What Are the Methods to Determine Surface Normal Direction at a Point?

[physiker99]

How do I find "surface normal direction" for a plane on a point with coordinates specified?

 

[ponjavic]

Do you have a plane equation or do you only have the coordinates of a point? Please be more specific.

 

[physiker99]

i need to find normal directions for r^2=9 and x+y+z^2=1 at the point (2,-2,1)

 

[gabbagabbahey]

i need to find normal directions for r^2=9 and x+y+z^2=1 at the point (2,-2,1)

First, neither of those surfaces are planes. The first is a spherical shell, and the second is a paraboloid.

Second, straight from wikipedia:

If[/PLAIN] a surface S is given implicitly as the set of points $(x,y,z)$ satisfying $F(x,y,z)=0$, then, a normal at a point $(x,y,z)$ on the surface is given by the gradient:

$$\mathbf{N}=\mathbf{\nabla}F$$

since the gradient at any point is perpendicular to the level set, and $F(x,y,z) = 0$ (the surface) is a level set of $F$.

I'd be shocked if this wasn't in your notes or textbook, and I strongly suggest you review the section of your text that covers this.

Anyways, just like wikipedia implies, to find a surface normal at a point, define $F$, take the gradient (in the appropriate coordinate system) and plug in the point.

 

[paranormal]

The surface normal direction for a plane on a point can be found by using the mathematical concept of vectors. A vector is a mathematical representation of a direction and magnitude. In this case, the surface normal direction represents the direction perpendicular to the plane at the specified point.

To find the surface normal direction, we can use the cross product of two vectors. The first vector should be the vector from the specified point to another point on the plane. The second vector should be a vector that lies on the plane. The cross product of these two vectors will result in a vector that is perpendicular to both vectors, which represents the surface normal direction.

Another way to find the surface normal direction is by using the gradient of the plane's equation at the specified point. The gradient is a vector that points in the direction of the steepest increase of a function. In this case, the plane's equation would represent the function, and the gradient at the specified point would represent the surface normal direction.

In summary, to find the surface normal direction for a plane at a specified point, one can use the cross product of two vectors or the gradient of the plane's equation at that point. Both methods will result in a vector that represents the direction perpendicular to the plane at the specified point.