Math For Measurements: Measuring Distances

== What is a Distance? ==
=== Introduction ===
Before we can talk about measuring distances and the role of distance measurements in map maling and the creation of spatial data, we need to define exactly what a distance is. Before we can do this we need to define and understand some other terms. These terms all have to do with geometry and you will see shortly how they are related to the definition of distance.
Note: Most text books on mathematics provide very dry and abstract definitions for mathematical terms. I am going to try to avoid this whenever possible. If I do give an abstract definition I will always try to follow it with a more concrete explanation, example, or illustration. That will hopefully help you understand the abstract definition.
=== Understanding the Definition of Distance - What is a point? ===
To understand the definition of a distance, we need to first understand the definition of a point.
Definition: Point
A single location in two dimensional or three-dimensional space that has no width, length, or height.
This definition is as about as abstract as you can get. Let's see if we can make some more sense of it with further explanation and an illustration.
Alternate Definition: Point
A point is simply a way to represent or symbolize a single spot in a system that humans have designed to measure off or partition some space they are interested in. 
Points are used to symbolize all sorts of things on maps. We might use a point to represent the location of an automobile accident, or the confluence of two rivers. Typically the things we represent on maps with a point really do have a width, height, and length, but it is easier to treat them as if they don't. For example, a city might stretch for miles over the surface of the Earth, and for hundreds of feet above and below the surface of the Earth. But on many types of maps at a large scale we might represent the same city with a singe point. (We will talk more about this concept in later sections of this chapter that deal with map scale.) This is known as modeling the real world, and it is something we do a lot when we create maps. For more information, see the “extra information” section on “Modeling With a Map”. 
Because points don't really have a width, length or height, we usually decorate them with some type of symbol or picture on our map. (For example, on treasure maps pirates marked the location of their loot with an “x”. You might also see a city represented by a solid or hollow circle.) The decisions about what type symbol or picture is used to represent a map is part of the science and art of cartography.
Note: Our definition of a point used the word “space”. You'll see that a lot of our other definitions that have to do with geometry use this word as well. Space is another abstract concept that can be hard to wrap your mind around. Humans are interested in all types of spaces. This could be a vast and immense space, like the Milky Way Galaxy, or a really small space, like the surface of a microchip. 
You can basically think of space as something that can be filled with things. 
For exmaple, outer space can be filled with stars, planets, and space dust. (It's called outer space because it is outside our planet.) The space in a kitchen might be filled with a stove, a dishwasher and a pantry. Space isn't always completely filled with things. It usually has a lot of “emptiness” where nothing is located. We like to call this emptiness empty space. Outer space has a lot of empty space. For example, there can be immense distances of empty space between planets or starts. Even a kitchen has some empty space. If it didn't, you wouldn't be able to walk from the pantry to the stove! Often times on a map we are interested in depicting how “things” fill up the “empty space”. To help us do this we often divide space up into a grid or other reference system that makes it easier to represent the locations of things in space. We will talk more about this in the section in this chapter on spatial reference systems. For know you just need to know that a point represents a single spot in this type of system that humans use to easily divide up or partition space.
A final illustration can help you to understand what a “point” is. If you were to take a blank sheet of paper, lay it on a hard surface like a table, and then lightly tap on it in a single spot with a pen or pencil the very small mark you would leave could be thought of as a point.
===Understanding the Definition of Distance – What is a line?===
To understand the definition of a distance, we also need to understand the definition of a line.
Definition: Line
An infinite series of connected points along definite path.
Alternate Definition: Line
A line is simply a way to represent a “path”, for lack of a better word. If you placed a pen or pencil on a piece of paper and moved the writing instrument the mark left on the paper would be a line.
We use lines to symbolize a variety of geographic features on maps. For example, you could use a line to represent a road, a river, the path of a vehicle, or the shoreline of a lake. It is important to remember that lines we commonly see on maps are a representation of a real world objects. Although a line may be styled with a certain width on a map for purposes of appearance, the reality is the shape of the feature represented by the line can be quite different. You can represent a river as a line, but a real river has width and depth. It is a three-dimensional object and not really a single line.
Technically, a line has no endpoints. An example of this is the line that represents the Earth’s axis of rotation. It has no start point or end point and can be thought of as extending forever into outer space from the top and bottom of the planet. This is a somewhat abstract concept, and most lines that we deal with in maps and GIS have a start point and end point. A line that has a definite starting and ending point is called a line segment. Typically people refer to line segments as “lines” and often times the terms “lines” and “line segments” are used interchangeably. I will do my best to avoid this practice in this chapter of the Free GIS Book, for the sake of clarity.
It is also important to note that lines and line segments are not always “straight”. Lines and line segments can be curves and portions of curves. For example, an arc of a circle could be called a line segment. We’ll call line segments that are “straight” straight line segments, and portions of curves and arcs curved line segments. Straight line segments and curved lines segments share common properties, like length and starting and ending points.
Now that we have spent some time describing lines and points, we can finally get to the definition of a distance.
===Definition of a Distance===
Definition: Distance
The length along a line or line segment between two points on the line or line segment.
What does it mean when we say "two points on the line or line segment"? How is a point "on" a line? If the line we mention in the definition of a distance represeneted a road, and the point "on the line" represented a huge boulder the size of your house, you would run smack into the boulder if you were driving on the road towards the point. If the point was "off of the line" you would dirve right by it. (You might stop to take pictures of it though.) There are mathematical tests that we can use to determine if a point is on or off of a line. I won't talk about this now. I'll save it for myself or another author in a future chapter on topology and spatial relationships.
==Measuring Distances==
Distances may be measured by the map maker in two primary ways. The map maker may measure distances directly, using some type of measuring instrument or observation, or they may measure distances indirectly by performing calculations using other direct measurements. 
===Direct Measurements===
Direct measurements of distance are made by the map maker with some type of measuring instrument or tool. Examples of these tools are a tape measure, a measuring wheel, or even by counting the paces a person walks between two points. Land surveyors can use a special instrument called a electronic distance meter, or EDM to measure distances by counting wavelengths of light beams that are bounced between the instrument and a target. Sonar pulses can be used to measure distances under water. Distances can also be measure in a similar way using RADAR. With the right equipment distances can be measured from aerial or satellite photographs and other remotely sensed images, and also from existing maps. (We will learn more about making measurements from maps in the section of this chapter on map scale.) 
===Indirect Measurments===
Indirect measurements of distance are based on calculations from other direct measurements. This often involves simple algebraic computations to determine the unknown distance from other known information and established mathematical principles. It can sometimes involve very complex calculations. For example, you can indirectly measure a distance between two points on the Earth’s surface established using GPS. The receivers accepting the GPS signals aren’t actually measuring the distance between the two points directly, but the distance can be computed from other direct measurements. (The distances measured from the receiver to the satellites are direct measurements.)
===Error In Distance Measurements===
It is important to remember that all measurements contain error, and this includes direct and indirect measurements of distance. For more information on the topic of measurement error, see the extra section entitled “Error in Measurments”.
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