axis?
DrPhil (not the TV guy)
If we knew what we were doing,we wouldn't have to do it
Yes, and then you can see there is a perfectly well-defined solution for z, the point of intersection, as a function of Z, the arbitrary point on the ellipsoid where you constructed the normal. Since the limits of Z are ħb, what are the limits for z? Does that satisfy the theorem that the point of intersection is "on the minor axis" of the ellipsoid?
DrPhil (not the TV guy)
If we knew what we were doing,we wouldn't have to do it
Bookmarks