Mapping Math: Introduction
This chapter of the Free GIS Book explains some of the math that is encountered when you work with digital maps, hard copy or paper maps, and other types of spatial data. In this chapter you will learn about the measurements that are used to create spatial data. This includes the math used to make these measurements and to manipulate them. After this we will talk about triangles and basic trigonometry. This knowledge will be important when we learn about 2D, or two-dimensional, coordinate geometry. (Raw measurements are converted to coordinate geometry, a form of information that can be used by most mapping programs.) We will extend our discussion of coordinate geometry to include 3D, or three-dimensional, coordinate geometry. Then we will talk in detail about map scale and some of the math involved in the layout of hard copy maps. We will finish the chapter by looking at some of the math behind a simple map projection.
This chapter does not include instruction on all of the math you will need to work with maps and other types of spatial data. It is not a math book. It does not cover in detail basic arithmetic or algebra. It does provide comprehensive coverage of basic trigonometry and basic coordinate geometry, but it is not an exhaustive treatment of these subjects. I've left lots of room for improvement and additional material.
You need to have several things with you to benefit fully from this chapter of the Free GIS Book. You'll need a pencil and some paper. You'll also need a calculator that has the ability to calculate the trigonometric functions sin, cosine, and tangent. (I'll explain where to find these buttons on your calculator in the section of this chapter that deals with trigonometry.) Your calculator should also be able to compute square roots. You should have a knowledge of basic arithmetic and a knowledge of basic algebra. (This means you should be comfortable adding, subtracting, multiplying, dividing and solving simple equations.)
Let me explain why I am taking the time to write this chapter of the Free GIS Book. I have personally found that there is a big difference between people that understand math, and people that can teach math in a way that is easily understood by others. It seems like there are more people in the first category than there are in the second. I struggled with math as a boy. This was partly due to my own abilities, but partly due to teachers that had little sympathy for those to whom math did not just “make sense”. My introduction to algebra as a young teenager was a surprise, because it was the first type of math I had encountered that I didn't despise. Even so, I struggled with algebra as I had all the other types of math before it. I would consistently score lower in math than in any other subject on standardized tests. I am telling you this because I want you to know that I understand what it is like to fight math and loose the brawl. I am finally winning this fight in my own life, and I now use math every single day. I hope this background will help me to be a better math teacher, and will make this a better chapter of the Free GIS Book.
I wanted to write some instructions on the math used in mapping that could be understood by all types of people, even by those who found math difficult. I personally think that traditional math books are horrible creatures that should be forever banned from the classroom. (My two favorite math books are “Idiot's Guide To Calculus” and “Calculus for Dummies.) I have also found that there is not a great deal of material available about the math used by mapping professionals, and certainly none as freely available as the content of the Free GIS Book. This frustration has been shared with me by other surveyors and Gis professionals. I hope this chapter will help to fill that void.
I found that I really enjoyed math and its ability to solve problems when I could see how it was practically applied in land surveying. That transition from purely concept to concept and application made a big difference for me as a math student. I want to show others how math concepts are applied to the making of maps and spatial data. Perhaps others will come to love math as I have when they see this.
All authors taint their written works with a bias or slant that is as unique as they are as individuals. I am no exception to this rule. I want to provide you with a little bit of information about myself, so that you can recognize and and understand my bias when you read this chapter of the Free GIS Book. I'm sure this bias will creep in despite by efforts to remain objective. This will also help the reader understand why some other mapping professionals may disagree with my style or protions of my material, and be justified in doing so.
First of all, the reader should understand that my day job is as a Land Surveyor. (I'm not a licensed land surveyor just yet. I'm still in training.) As I mentioned earlier, I use math in my work every day, and in my world trigonometry rules as King. I deal mostly with the making and analyzing measurements, and with the act of creating spatial data. As a result most of the material in this chapter focuses on the math used in those processes. Land Surveyors also tend to be a little more particular about the accuracy and precision of spatial data than other mapping professionals, and this may pickyness may be evident in this chapter occasionally.
Secondly, I should let you know that most of my experience with GIS is as a programmer or software developer. (Although I have embarked on a couple of projects that I hope will help to remedy this.) This means my expertize in the math behind spatial analysis will be a little weaker. (What a great opportunity this would be for a fellow mapping professional with a noble heart to step in as a contributing author of this chapter...) I still understand the concepts of this area of GIS and may be brave enough to talk about it in additional sections of the chapter. We'll have to see how things go.
Finally, I must tell you that I'm not a highly educated man by the standards of “modern society”, for lack of a better term. I learned about land surveying and GIS at my community college in Montana, but did not have any “higher” education after that. All of my other knowledge has been gained through practical experience on the job or by self instruction and volunteer projects. I personally think that “higher” education can be overrated, and that practical experience and self instruction can be just as, or more, beneficial. At any rate, I have no fancy degrees, diplomas, certifications, or titles, and I don't have any letters that follow my name. I thought I would warn those among my readers who feel that is important.
Although I hope it appeals to all kinds of people, I am targeting 2 types of people specifically with this chapter. The first type of person is new to land surveying, GIS, and mapping. The second type of person may have some experience in one of these professions, but would like to tackle more of the math behind the work they do. For everyone else, this material will hopefully serve as a good review.
I'd like to finish up with some words about the style and format of this chapter. There hasn't been a lot of material written for the Free GIS Book at this point in time, and I feel like a pioneer somewhat. I have an idea of what direction I want to head, but now definitive guide on how to get there.
The style of this chapter will be very informal. I've already mentioned how I feel about stuffy and mechanical math text books, and I definitely won't be writing a chapter that would be accepted by one of those publishers. I'm going to go out of my way to do the following in this chapter:
[1] Show every single step when solving a math problem.
[2] Clearly define and explain unfamiliar terms.
[3] Clearly indicate and highlight important terms and formulas.
[4] Explain the concept and practical application of the math, not just have you memorize rules and formulas.
[5] Use lots of diagrams and pictures.
The informal style I hope to use does not mean that I want to have mistakes or errors. I want this chapter to be as grammatically and technically correct as possible. I also don't want any math mistakes. (Yes, I do make math mistakes and I have seen them in other math books.) If you find a mistake, please send an e-mail to sunburned.surveyor@gmail.com and let me know about it. I'll get it fixed as soon as I can. While we are on the subject, let me also add that you are welcome to send me suggestions and ideas for the chapter. Also do not hesitate if you want to help me with the writing of the chapter, or with translation into a language other than English.
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