Normal vector at a point on a sphere/ellipsoid

**scorpion007** · September 1st 2006, 10:00 PM

Hi,

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

**CaptainBlack** · September 1st 2006, 10:41 PM

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

**scorpion007** · September 1st 2006, 10:48 PM

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?

**CaptainBlack** · September 1st 2006, 11:48 PM

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

**Rebesques** · September 2nd 2006, 01:19 AM

Don't forget to divide by its length

(...u said normal!)

**CaptainBlack** · September 2nd 2006, 03:34 AM

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

Thread: Normal vector at a point on a sphere/ellipsoid

LinkBack

Thread Tools

Display

Normal vector at a point on a sphere/ellipsoid

Similar Math Help Forum Discussions

Finding the normal vector and the distance of point P to the plane

Using a normal vector and point to find distance and the closest point

Normal vector at a point on a surface

Finding point on sphere furthest from point in space

Line Intersection Point on Ellipsoid

Search Tags