# Thread: Point in Tetrahedron test

1. ## Point in Tetrahedron test

Hi,

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

2. 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.

3. 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

4. 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.

5. 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?

6. 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.

7. 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

8. 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?