# Math Help - Normal vector at a point on a sphere/ellipsoid

1. ## Normal vector at a point on a sphere/ellipsoid

Hi,

Can someone please let me know how I would go about finding a normal vector at any point on a sphere/ellipsoid?

2. Originally Posted by scorpion007
Hi,

Can someone please let me know how I would go about finding a normal vector at any point on a sphere/ellipsoid?
If the (any) surface is defined implicitly by an equation of the form:

$
F(x,y,z)=0
$

then the gradient of $F$:

$
\nabla F(x,y,z)
$

defines the direction of the normal. In general there will be ambigity
in the sense of the outward and inward directions of the normal (but with
the usual forms of the equation of the sphere of ellipse you should get
an outward normal).

RonL

3. Originally Posted by CaptainBlack
If the (any) surface is defined implicitly by an equation of the form:

$
F(x,y,z)=0
$

then the gradient of $F$:

$
\nabla F(x,y,z)
$

defines the direction of the normal. In general there will be ambigity
in the sense of the outward and inward directions of the normal (but with
the usual forms of the equation of the sphere of ellipse you should get
an outward normal).

RonL
Okay, so for a sphere with equation:

$F(x,y,z) = x^2 + y^2 + z^2 - r^2 = 0$

I just need to take a derivative? How do I do that with 3 variables?

4. Originally Posted by scorpion007
Okay, so for a sphere with equation:

$F(x,y,z) = x^2 + y^2 + z^2 - r^2 = 0$

I just need to take a derivative? How do I do that with 3 variables?
Gradient not derivative. It is the vector:

$\nabla F(x,y,z)=\left[ \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right]$
$
=\left[2x,\ 2y,\ 2z\right]
$

RonL

5. Don't forget to divide by its length

(...u said normal!)

6. Originally Posted by Rebesques
Don't forget to divide by its length

(...u said normal!)
He actually said "a normal", but the point is well worth making anyway

RonL