Types of Graphs

I hate to break it to you, but all graphs are not created equal. I realized this recently, wrote an article, and had my computer crash. The article vanished, but I’m determined to get this idea on the web. It’s a simple but interesting concept.

Types of Graphs

Let’s start with the types of graphs. Just as the universe can be divided into “bananas” and “non-bananas”, I’m going to divide graphs into “map” and “non-map” graphs.

  • “Map graphs” are ones where the x and y axes correspond to distance: feet, meters, whatever you want. Thus, an x-y coordinate represents a location of some sort.
  • “Non-map graphs”, or all other graphs, are simply graphs where the coordinates do not imply distance, and instead the axes indicate production, time or some other variable. Some non-map graphs are hybrid graphs where one axis is distance and the other is not.

We’ll see that these graphs types are not the same and it’s important to know the difference.

Graph Examples

Consider this typical example of a map graph:

map_graph_regular.gif

Suppose we want to get from the proverbial point A (1, 1) to point B (4, 5). Let’s say there’s some cost associated with the trip: time spent, gas used, or any other expense you want. Our goal is to minimize this expense by driving as little as possible.

We know the shortest distance between two points is a straight line. Using Pythagoras’ theorem, we figure out this distance is 5 units (making a 3-4-5 right triangle).

If we decided to drive along in a triangle, going straight across to (4,1) and then from (4,1) to B, it would be a distance of 7 units. Obviously, going from A to B directly is the least expensive path.

Now consider this typical “non-map” graph showing the number of cars and trucks at a rental company:

purchase_graph_paths.gif

Suppose the company wants to increase its inventory. Currently it is at point A, with one car and one truck. It wants to get to point B (4 cars, 5 trucks). The cost of the “trip” is simply the cost of buying the new cars and trucks. How should it buy its vehicles?

This isn’t a very hard problem. We can see that no matter what “path” they take to B, the cost is the same — they end up buying 3 cars and 4 trucks. They can buy all their cars then all their trucks, vice-versa, or anything in between. They can even buy more cars and trucks than they need and sell them (assuming they can get the same price), which gives us the windy path. If they buy all of one type and then the other, they get a path shaped as a triangle.

Now, why are these two examples different? The simple reason is that distance does not matter in non-map graphs. Quite simply, if a map doesn’t have distance on its axes, then “distance” in the graph does not matter at all. Pythagoras’ theorem does not help us. Knowing that the shortest distance between two points is a line does not help us. We can’t escape the fact that we must end up with 4 cars and 5 trucks. But is this the end of the story?

Nope, it’s not the end of the story

It appears that in map graphs we want to take a direct route. In non-map graphs, we are stuck and can’t take a shortcut. Now consider this map graph, which shows terrain like mountains:

map_graph_mountains.gif

We still want to get from A to B, but now we must cross the mountains to do so. Suppose that in the mountains, the cost of traveling is 10 times normal. Suddenly the direct route doesn’t make as much sense. Taking the triangle route along the sides (from A to (4,1) to B) will be cheaper.

The lesson: On distance graphs where the cost varies from point to point, the cheapest path may no longer be a straight line. This enters the realm of math and physics problems, where you are measuring distance, but there is another effect (like a field) that changes the cost from point to point. Future articles will talk about this more.

Advanced example: Light waves

A great example of a hybrid graph (distance on one axis but not the other) is how light waves are usually portrayed. Here’s an example of a wave from Wikipedia:

Wave.png

It’s a great image, but has the potential for confusion. Light waves don’t travel up and down like a rollercoaster, but it’s very easy to think that after looking at the graph.

In fact, what the graph is saying is that as the wave moves along, its displacement (strength) varies (the amplitude is the maximum value of the displacement). The light itself can take the same path. To show this, I added a line to the graph where the brightness (displacement) varies:

wave_amplitude_line.png

Notice how the brightness of the bar corresponds to the displacement of the wave. At the peaks it is white (full brightness = max strength in positive direction), and in the valleys it is black (full darkness = max strength in negative direction). In the middle, at the zero points, the displacement is “neutral” and has no strength (by the way, our distinctions of what’s “positive” and “negative” are arbitrary; we just have to pick a definition and stick to it).

That is the concept the graph is trying to convey, but it can be tough to get our brain to look at the graph and not think of a wave as wiggling up and down like a roller-coaster.

Sure, waves can wiggle up and down (just like most everything else in the universe), but that’s not the point of the diagram. The point is that the strength wiggles up and down, even for things moving in a straight line.

For you science geeks, yes, the true and mysterious nature of light appears to be an interesting particle-wave duality, where the strength of a particle varies with distance and shows other wave properties, but just hang with me for the sake of example.

Intuition

  • Distance matters on map-graphs. Although a straight line may be the shortest path, it’s not necessarily the “cheapest”. Be careful.
  • On other maps, distance on the graph doesn’t mean location. The axes don’t measure distance, so they paths along the graph don’t correspond to locations travelled. It’s easy to confuse objects taking winding paths on the graph as taking a winding path in the real world.
Kalid Azad loves those Aha! moments when an idea finally clicks. BetterExplained believes in learning through intuition, not blind memorization, and is honored to serve 250k readers each month.

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18 Comments

  1. Interesting take on graphs. After reading the title, I was hoping that you were going to explain logarithmic graphs. Maybe in a future article?

    typo: “time spend”

  2. Two cautions about your comments on representation of waves: your employment of the word “amplitude” differs from the standard technical usage. What varies is not the amplitude but what the Wikipedia figure labels “displacement.” “Amplitude” is generally used to denote the maximum value of the displacement (or electric field strength or whatever other quantity is “waving”). That’s why the figure shows amplitude by drawing a line under a peak, rather than some other arbitrary location.

    The other quibble I have is that when I think of that graph as representing a classical light wave (in which case we replace “displacement” with “electric field”) I consider BOTH the peaks and valleys to be where the field is “strongest.” Suppose we’re looking at linear-polarized light and the vertical axis is some component of the electric field. The difference between the peaks and valleys is NOT that the field is weaker in the valleys, it’s that it has the opposite direction. If I were to say the field is “weakest” anyplace it would be where the wave intersects the horizontal axis. But even then I have to remember that the magnetic field component (directed either into or out of the page) is at its greatest magnitude for those points.

  3. Hi John, thanks for the comment! I agree with both points, I was being loose with the terminology. Amplitude (max) vs. displacement (current) is a good distinction.

    Also, I agree light is not “weakest” in the valleys — it’s simply the most negative (measured with respect an arbitrary positive/negative distinction). I’ll make the corrections in the article — appreciate the feedback!

  4. I would like to hear about logarithmic graphs as well. And what is the criterion for using log-log graph paper vs. semi-log graph paper? Thank you very much for your article on the demystifying of the natural logarithm, by the way.

  5. @Anonymous: Thanks for the comment, that’s a good idea for a future article. You’re welcome for the ln article, it was fun to write.

  6. You should probably include some simple statistical graphs such as frequency polygon graphs, histograms, ogives, and so on.

  7. Damn, the buttons are sensitive. I’m not even done.

    You should probably include some analogies for the light wave graph, if not for all of your “lessons”.

    E.g, “where the strength of a particle varies with distance and shows other wave properties,” kind of like how uncontrollable or varied it is the amount of force acted while you are running. Left leg could stomp hard, while the right leg use much less than that, and moves on like a cycle. The reverse could also happen “, but just hang with me for the sake of example.”

    There. I know there is a flaw somewhere, though…

  8. I use triangles to determine position to create a scale map, not x-y coordinates. If there is one known distance on a landscape, say it is from point A to point B, and all landscape objects can be seen from point A and from point B, then a scale map can be drawn. Draw point A and B on paper, ratio of distance from point A to point B and line length of A-B on drawing will become scale of map. Imaginery line between landscape objects to point A and to point B will determine objects position on drawing or map. No need protractor, nor compass. No need to visit and measure landscape objects distance. Please check below address:

    http://maruzar.blogspot.com/2011/12/drawing-simple-scale-map-by-triangle.html

  9. For the mountain example:

    You can just think about a x, y map where you need to minimise your COST function which will depend from x and y (thus C(x,y) ). Therefore you could show a graph with only x,y axis that represents where the function lives (the spatial points you want to physically be in) and after that a 3D graph of the hypothetical cost function with it’s minimum cost path.

  10. Things moving in straight line can also have varying strength/propery…. Nice. Thanks Kalid.

  11. this information is not helpful at all it only tells you is what a line graph is and you call your selves better explanied

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