Two Dimensional Coordinate Plane with Distance as Third Dimension

In summary, the conversation discusses the creation of a three-dimensional space by adding a third dimension to a two-dimensional finite plane. The resulting shape is a rectangular solid with a pyramid removed from one end if the plane is a square, or a cylinder with a cone-shaped cut-out if the plane is a circle. The height of the object can either be a constant maximum distance or vary according to the maximum distance between points. The one-dimensional case can be visualized as a cone, and the two-dimensional case can be visualized as a scalloped box with parabolic sides. The significance of this "state space" with position and distance is questioned.
  • #1
crastinus
78
9
TL;DR Summary
What shape will this 3-D object have?
Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have?

I've worked it out some myself, but I don't think I quite understand how to do it in the best way. Obviously, the resulting shape is some kind of rectangular solid. What I get when I think about this is a rectangular solid with a pyramid removed from one end.

How would it be different if we made our finite plane a circle? Then the resulting 3-D object would certainly be a cylinder of some kind.

Thanks.
 
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  • #2
Possible distances between any points range from 0 to sqrt(2) times the length of your square, and from 0 to the diameter of your circle. Does that mean the object just has a constant height equal to that maximal distance? Or does every point get a height according to its maximal distance to other points in the object (making a pyramid out of the square and a cylinder with a cone-shaped cut-out out of the circle)? Or something else?
 
  • #3
If you first consider a one dimensional case along the x-axis then each x value would be its distance from zero. Plotting the x,dist on on an xy plot would give the line y=x

Extending to your 2D case is equivalent to rotating the y=x about the y-axis giving a cone. Considering you have a square then youll get scalloped box in 3D where the scallops are parabolic from the definition of cutting a cone with a plane parallel with its central axis along the y direction. The planes are the sides of the square which become the sides of the box when extended in 3D.

Did i say that right?
 
  • #4
Thanks for the responses! I just keep thinking about it.

This is a sort simple "state space" with just position and distance. I wonder if there is any significance to it.
 

1. What is a Two Dimensional Coordinate Plane with Distance as Third Dimension?

A Two Dimensional Coordinate Plane with Distance as Third Dimension is a mathematical concept that extends the traditional two-dimensional coordinate plane to include a third dimension, which represents distance or depth. This allows for the representation of three-dimensional objects in a two-dimensional space.

2. How is the third dimension, distance, represented in this coordinate plane?

The third dimension, distance, is represented by adding a numerical value to each point on the two-dimensional coordinate plane. This value is typically represented by a number or letter, and it indicates the distance of that point from the plane or origin.

3. What is the purpose of using a Two Dimensional Coordinate Plane with Distance as Third Dimension?

The purpose of using this type of coordinate plane is to simplify the representation of three-dimensional objects in a two-dimensional space. It allows for easier visualization and analysis of complex objects and their relationships.

4. How is this coordinate plane used in scientific fields?

This coordinate plane is commonly used in fields such as physics, engineering, and geography to represent three-dimensional objects and their relationships in a two-dimensional space. It is also used in computer graphics and mapping applications.

5. Are there any limitations to using a Two Dimensional Coordinate Plane with Distance as Third Dimension?

While this coordinate plane is useful for representing three-dimensional objects in a two-dimensional space, it does have limitations. It cannot accurately represent curved or irregular objects, and it may not be suitable for precise measurements or calculations.

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