Math For Measurements: Measuring Distances - Revision history

Neteler: Reverted edits by Ptère (talk) to last revision by Sunburned Surveyor

2012-04-10T12:53:53Z

Reverted edits by Ptère (talk) to last revision by Sunburned Surveyor

Wiki-Ptère: /* Understanding the Definition of Distance - What is a point? */

2012-04-10T12:41:10Z

Understanding the Definition of Distance - What is a point?

Wiki-Sunburned Surveyor: /* Measuring Distances */

2007-03-19T18:14:22Z

Measuring Distances

Wiki-Sunburned Surveyor at 19:25, 25 February 2007

2007-02-25T19:25:08Z

Wiki-Sunburned Surveyor at 19:17, 25 February 2007

2007-02-25T19:17:12Z

Wiki-Sunburned Surveyor at 19:57, 29 January 2007

2007-01-29T19:57:13Z

Wiki-Sunburned Surveyor at 19:56, 29 January 2007

2007-01-29T19:56:39Z

Wiki-Sunburned Surveyor: /* Understanding the Definition of Distance - What is a ''point'' */

2007-01-14T17:42:42Z

Understanding the Definition of Distance - What is a ''point''

Wiki-Sunburned Surveyor at 17:42, 14 January 2007

2007-01-14T17:42:12Z

Wiki-Sunburned Surveyor at 17:03, 14 January 2007

2007-01-14T17:03:44Z

New page

'''What Is A Distance'''

'''Horizontal Distances'''

'''Vertical Distances'''

'''Distances On The Earth’s Surface'''

Navigation

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 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 <span class="plainlinks">[http://www.jlm-diffusion.com/reseau-revendeurs-partenaires/cuisine-nantes.html<span style="color:black;font-weight:normal; text-decoration:none!important; background:none!important; text-decoration:none;">Kitchen</span>] 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.
+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.

@@ Line 27: / Line 27: @@
 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.
+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 <span class="plainlinks">[http://www.jlm-diffusion.com/reseau-revendeurs-partenaires/cuisine-nantes.html<span style="color:black;font-weight:normal; text-decoration:none!important; background:none!important; text-decoration:none;">Kitchen</span>] 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.

@@ Line 60: / Line 60: @@
 ==Measuring Distances==
-We can often use an instrument like a tape measure to measure the distance between two points on a straight line segment. It is a little trickier to measure the distance along curved line segments. You might measure that distance by rolling a wheel over the line and counting the revolutions. More frequently we use straight line measurements and some known mathematical properties of curves to measure curved distances. We will talk more about this in the section of this chapter entitiled "Measuring Distances Along Arcs and Curves".
+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.
 == Navigation ==

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 == What is a Distance? ==
-=== Definition of 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''.
@@ Line 58: / Line 58: @@
 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===
+==Measuring Distances==
 We can often use an instrument like a tape measure to measure the distance between two points on a straight line segment. It is a little trickier to measure the distance along curved line segments. You might measure that distance by rolling a wheel over the line and counting the revolutions. More frequently we use straight line measurements and some known mathematical properties of curves to measure curved distances. We will talk more about this in the section of this chapter entitiled "Measuring Distances Along Arcs and Curves".

← Older revision		Revision as of 19:17, 25 February 2007
Line 58:		Line 58:
	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.		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.

−	We can often use an instrument like a tape measure to measure the distance between two points on a straight line segment. It is a little trickier to measure the distance along curved line segments. You might measure that distance by rolling a wheel over the line and counting the revolutions. More frequently we use straight line measurements and some ~~math~~ to measure curved distances. We will talk ~~a lot~~ more about this in ~~a later~~ section.	+	===Measuring Distances===
		+
		+	We can often use an instrument like a tape measure to measure the distance between two points on a straight line segment. It is a little trickier to measure the distance along curved line segments. You might measure that distance by rolling a wheel over the line and counting the revolutions. More frequently we use straight line measurements and some known mathematical properties of curves to measure curved distances. We will talk more about this in the section of this chapter entitiled "Measuring Distances Along Arcs and Curves".

← Older revision		Revision as of 19:57, 29 January 2007
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	Now that we have spent some time describing lines and points, we can finally get to the definition of a distance.		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		Definition: Distance

← Older revision		Revision as of 19:56, 29 January 2007
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	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.		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: 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.
		+
		+	We can often use an instrument like a tape measure to measure the distance between two points on a straight line segment. It is a little trickier to measure the distance along curved line segments. You might measure that distance by rolling a wheel over the line and counting the revolutions. More frequently we use straight line measurements and some math to measure curved distances. We will talk a lot more about this in a later section.
		+

	== Navigation ==		== Navigation ==

@@ Line 6: / Line 6: @@
 '''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'' ===
+=== 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.