Point inside a tetrahedron with vectors

[Alettix]

Homework Statement

As part of a longer problem:
"Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c."

Homework Equations

I am not sure... vector addtion?

The Attempt at a Solution

I don't really know how to attack this problem. I was thinking that all points inside the tetrahedron must be able to be decomposed into vectors along a,b,c provided that these are not coplanar (which they cannot be if they form a tetrahedron):
r = αa + βb + γc
However, some restriction must be placed on the coefficients. I think 0≤α+β+γ≤1 is a condition, but it is not sufficient (imagine for instance |a| >> |b| and let α=-1.5 and β=2, clearly r can get outside the tetrahedron).

I feel very lost regarding this problem and appreciate any help and guidance. :)

 

[BvU]

r = αa + βb + γc

Is a good start.
To find conditions for the coefficients, practice with a triangle, e.g in the x,y plane ... r = αa + βb

 

[Ray Vickson]

Homework Statement

As part of a longer problem:
"Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c."

Homework Equations

I am not sure... vector addtion?

The Attempt at a Solution

I don't really know how to attack this problem. I was thinking that all points inside the tetrahedron must be able to be decomposed into vectors along a,b,c provided that these are not coplanar (which they cannot be if they form a tetrahedron):
r = αa + βb + γc
However, some restriction must be placed on the coefficients. I think 0≤α+β+γ≤1 is a condition, but it is not sufficient (imagine for instance |a| >> |b| and let α=-1.5 and β=2, clearly r can get outside the tetrahedron).

I feel very lost regarding this problem and appreciate any help and guidance. :)

Any point ##\vec{r}## in the tetrahedron can be written as ##\vec{r} = p_0 \vec{0} + p_a \vec{a} + p_b \vec{b} + p_c \vec{c}## with ##p_0, p_a, p_b, p_c \geq 0## and ##p_0 + p_a + p_b + p_c = 1##. In other words, a necessary and sufficient condition is that ##\vec{r} = p_a \vec{a} + p_b \vec{b} + p_c \vec{c}## with ##p_a,p_b,p_c \geq 0## and ##p_a + p_b + p_c \leq 1##.

If some ##p_0, p_a,p_b,p_c## is < 0 or > 1 the point ##\vec{r}## is not in the tetrahedron.

Se, eg.,
http://mathworld.wolfram.com/ConvexHull.html

 

[BvU]

PF doesn't provide full answers but helps folks find them by themselves?

 

[haruspex]

PF doesn't provide full answers but helps folks find them by themselves?

Yes, Ray's post was rather a major assist, but I am not sure whether it is what the question is after. The existence of three parameters satisfying certain constraints is hardly a straightforward test to apply.

 

[Ray Vickson]

PF doesn't provide full answers but helps folks find them by themselves?

I agree, but my post still left quite a bit for the OP to do. That being said, I did wonder myself whether I said too much.

The problem is that the basic topic involved is that of "convexity", and most of the on-line sources to not do a good job of dealing with the OP's problem at an introductory level; they often deal with geometric rather than algebraic aspects. Many of the sources either do not deal with the OP's problem at all or else deal with it as an advanced computer science topic, and not as introductory material.

 

[Alettix]

Is a good start.
To find conditions for the coefficients, practice with a triangle, e.g in the x,y plane ... r = αa + βb

From geometric considerations in a triangle, I can justify the constraints described by Ray. How do I know that this can be generalised to several dimensions? And is there a way of proving this purely algebraically?

 

[haruspex]

ow do I know that this can be generalised to several dimensions

For a proof, you have to start from a definition of the interior of a polytope. The link Ray provided defines convex hull as points which can be expressed in that way. If you can think of a more fundamental definition, you could try proving it from there, but it seems reasonable to me to go with that definition of convex hull and say that is what you mean by the interior volume.

 

[Alettix]

For a proof, you have to start from a definition of the interior of a polytope. The link Ray provided defines convex hull as points which can be expressed in that way. If you can think of a more fundamental definition, you could try proving it from there, but it seems reasonable to me to go with that definition of convex hull and say that is what you mean by the interior volume.

All of this sounds very exciting, but is probably well above first year physics undergrad maths, isn't it? I think I shall return to this question when I have met more of these concepts in my education. Thank you for your help! :)

 

[haruspex]

All of this sounds very exciting, but is probably well above first year physics undergrad maths, isn't it? I think I shall return to this question when I have met more of these concepts in my education. Thank you for your help! :)

Not sure, certainly not beyond 1st year undergrad maths.
You were very close to this with your first post. You just had to take your observation about α =-1.5 further and discover that any negative coefficient would be a problem.
As to whether this constitutes the answer to the question posed, I am not sure. As I wrote, it does not offer a prescriptive test. Given the corners of a tetrahedron and a vector r, how would I go about determining whether such coefficients exist? To take it further, you might need to figure out how to calculate such coefficients.