Hi,

Please have a look on the attachment. It shows a graph for x^2 - z^2 = 1. Could you please help me to understand that why the variable y is absent from the equation altogether although from the graph it's clear that y is not zero? Thank you.

 

although from the graph it's clear that y is not zero

Ever? Are you sure?

 

Ever? Are you sure?

Yes, it is zero at the point where hyperbolic cylinder cuts thru the x-axis.

 

Yes, it is zero at the point where hyperbolic cylinder cuts thru the x-axis.

Ok. So there is a solution when y == 0. Are there solutions for y != 0?

Asked a different way: can you think of values of y where there'll be no solution?

 

Sorry, I'm confused. I'm not seeking any solutions. In other words, I was trying to say that I cannot envision how the shown graph could be generated using the equation x^2 - z^2 = 1. Thank you.

 

Sorry, I'm confused. I'm not seeking any solutions. In other words, I was trying to say that I cannot envision how the shown graph could be generated using the equation x^2 - z^2 = 1. Thank you.

The surfaces in the graph are the "solutions". Are there any solutions dependent on a particular value of y?

 

Thank you.

I think that I get your point. The surfaces shown in the graph do not represent the graph for equation x^2 - z^2 = 1. If we take x and z coordinates of each point (x,y,z) on the surfaces, they will satisfy the equation x^2 - z^2 = 1. Do I have it right?

 

I think the better way to look at it is to look at the xz plane (set y = 0). Now pick the y=1 plane. Does the graph look ANY different than the y = 0 plane? Or the y = -27.54 plane? Or ANY plane that is parallel to the xz plane?

It is analagous to taking an equation line y = mx + b and plotting in the x,y,z space. Notice that there's no z in the equation, but that doesn't prevent us from plotting y = mx + b in 3-D space on a set of x,y,z axes.

 

Hi,

Please have a look on the attachment. It shows a graph for x^2 - z^2 = 1. Could you please help me to understand that why the variable y is absent from the equation altogether although from the graph it's clear that y is not zero? Thank you.

Hello there,

Welcome to the world of symmetry. Symmetry is an amazing quality of lots of stuff in the universe and what makes us actually be able to understand anything at all.

As strange as this might sound, imagine we have a box (in inches) 10x12x6. What allows us to get a grasp on how big this box really is. It is the ability to recognize each dimension independently knowing that the others are always constant in their given directions.

The amazing thing about symmetry is that it allows understanding things that are otherwise much harder to understand. Take that box for example, rotate it so that no measurement length is along any axis like x,y, or z. Now try to figure out how big it is by calculating the slopes and such to come up with the three linear measurements. A lot harder right?

Now with the case in point, it is a curve, so we cant find symmetry in all directions can we? Or maybe we can. But taken directly we can only find one dimension symmetry and that is along the y axis. That means that nothing interesting happens along the y axis that affects the curve in a way that can not be described by the other variables alone. Hence we have symmetry in that direction so we can describe the curve without the need for one of the variables that doesnt change anything anyway. The context however is 3d so we must recognize that there still is a third dimension and that is y.
What we might delve into is some limit of y, such as how wide the graph is, but for the lack of such a constraint we imagine y going infinitely in both directions plus and minus so we dont need to mention 'y' at all, although again this is under the context of 3d not 2d and that must be apparent.

Sometimes symmetries are not apparent from the graph or equation. In that case we look for symmetries in order to simplify things. Take your curve again for example and rotate it by 45 degrees (arbitrary rotation angle) along any axis exceot the y axis.. Now 'y' does matter. So what if we were presented with that graph instead? Would we have to include 'y'? Well, if we rotate the new graph by -45 degrees we get 'rid' of the variable 'y', and so we might be able to use that to simplify the understanding of the curve which we see in the original curve. The rotation of -45 degrees may not matter at all because when we look at things in 3d we most often do not pay attention to what direction we are looking at it from. A car for example, we describe it as a car, but we know it will look different from every angle yet we dont always need that information to know what a car is.

This is something that is used by scientists and mathematicians all the time. It's such a broad area it's hard to describe in one post. Even in curve fitting, there are ways to rotate matrixes so that we can isolate the behavior of a variable in an equation by aligning it with an axis, and even align several variables so that we can understand the effect each one has by itself, without considering the other variables. In that way we can minimize along all axes in one operation.

So the short answer is dimensional symmetry and it's consequence on mathematical simplification.

 

Thank you, everyone.

@MrAl: Your narrative was helpful as always to see things from a different perspective.