Question about Normal to an Ellipsoid?

[Omid82]

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 /////

 

[DrPhil]

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: $\displaystyle x^2/a^2 + y^2/a^2 + z^2/b^2 - 1 = 0$

Normal vector: $\displaystyle \vec{n} = (2x/a^2)\ \hat{x} + (2y/a^2)\ \hat{y} + (2z/b^2)\ \hat{z}$

Let $\displaystyle (X, Y, Z)$ be a point on the ellipsoid. The equation of the normal line through that point is

........$\displaystyle (2X/a^2)\ (x - X) + (2Y/a^2)\ (y - Y) + (2Z/b^2)\ (z - Z) = 0$

........$\displaystyle (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - \left[ X^2/a^2 + Y^2/a^2 + Z^2/b^2\right] = 0$

........$\displaystyle (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - 1 = 0$

Can you show that that line intersects the z-axis?

 

[Omid82]

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: $\displaystyle x^2/a^2 + y^2/a^2 + z^2/b^2 - 1 = 0$

Normal vector: $\displaystyle \vec{n} = (2x/a^2)\ \hat{x} + (2y/a^2)\ \hat{y} + (2z/b^2)\ \hat{z}$

Let $\displaystyle (X, Y, Z)$ be a point on the ellipsoid. The equation of the normal line through that point is

........$\displaystyle (2X/a^2)\ (x - X) + (2Y/a^2)\ (y - Y) + (2Z/b^2)\ (z - Z) = 0$

........$\displaystyle (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - \left[ X^2/a^2 + Y^2/a^2 + Z^2/b^2\right] = 0$

........$\displaystyle (X/a^2)\ x + (Y/a^2)\ y + (Z/b^2)\ z - 1 = 0$

Can you show that that line intersects the z-axis?

Thank you , but for showing that this line intersects the z axis , i should put x=y=0 ? Thank you again.

 

[DrPhil]

Thank you , but for showing that this line intersects the z axis , i should put x=y=0 ? Thank you again.

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 ±b, what are the limits for z? Does that satisfy the theorem that the point of intersection is "on the minor axis" of the ellipsoid?