Thread Subject:

Angle between plane and its normal

Hi!
I was operating on the triangulation of the holes in a stl file. There i need to compute if the normals of the triangulated facets are pointing outside the plane( away from the body), or inside the plane. In mathematical terms, i need to find if a normal is making an angle of 90 or -90, with its surface. The surface is 3 dimensional and may be closed, so the function needs to be generalized. In matlab i found no function, that can tell me that the angle is 90 or -90 (considers the sense of traversing the angle). If anyone knows, please help.

"Archak Goel" wrote in message <jqcau4$rg3$1@newscl01ah.mathworks.com>...
> Hi!
> I was operating on the triangulation of the holes in a stl file. There i need to compute if the normals of the triangulated facets are pointing outside the plane( away from the body), or inside the plane. In mathematical terms, i need to find if a normal is making an angle of 90 or -90, with its surface. The surface is 3 dimensional and may be closed, so the function needs to be generalized. In matlab i found no function, that can tell me that the angle is 90 or -90 (considers the sense of traversing the angle). If anyone knows, please help.

This is impossible to tell, in the sense that a planar surface
has no preferred positive or negative direction in terms of
it as a simple plane.

However, in context, a plane can be construed to have a
positive or negative direction for the normal. For example,
if you are generating the plane from a pair of vectors, thus
using a cross product (and therefore the right hand rule),
the normal vector will have a sign that depends on the
order of the two vectors supplied.

I'd suggest that what you usually want here is to know the
orientation of the normal vector. That orientation can be
inferred rom a dot product. If a given point is known to be
on a specific side of the plane, then you can infer a direction.
For example, suppose you have a three dimensional simplex,
defined by four vertices, {A,B,C,D}.

The normal vector N to the plane that contains the facet ABC
is given by

   N = cross(A-B, A-C)

where cross is the cross product. Does this normal vector
point inwards or outwards from the simplex? A simple solution
is given by the dot product

   dot(D-A,N)

If this dot product is positive, then the facet normal vector
is inward pointing relative to the simplex. If it is negative,
then the normal vector points outwards.

All of this never required any explicit computation of any
angle.

John

"John D'Errico" <woodchips@rochester.rr.com> wrote in message <jqd5l1$a4b$1@newscl01ah.mathworks.com>...
> "Archak Goel" wrote in message <jqcau4$rg3$1@newscl01ah.mathworks.com>...
> > Hi!
> > I was operating on the triangulation of the holes in a stl file. There i need to compute if the normals of the triangulated facets are pointing outside the plane( away from the body), or inside the plane. In mathematical terms, i need to find if a normal is making an angle of 90 or -90, with its surface. The surface is 3 dimensional and may be closed, so the function needs to be generalized. In matlab i found no function, that can tell me that the angle is 90 or -90 (considers the sense of traversing the angle). If anyone knows, please help.
>
> This is impossible to tell, in the sense that a planar surface
> has no preferred positive or negative direction in terms of
> it as a simple plane.
>
> However, in context, a plane can be construed to have a
> positive or negative direction for the normal. For example,
> if you are generating the plane from a pair of vectors, thus
> using a cross product (and therefore the right hand rule),
> the normal vector will have a sign that depends on the
> order of the two vectors supplied.
>
> I'd suggest that what you usually want here is to know the
> orientation of the normal vector. That orientation can be
> inferred rom a dot product. If a given point is known to be
> on a specific side of the plane, then you can infer a direction.
> For example, suppose you have a three dimensional simplex,
> defined by four vertices, {A,B,C,D}.
>
> The normal vector N to the plane that contains the facet ABC
> is given by
>
> N = cross(A-B, A-C)
>
> where cross is the cross product. Does this normal vector
> point inwards or outwards from the simplex? A simple solution
> is given by the dot product
>
> dot(D-A,N)
>
> If this dot product is positive, then the facet normal vector
> is inward pointing relative to the simplex. If it is negative,
> then the normal vector points outwards.
>
> All of this never required any explicit computation of any
> angle.
>
> John


Thanks John for your explanation. That means I have to work for an algorithm to generate the points , respect to which I can determine the normal direction.