Math For Measurements: Measuring Distances

Definition of Distance
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.

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