Contour Maps

This is a contour map labeled according to acc...Calculus is filled with complexities, many of which are just plain absurd to deal with in the form of a blog, and while they may not ONLY be able to be taught in the classroom, having 20 or 120 other heads all equally confused as you are does help. Contour Maps are one of the sections that are important, but not at all difficult, so lets get to work.

You are a person, in three dimensions, so we perceive length, width and depth in a way that is pragmatic and useful, but for the sake of demonstrations you are a two dimensional person, and you are going to be looking along your surface as a three dimensional person drops through. You want to tell your friend about this, and so you have set up a series of cameras and are able to get the outline of a shape at an arbitrary time interval. Someone drops a cube through. (Oh, by the way, things are also primitive and black and white.)

CubeWhile it’s falling through you will see a square, being that you are two dimensional, but when its not you wont see a damned thing, just a while space. You have no data about this item, other than at some points in time it was perceived as a square. No matter the time you choose for your 2D photograph you will see a square of the exact same dimensions, which is pretty boring.

ShapesWhat does our 2D representation of you see when we drop a more interesting shape through your plane? Believe it or not you are doing something you may have already done in a previous class, called a trace. Over a fixed interval you take a trace of a parametric function. the two diagrams you see on the side describe two completely different shapes. In two dimensions you will see a squared dome or something similar to a table leg, the only variable here is time, and time tells you the direction its pointing as it falls through our plane. This may sound like absurdity and or complication, but it will become more useful in just a moment.


Lets look at a crummy drawing of a graph to help things. At equal intervals you will see that the displacement is not equal. The fact that it increases tells you that this is a concave down function. Since our contour maps are on a fixed time interval, assuming that the shapes are moving into the center of the squares as time goes on, you should be able to see the top right object as concave down, in the same way that our graph is concave down. The displacement in the graph on the left is equivalent to the trace, and thus the space between any two pairs of displacement lines and likewise the space between two traces is increasing.

If you look at the second one you should be able to see the table leg is becoming thinner, so its pointing up, like a stalagmite, and if time goes out you are looking at a stalactite. You can take the featured picture up top and picture it this way, you drop a mountain range through a plane and at a given time interval you take a trace of the mountains that are piercing the plane.

This is an exercise in 3D concavity and contour drawing, but its not going to end here. Next up I have an extension of vectors that may be a good read as well.