Curve

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An example of a curve.

In mathematics, the term curve is, generally speaking, an object similar to a straight line but which is not required to be straight or required to be a line. More precisely, a curve is a compact open unbounded pseudo-oriented mostly-connected probably-squiggly extra-homotopic inter-diffeomorphic exact-but-not-closed 1.1-dimensional submanifold of a vector space. All these terms have well defined meanings in topology and they work in perfect harmony to bring you the amazing mathematical achievement of an object similar to a straight line but which is not straight and is not a line.

Criteria for Straight Line-y-ness

Before you learn about curves, you should learn about straight lines because, as explained above, curves are straight lines that are not straight or lines.

Straight Lines are Straight

Sit down at a table. Ensure you are not drunk or high. Call Descartes and ask him if you perceptions are clear and distinct. Now look at the alleged straight line. Is is straight? If so, and if you have ensured the clarity and distinctness of your perceptions, then the alleged straight line is, in fact, a straight line. If not, then the alleged straight line is not straight or a line. Or the evil deceiver is deceiving you. Call Descartes again to make sure.

Straight Lines are Linear

You may run into a deep mathematical obstruction when applying the preceding criterion to an alleged straight line. For example, you may be sitting near a black hole that curves spacetime too much for you to be able to evaluate the straightness of the alleged straight line. Also, there is no cell phone reception near black holes so you will have difficulty contacting Descartes. So now you should try to apply the linearity criterion. The linearity criterion states that linearity implies straight line-y-ness. In order to apply this criterion, randomly pick a few thousand points on the alleged straight line (but be quick, lest you are sucked into the black hole before your task is complete!) Now if these points are collinear, then the alleged straight line has a high probability of being an actual straight line.

Straight Lines are Bouncy-Homotopic

No knows what this means, but it may turn out to be useful if all other criteria fail.

Relaxing the Conditions of Straightness and Line-y-ness

In order you rigorously relax the conditions of straightness and line-y-ness, you must define a rank-3 homotopy between curves and lines. No one knows how to do this, so topologists generally stare curves into submission. This is similar to proof by intimidation except that you are intimidating the mathematical concept itself. Recent developments have succeeded in developing nice mathematical methods for rank-1 and rank-2 homotopies, though. It may not be long until some genius comes along and develops a general rank-3 homotopy.

Rank-1 Homotopy

Push the curve around a little bit. If you can make it look like a line, proudly declare that you have derived the rank-1 homotopy between the curve and the line. If you can do this, then a deep result of topology guarantees that your curve might be sort of a pseudo-line. Maybe.

Rank-2 Homotopy

Fashion your curve into a lasso and try to catch some points. If you are in three dimensions, you will fail miserably. This is proved using (1) a closed-but-not-exact form (2) Stokes' theorem (3) cowboys. You integrate the closed-but-not-exact form around the boundary of the cowboy and note that this is congruent to the lasso homotopy, which implies cheese. You then bribe everyone with the cheese to distract them from the fact that you haven't proven anything.