The Model Represented by CurvedLand

One way to understand curved space is through embedding. Embedding is the process of representing a curved space by showing its mathematical equivalent in the context of a higher-dimensional space; for example, you could imagine that some 2-dimensional curved space could be modeled as a curved surface in a 3-dimensional flat space.

While embedding is extremely useful, because it allows people to use physical intuition to understand how objects behave in curved space, it has some limitations. Embedding inspires the idea that curved spaces must exist in higher-dimensional flat spaces; it makes us think of curved space as simply a byproduct of a space being bent in a higher dimensional flat space. However, this is not the case; a space can simply be curved without existing in a higher dimensional space. In fact, some types of curved space cannot be shown using the embedding technique; one example of this is space of constant negative curvature.

2D Space
of Constant
Positive Curvature
The applet has been created to try to show curved space without the embedding model. The applet shows 2-dimensional curved space of constant positive curvature; in terms of embedding, this is mathematically equivalent to the surface of a sphere. Positive curvature simply means that lines that appear to be parallel at one point start to converge if you move in either direction (in negative curvature, the lines would diverge). If you want to use your physical intuition, think about this situation on the surface of a globe; the longitude lines appear parallel at the equator, but they begin to converge as you move towards either pole. You can also see this in the applet: draw one great circle by turning on the trail and walking in a straight line until you get back where you started, and then move one step to the side and walk another great circle; you start out walking parallel to the original line, but eventually, the two paths cross each other.

The applet shows the 2-dimensional space in the same way that a mapping program would show a town or state. You see the shape and position of objects in the space as you would see them from your perspective at the center of the screen, but you have a "bird's eye view" of the whole situation. Your view is not limited to only the things you would actually physically see were you at that point in space. For example, if you were inside a building, a traditional mapping program would show you not only the inside of the building (which you could see), but also the positions of objects outside the building and down the street that you could not see from your current position. The same is true for the applet. In a 2-dimensional space of constant positive curvature, you would never see past the image of yourself visible in every direction; your line of sight will follow great circles around the spherical embedding model, and will converge back on you eventually. Nevertheless, the applet allows you to do so by taking you outside of the space and giving you a bird's eye view.

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Copyright (C) 2010 Stephanie Erickson, Gary Felder
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