Understanding 3D Planes & Hyperplanes: Intuition & Practical Explanation

[s3a]

In the following two problems I am trying to get a deeper intuition of, the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well. Can someone please show me why, practically, in the context of these problems?

Question with hyperplane:
Find an equation of the hyperplane H in R^4 that passes through P(3, -4, 1, -2) and is normal to u = [2,5,-6,-3].

Question with (regular) plane:
Find an equation of the plane H in R^3 that contains P(1,-3,-4) and is parallel to the plane H' determined by the equation 3x - 6y + 5z = 2.

Any input would be greatly appreciated!

 

[Ray Vickson]

In the following two problems I am trying to get a deeper intuition of, the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well. Can someone please show me why, practically, in the context of these problems?

Question with hyperplane:
Find an equation of the hyperplane H in R^4 that passes through P(3, -4, 1, -2) and is normal to u = [2,5,-6,-3].

Question with (regular) plane:
Find an equation of the plane H in R^3 that contains P(1,-3,-4) and is parallel to the plane H' determined by the equation 3x - 6y + 5z = 2.

Any input would be greatly appreciated!

What is a hyperplane in R^n? (I know the answer, but I want you to spell it out.) What would be the general form of equation for a hyperplane in R^4? What do you get if you use the given data?

If you do not know what a hyperplane is, or what is its equation, you need to fill in that background knowledge first. Consulting a textbook or course notes (or even Google) would be step 1.

 

[s3a]

Hi, Ray.

What is a hyperplane in R^n? (I know the answer, but I want you to spell it out.) What would be the general form of equation for a hyperplane in R^4? What do you get if you use the given data?

A hyperplane, H, in R^n is the set of points (x_1, x_2, ..., x_n) that satisfy a linear equation

a_1 x_1 + a_2 x_2 + ... + a_n x_n = b

where the vector u = [a_1, a_2, ..., a_n] of coefficients is not zero (and is the normal vector).

 

[Michael Redei]

That's a good definition. Now, can you use Ray's suggestion and write down the general form of an equation for a hyperplane in R⁴? Can you get the coefficients a₁,... from the data in your initial problem? How about the number b?

 

[s3a]

Sorry, I forgot to write that part!:

a_1 x_1 + a_2 x_2 + a_3 x_3 a_4 x_4 = b

I actually know how to do both problems; I'm just trying to understand what a hyperplane is and the (n – 1) dimensionality of it as well as what the differences between it and a (regular) plane are.

To answer you though,

Question with hyperplane:
2x_1 + 5x_2 – 6x_3 – 3x_4 = –26

Question with (regular) plane:
3x – 6y + 5z = 1

 

[haruspex]

the plane has 3 variables and is 3 dimensional

Not really. A plane in 3 space is essentially a 2 dimensional manifold, but interpreted as embedded in 3 dimensions. Once you take that view it all becomes consistent.

 

[s3a]

Could you elaborate please?

 

[Dick]

Could you elaborate please?

Can you elaborate on why you think "the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well". You may be thinking of dimension differently in those two cases.

 

[Bipolarity]

Another way to thing about it is that a hyperplane of n-space is an (n-1)dimensional affine subspace of n-space.

So the hyperplane of a line is a point; the hyperplane of a plane is a line; the hyperplane of ambient 3-space is a plane, the hyperplane of 4-space would be an ambient 3-space... and so on. It becomes impossible to visualize in higher dimensions, so just use the lower dimensions to visualize.

A plane in 3-space has two free variables because it is constrained by exactly one equation in the 3 variables of the space.
In the same way, a hyperplane in n-space has (n-1) free variables because it is constrained by exactly one equation in the n variables of the space.
Thus, the hyperplane, having (n-1) free variables, is an (n-1) dimensional affine subspace of n-space.

BiP

 

[s3a]

Can you elaborate on why you think "the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well". You may be thinking of dimension differently in those two cases.

I am very confused about this. I was just thinking that, for “normal” constructs, each additional variable implies an additional dimension and that the hyperplane has one less dimension than the amount of variables.[/QUOTE]

Another way to thing about it is that a hyperplane of n-space is an (n-1)dimensional affine subspace of n-space.

So the hyperplane of a line is a point; the hyperplane of a plane is a line; the hyperplane of ambient 3-space is a plane, the hyperplane of 4-space would be an ambient 3-space... and so on. It becomes impossible to visualize in higher dimensions, so just use the lower dimensions to visualize.*

A plane in 3-space has two free variables because it is constrained by exactly one equation in the 3 variables of the space.
In the same way, a hyperplane in n-space has (n-1) free variables because it is constrained by exactly one equation in the n variables of the space.
Thus, the hyperplane, having (n-1) free variables, is an (n-1) dimensional affine subspace of n-space.

BiP

I don't understand what “affine” means. Okay so, when used in an English sentence, I must “take a hyperplane of something” rather than just having a hyperplane be a hyperplane like a plane is plane?

I don't see how a plane (in three dimensions) is constrained.

Did you just imply that a plane is also an (n – 1) dimensional construct (as opposed to just a hyperplane)?

Sorry for the dumb question(s) but, I am extremely lost with this all.

 

[Michael Redei]

A plane in 3-dimensional space consists of points with three coordinates (x,y,z). Since not each combination of these three is allowed, they're "restrained" by a linear equation, and such a plane is said to have the dimension "1 less than that of 3-dimensional space", i.e. dimension 2 (not 3).

It may help to remember that such a plane is just "a" plane, and not the one you think of as having just two coordinates (x,y) for each point. Each such plane in 3-dimensional space has all other properties of "the" plane, but they are not the same.

As to hyperplanes: they are just subsets of any n-dimensional space restricted by one linear equation. So in 4-dimensional space, a hyperplane has dimension 3, but in 17-dimensional space it would have dimension 16 etc.

And in 3-dimensional space, a hyperplane is any set of points satisfying one linear equation. This is, again, just "a" plane.

If you consider "the" plane with points (x,y), a linear equation describes a line, which is a hyperplane in the plane.

So, in a nutshell: "Hyperplane" doesn't mean "hyper = bigger (in dimension) than a plane", but "like a plane is to 3-dimensional space, but hyper = generalised to any number of dimensions".

 

[haruspex]

I don't understand what “affine” means.

It means linear, but not necessarily through the origin.
If y = mx, we can say y is proportional to x, but if y = mx+c then it's an affine relationship.

I don't see how a plane (in three dimensions) is constrained.

Any relationship between co-ordinates constitutes a constraint. In 2D, the set of points satisfying the constraint x²+y²≤a² is a disc. The set of points satisfying the constraint x²+y²=a² is a circle. Mathematically, a circle in 2D is considered a one dimensional "manifold". If you are constrained to be on it, then at any given point on it there is only ever one dimension you can move in. We can only picture it in two or more dimensions, so it's natural to think of it as being 2 dimensional, but really that's because our ability to visualise is largely limited to embedding things in Euclidean space. In the same way, a 2-dimensional plane can be embedded in Euclidean 3-space, maybe orthogonal to an axis, maybe not, but it remains in essence a 2D object.

 

[s3a]

I believe that my confusion is more simple than all that. If the answer to my upcoming question below is “Yes.” then, I've overcome the trouble I've had when I initially started this thread.

Is it 100% correct to say that a hyperplane is just a term referring to any (n – 1) dimensional object in an n dimensional space whereas a (regular) plane is simply a hyperplane with the specific case of n = 3?

(Please provide a strict “Yes.” or “No.” prior to any potential elaborations.)

 

[Bipolarity]

I believe that my confusion is more simple than all that. If the answer to my upcoming question below is “Yes.” then, I've overcome the trouble I've had when I initially started this thread.

Is it 100% correct to say that a hyperplane is just a term referring to any (n – 1) dimensional object in an n dimensional space whereas a (regular) plane is simply a hyperplane with the specific case of n = 3?

(Please provide a strict “Yes.” or “No.” prior to any potential elaborations.)

Yes, with the added condition that the "object" must flat (or affine), i.e. linear.
If the object is not flat, it is not really considered a hyperplane.

For example, a sphere is a (3-1)dimensional object in 3 dimensional place. But it is not a hyperplane of 3-space, because it is not flat. This is obvious from the fact that the equation of a sphere is not linear.

But a plane in 3-space is indeed a hyperplane because it is flat, i.e. defined by a linear equation.

BiP

 

[Michael Redei]

Yes, with the added condition that the "object" must flat (or affine), i.e. linear.
If the object is not flat, it is not really considered a hyperplane.

That applies only to affine spaces, of course. In non-affine spaces the concept of "flatness" makes little sense.

 

[s3a]

Are hyperplanes with no adjective specified implicitly assumed to be affine? What I mean by this, if it's unclear, is that Wikipedia, for example, mentions affine hyperplanes, vector hyperplanes and projective hyperplanes, would a hyperplane automatically be assumed to be affine if an underlined part from above is not specified?

I would also like to understand/visualize how a(n) (affine) hyperplane separates the particular space into two half-spaces.

According to Wikipedia, it's above or below the constant term in the hyperplane equation but, I can't visualize that. (I'm referring to the two- and three-dimensional spaces since I expect to not be able to visualize four dimensions and more.)

 

[haruspex]

Are hyperplanes with no adjective specified implicitly assumed to be affine? What I mean by this, if it's unclear, is that Wikipedia, for example, mentions affine hyperplanes, vector hyperplanes and projective hyperplanes, would a hyperplane automatically be assumed to be affine if an underlined part from above is not specified?

Projective hyperplanes are not Euclidean hyperplanes. They include a presentation of the 'points at infinity'. Certainly an unqualified hyperplane should be assumed to be affine.

I would also like to understand/visualize how a(n) (affine) hyperplane separates the particular space into two half-spaces.

According to Wikipedia, it's above or below the constant term in the hyperplane equation but, I can't visualize that. (I'm referring to the two- and three-dimensional spaces since I expect to not be able to visualize four dimensions and more.)

An affine hyperplane is defined by v.x = a, where v is a given vector and the dot indicates dot product. It separates the space into the regions v.x < a and v.x > a. Any continuous path from a point x₁ satisfying v.x₁ < a to a point x₂ satisfying v.x₂ > a necessarily passes through some point x for which v.x = a

 

[Bipolarity]

Are hyperplanes with no adjective specified implicitly assumed to be affine? What I mean by this, if it's unclear, is that Wikipedia, for example, mentions affine hyperplanes, vector hyperplanes and projective hyperplanes, would a hyperplane automatically be assumed to be affine if an underlined part from above is not specified?

I would also like to understand/visualize how a(n) (affine) hyperplane separates the particular space into two half-spaces.

According to Wikipedia, it's above or below the constant term in the hyperplane equation but, I can't visualize that. (I'm referring to the two- and three-dimensional spaces since I expect to not be able to visualize four dimensions and more.)

I don't know about higher maths, but in linear algebra, my professor usually means "affine" hyperplane when he says hyperplane since most of my linear algebra deals with linear equations. In other branches of math, it is probably ambigious to just say hyperplane.

z = 0 represents the xy-plane, which is a hyperplane of [itex]ℝ^{3}[/itex]. Above this plane is all the points in [itex]ℝ^{3}[/itex] for which z>0. On the other hand, below this plane is all the points in [itex]ℝ^{3}[/itex] for which z<0. In the equation z=0, the 0 is the constant term.

BiP

 

[Michael Redei]

Projective hyperplanes are not Euclidean hyperplanes. They include a presentation of the 'points at infinity'. Certainly an unqualified hyperplane should be assumed to be affine.

That depends on the circumstances. When you're talking mainly about projective spaces, projective lines, projective planes etc., you'll usually omit the word "projective", since it's understood. And so a mere "hyperplane" would be taken to be a projective one.

 

[Michael Redei]

An affine hyperplane is defined by v.x = a, where v is a given vector and the dot indicates dot product. It separates the space into the regions v.x < a and v.x > a.

That only works if your scalar field is ordered, i.e. ##\mathbb Q## or ##\mathbb R##.

 

[haruspex]

That only works if your scalar field is ordered, i.e. ##\mathbb Q## or ##\mathbb R##.

Good point. So if the field is ##\mathbb C##, a hyperplane could not be said to separate the space into two regions. In one dimension, a point on the real line or in the complex 'plane' is a hyperplane, but only fragments the space in the real case.

 

[Michael Redei]

Good point. So if the field is ##\mathbb C##, a hyperplane could not be said to separate the space into two regions. In one dimension, a point on the real line or in the complex 'plane' is a hyperplane, but only fragments the space in the real case.

This is proving to be most thought-provoking. What you're saying is that the set ##\mathbb C##, which we draw on a sheet of paper as if it were a "plane", is really just a line (a one-dimensional vector space over the complex numbers).

Now I wonder whether the following construction is new (and/or just nonsense):

If we take the complex plane ##\mathbb C^2 = \{(z_1,z_2) : z_1,z_2\in\mathbb C\}##, a hyperplane will resemble ##\mathbb C_{z0} := \{(z,0)\in\mathbb C^2\} = \{(z_1,z_2)\in\mathbb C^2:z_2=0\}##. This hyperplane doesn't divide the plane ##\mathbb C^2## into two parts.

Looking at ##\mathbb C^2\cong\mathbb R^4## and ##\mathbb R_{xx00} := \{(x_1,x_2,0,0)\in\mathbb R^4\}##, we know that ##\mathbb R_{xx00}## isn't a hyperplane in ##\mathbb R^4## and so can't divide this space in two either (but for a different reason).

However, ##\mathbb R_{xxx0} := \{(x_1,x_2,x_3,0)\in\mathbb R^4\}## is a hyperplane in ##\mathbb R^4## and does split ##\mathbb R^4##. So we could look at ##\mathbb C_{zx0} := \{(z,x) : z\in\mathbb C, x\in\mathbb R\}##. This isn't a hyperplane in ##\mathbb C^2##, and it isn't even a vector-subspace of ##\mathbb C^2##, but it does split ##\mathbb C^2## in two, doesn't it?

Is there a name for such sets like ##\mathbb C_{zx0}##? I'd be tempted to call it a 1.5-dimensional subspace of ##\mathbb C^2##.

 

[Dick]

This is proving to be most thought-provoking. What you're saying is that the set ##\mathbb C##, which we draw on a sheet of paper as if it were a "plane", is really just a line (a one-dimensional vector space over the complex numbers).

Now I wonder whether the following construction is new (and/or just nonsense):

If we take the complex plane ##\mathbb C^2 = \{(z_1,z_2) : z_1,z_2\in\mathbb C\}##, a hyperplane will resemble ##\mathbb C_{z0} := \{(z,0)\in\mathbb C^2\} = \{(z_1,z_2)\in\mathbb C^2:z_2=0\}##. This hyperplane doesn't divide the plane ##\mathbb C^2## into two parts.

Looking at ##\mathbb C^2\cong\mathbb R^4## and ##\mathbb R_{xx00} := \{(x_1,x_2,0,0)\in\mathbb R^4\}##, we know that ##\mathbb R_{xx00}## isn't a hyperplane in ##\mathbb R^4## and so can't divide this space in two either (but for a different reason).

However, ##\mathbb R_{xxx0} := \{(x_1,x_2,x_3,0)\in\mathbb R^4\}## is a hyperplane in ##\mathbb R^4## and does split ##\mathbb R^4##. So we could look at ##\mathbb C_{zx0} := \{(z,x) : z\in\mathbb C, x\in\mathbb R\}##. This isn't a hyperplane in ##\mathbb C^2##, and it isn't even a vector-subspace of ##\mathbb C^2##, but it does split ##\mathbb C^2## in two, doesn't it?

Is there a name for such sets like ##\mathbb C_{zx0}##? I'd be tempted to call it a 1.5-dimensional subspace of ##\mathbb C^2##.

Your ##\mathbb R_{xxx0} := \{(x_1,x_2,x_3,0)\in\mathbb R^4\}## isn't a subspace of C^2. It's not closed under scalar multiplication by elements of C.

 

[haruspex]

the set ##\mathbb C##, which we draw on a sheet of paper as if it were a "plane", is really just a line (a one-dimensional vector space over the complex numbers).

Yes, the distinction between the complex 'plane' and 2D Euclidean space often confuses people.

we could look at ##\mathbb C_{zx0} := \{(z,x) : z\in\mathbb C, x\in\mathbb R\}##. This isn't a hyperplane in ##\mathbb C^2##, and it isn't even a vector-subspace of ##\mathbb C^2##, but it does split ##\mathbb C^2## in two, doesn't it?

Yes, and in the same way you can draw a line in the complex 'plane', but its equation doesn't have a nice formulation in complex variables. It involves treating the real and imaginary parts separately.

 

[Michael Redei]

Yes, and in the same way you can draw a line in the complex 'plane', but its equation doesn't have a nice formulation in complex variables. It involves treating the real and imaginary parts separately.

What's "nice" though? I quite like ##\{(z_1,z_2)\in\mathbb C^2 : \overline{z_2}=z_2\}##.

 

[haruspex]

What's "nice" though?

As in analytic.

 

[HallsofIvy]

Yes, planes and hyperplanes are necessarily flat. That is part of the definition. The sphere [itex](x_1- a_1)^2+ (x_2-a_2)^2+ \cdot\cdot\cdot+ (x_n- a_n)^2= r^2[/itex] in Rⁿ is an n-1 dimensional surface but is not a "hyperplane". A hyperplane, like a line in two dimensions or a plane in three, extends to infinity and has two sides. You cannot get from a point on one side to a point on the other side without crossing it.

 

[s3a]

When looking here ( http://en.wikipedia.org/wiki/Hyperplane#Affine_hyperplanes ), what is b exactly?

I don't just mean "a constant". For example, in R^2, let's assume (1)(x_1) + (1)(x_2) = (1)(x) + (1)(y) = 0, the space is separated to x + y > 0 and x + y < 0. What exactly is above or below 0? When dealing with z > 0 or z < 0 (from a previous post), it makes sense to me that it means more in the positive direction of the z axis or more in the negative direction of the z axis but, what about in the example I've given above?

(I get it geometrically now but, I just want to get the number/inequality thing I asked above.)

 

[Michael Redei]

Earlier in this thread you defined a hyperplane H to be the set of all vectors x with vx = b for some scalar b. The vector v here is orthogonal to H. If you look at the line of all points sv with scalars s, you'll find that it meets H in a point P, and both P and H lie at a distance of |b| from the origin. If b > 0, then all points x with vx < b lie on the same side of H as the origin does, and all points x with vx > b lie on the side that doesn't contain the origin.