# Two Dimensional Coordinate Plane with Distance as Third Dimension

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• crastinus
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.
crastinus
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.

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?

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?

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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