Thread: Intersections of hyperplanes

I didn't know where to post this, basically I'm trying to figure out ways of finding the intersection between to hyperplanes in a given dimension, It's easy to find the intersection of:

Two lines in the x,y plane

Two planes in the x,y,z realm

Two realms in the x,y,z,w flune

it's little bit hard (I don't know how) to find the interstection of two lines in the x,y,z realm

I thought maybe it would be easy with vector equations (because they have one equals sign)

lines in x,y,z have two equals signs

I've been trying to find the intersection of two planes in x,y,z,w (I know it won't always be possible, but it should be possible to get the equation of a line)

Are there any techniques for this type of thing

Originally Posted by wintersolstice

I didn't know where to post this, basically I'm trying to figure out ways of finding the intersection between to hyperplanes in a given dimension, It's easy to find the intersection of:

Two lines in the x,y plane

Two planes in the x,y,z realm

Two realms in the x,y,z,w flune

it's little bit hard (I don't know how) to find the interstection of two lines in the x,y,z realm

I thought maybe it would be easy with vector equations (because they have one equals sign)

lines in x,y,z have two equals signs

I've been trying to find the intersection of two planes in x,y,z,w (I know it won't always be possible, but it should be possible to get the equation of a line)

Are there any techniques for this type of thing

Just use linear algebra

Basically you have two equations

$\displaystyle a_1x+a_2y+a_3z+a_4w=a_5$ and $\displaystyle b_1x+b_2y+b_3z+b_4w=b_5$

This has system can be written as an augmented matrix

$\displaystyle \begin{bmatrix} a_1 & a_2 & a_3 & a_4 & a_5 \\ b_1 & b_2 & b_3 & b_4 & b_5 \end{bmatrix}$

Now just get this to reduced row form and you will have three free variables and the solution will have three vectors that make up your surface in 4d space. You can use this in $\displaystyle \mathbb{R}^n$ for any $\displaystyle n \in \mathbb{Z}^{+}$

You will get no solution if the 2nd row is all 0's except for the last entry.

I hope this helps

Originally Posted by TheEmptySet

Just use linear algebra

Basically you have two equations

$\displaystyle a_1x+a_2y+a_3z+a_4w=a_5$ and $\displaystyle b_1x+b_2y+b_3z+b_4w=b_5$

This has system can be written as an augmented matrix

$\displaystyle \begin{bmatrix} a_1 & a_2 & a_3 & a_4 & a_5 \\ b_1 & b_2 & b_3 & b_4 & b_5 \end{bmatrix}$

Now just get this to reduced row form and you will have three free variables and the solution will have three vectors that make up your surface in 4d space. You can use this in $\displaystyle \mathbb{R}^n$ for any $\displaystyle n \in \mathbb{Z}^{+}$

You will get no solution if the 2nd row is all 0's except for the last entry.

I hope this helps

Unfortuanalty no

first - I don't understand the use of an augmented matrix

second - All this says is how to find the intersection of two realms in x,y,z,w, I already said that was easy (I know how to do that.

Sorry

finding the intersecxtion of 2 "2D" planes in a "4D" plane (and similar) is what I want

a general way of finding any to types of plane in any dimensions

There probably isn't a universal way for all of these, just some stratigies for some might help me find more

I'm not sure I understand your question. You said the "intersection of two hyperplanes". A hyperplane in n dimensions is a linear manifold of dimension n- 1, not just any hyperplane. In particular, any hyperplane in $\displaystyle R^3$ is a plane and two planes (unless parallel) intersect in a line. A line is NOT a hyperplane in $\displaystyle R^3$ and, in general, two lines do NOT intersect in $\displaystyle R^3$.

If, by "hyperplane", you mean any linear manifold, and you are working in, say, n dimensions and want to find the intersection between a 2 dimensional linear manifold and a 4 dimensional manifold, then about the best you can do is put the equations for one into the equations for the other and try to solve. Of course, it might well happen that they do NOT intersect.

In n dimensions, you can write a 2 dimensional linear manifold as n linear equations in two parameters. You can write a 4 dimensional manifold as n linear equations in 4 parameters. Setting the coordinates equal, you will have n linear equations in 4+ 2= 6 parameters. If n= 6 then you can expect to solve those for one point of intersection. If n< 6, you will have fewer equations than parameters and will have to solve for 6- n of them in terms of the other n. That is, the intersection will be a 6- n dimensional linear subset of the 4 dimensional manifold (of course n must be greater than or equal to 4 so 6- n is less than or equal to 2). If n> 6 you will have more equations than parameters and, in general, the two manifolds do not intersect.

Originally Posted by HallsofIvy

I'm not sure I understand your question. You said the "intersection of two hyperplanes". A hyperplane in n dimensions is a linear manifold of dimension n- 1, not just any hyperplane. In particular, any hyperplane in $\displaystyle R^3$ is a plane and two planes (unless parallel) intersect in a line. A line is NOT a hyperplane in $\displaystyle R^3$ and, in general, two lines do NOT intersect in $\displaystyle R^3$.

If, by "hyperplane", you mean any linear manifold, and you are working in, say, n dimensions and want to find the intersection between a 2 dimensional linear manifold and a 4 dimensional manifold, then about the best you can do is put the equations for one into the equations for the other and try to solve. Of course, it might well happen that they do NOT intersect.

In n dimensions, you can write a 2 dimensional linear manifold as n linear equations in two parameters. You can write a 4 dimensional manifold as n linear equations in 4 parameters. Setting the coordinates equal, you will have n linear equations in 4+ 2= 6 parameters. If n= 6 then you can expect to solve those for one point of intersection. If n< 6, you will have fewer equations than parameters and will have to solve for 6- n of them in terms of the other n. That is, the intersection will be a 6- n dimensional linear subset of the 4 dimensional manifold (of course n must be greater than or equal to 4 so 6- n is less than or equal to 2). If n> 6 you will have more equations than parameters and, in general, the two manifolds do not intersect.

I'm talking about two "sub-planes" inside a larger plane (of co-ordinate space) (which I think implies the latter of your options)

Basically I know that planes don't aways intersect even when they're no parallel, but with two lines in realm space it is possible to find a point (even if they don't intersect) you substitute the point into equations to see if the point is on the lines

I wanted to know how to do something similar for higher planes and manifolds, I was aware that you use the equations and substitute them into each other, but I don't know the procedure for doing that

It doesn't matter if there "parametric" or "cartesian" (with mutiple equal signs like that of a line in realm space) I just need some form of algorithm for finding the intersections of a pair of r-planes inside a n-space (co-ordinate space) where n>r