# Point in Tetrahedron test

• Sep 11th 2006, 03:26 AM
scorpion007
Point in Tetrahedron test
Hi,

How would I go about determining whether a point P is enclosed by a tetrahedron A,B,C,D?
• Sep 11th 2006, 08:24 AM
ThePerfectHacker
This is what I do not understand.
You are given a tetrahedron and asked to determine if a point is inside?
You cannot, because it can be inside or outside.
~~~~
Or are you given a certain tetrahedron surface defined as,
f(x,y,z)=C
And you are given P(x_0,y_0,z_0)
And asked to determine if P is inside f?
If so,
It looks very similar to that polygon test (whatever it is called) for 2-d.
• Sep 11th 2006, 08:45 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
This is what I do not understand.
You are given a tetrahedron and asked to determine if a point is inside?
You cannot, because it can be inside or outside.
~~~~
Or are you given a certain tetrahedron surface defined as,
f(x,y,z)=C
And you are given P(x_0,y_0,z_0)
And asked to determine if P is inside f?
If so,
It looks very similar to that polygon test (whatever it is called) for 2-d.

You are given a tetrahedron with vertices A, B, C, D, and you want to know
if the point P is inside the tetrahedron.

one way of doing this is given at PNPOLY - Point Inclusion in Polygon Test - WR Franklin (WRF)

RonL
• Sep 11th 2006, 08:48 AM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlack
You are given a tetrahedron with vertices A, B, C, D, and you want to know
if the point P is inside the tetrahedron.

I do not have a general method to do this.
But I would create 4 linear inequalities involving x,y,z
And see if P satisfies all 4.
Again this is not a general method.
You need to adjust it to problem to problem.
• Sep 11th 2006, 06:45 PM
scorpion007
I guess I'm looking for a way to determine this using simple vector calculations (cross/dot products).

What if point P was always (0, 0, 0) would that make the calculation easier?
• Sep 12th 2006, 02:09 AM
Quick
Quote:

Originally Posted by scorpion007

What if point P was always (0, 0, 0) would that make the calculation easier?

If that's the case, then all you have to do is see if the vertices are in different quadrants.
• Sep 12th 2006, 05:00 AM
CaptainBlack
Quote:

Originally Posted by scorpion007
Hi,

How would I go about determining whether a point P is enclosed by a tetrahedron A,B,C,D?

Choose and arbitary unit vector u and consider the ray P+a u, if this meets
the faces at only one internal point (of a face) for positive a then P is an
internal point of the tetrahedron, if it meets an edge or a vertex, then
perturb the unit vector to avoid this.

(Such a ray from an external point will meet the faces either twice or zero
times)

RonL
• Sep 17th 2006, 09:38 PM
malaygoel
Quote:

Originally Posted by scorpion007
Hi,

How would I go about determining whether a point P is enclosed by a tetrahedron A,B,C,D?

Are the points given in a coordinate system?