Two Dimensional Coordinate Plane with Distance as Third Dimension

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Summary:

What shape will this 3-D object have?

Main Question or Discussion Point

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.
 

Answers and Replies

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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?
 
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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?
 
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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.
 

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