Measurements and Scale

Intro to Mapping Science

Overview

• Measurements– Review of linear and areal measurement

systems– Rationale behind linear and areal unit

definitions• Scale

– The Representative Fraction– Use on a map– Defining when making a map

Linear and Areal

• Linear – measured distance along a line

• Areal – size of a portion of a surface

Imperial Units

• Most of the measurement units we in the US are accustomed to are a legacy of measuring distances using the human body.

• These Imperial or English units are inches, feet, yards and miles.

Defining an Inch

• Based on barleycorns: "three grains of barley, dry and round, placed end to end, lengthwise"

• Based on the average width of a thumb at the base of the fingernail.

• 1 inch is 25.4 mm• 1 foot is 12 inches

Defining a Foot

• Anecdotal evidence points to the "foot" based on Henry I's actual foot.

• A large majority of the population has feet smaller than 1 foot in length.

• However, with footwear on, a human foot with shoe is close to 1 foot in length.

• Useful in ad-hoc measurement – "pacing"

Defining Yards and Miles

• 1 Yard = 3 feet– A yard is roughly one stride when walking

quickly.• A mile was originally 1,000 paces (2,000

steps)• 1 Mile = 5,280 feet or 1,760 yards• 1 Mile = 8 furlongs (= 660 feet)

Nautical Miles

• A nautical mile is 1 minute along a great circle. Also referred to as "1 minute of latitude along a meridian."

• Measuring distances at sea along great circle routes.

• Used in GPS tracking, polar exploration…• Any distance travelled around a sphere.• 1 nmi = 6080.20 feet.

Remember – measurement systems can be entirely arbitrary! As long as the base units are agreed upon, you could use whatever you want to measure the Earth!Some admittedly make less sense than others…

One more… Smoots.• Named for Oliver Smoot,

who was used as a measuring stick to measure the Harvard Bridge.

• A smoot is 5 feet, 7 inches - his height.

• Literally picked him up and carried him the length of the bridge.

Metric (SI) System

• Straightforward, standardized system of measurement, with defined base units.

• Decimalized units – based on powers of 10.• The base linear unit of the Metric system is

the meter.

Why the metric system?

• Easy to use• Easy to convert• Standard system used around the world.• Who doesn't use the metric system?

Meter

• Originally defined as 1/10,000,000th of the distance between the North Pole and the Equator, through Paris, France.

• Think back to UTM northings – measured north from the Equator.

• Now defined as distance light travels in 1/299,792,458 of a second.

SI Prefixes

• The metric system is based on multiples of ten.

• Units are either multiplied or divided by a multiple of ten to form a new unit of measurement.

• Prefixes for specific multiples are appended to the base unit of measurement.

SI Prefixes

• Going smaller…– deci-, centi-, milli-, micro-

1 meter = 1 meter1 decimeter = 1/10th meter

= 0.1 meter1 centimeter = 1/100th meter

= 0.01 meter1 millimeter = 1/1,000th meter

= 0.001 meter1 micrometer = 1/1,000,000th meter

= 0.000001 meter

SI Prefixes

• Going larger…– deca-, hecto, kilo-, mega-

1 meter = 1 meter1 decameter = 10 meters1 hectometer = 100 meters1 kilometer = 1,000 meters1 megameter = 1,000,000 meters

SI Prefixes

• The metric system is incredibly easy to convert between different scales of measurement.

• Just move the decimal point.

0 0 0 1 0 0 0kilo- hecto- deca- base

unit (mete

rs)

deci- centi- milli-.

Defining areas

• Imperial– Square feet– Acres– Square miles

• Metric– Square meters– Hectares– Square kilometers

Square feet

• 1 square feet is defined as a square having all sides measure 1 feet in length.

• A square foot does not have to actually be a square shape.

• Multiply the length by the width:– 1' x 1' = 1 square foot– 2' x 5' = 10 square feet– .25' x 4' = 1 square foot

Defining an Acre

• 43,560 square feet.• A square 208' 9" on each side.• One furlong (660ft) and one chain (66ft).

– Amount 1 person with one ox could effectively plough in one day.

– This application of the acre helped establish it as standard area measurement in agriculture, planning and taxation.

Square Miles

• The area enclosed by a square that measures one mile on each side.

• 27,878,400 Square Feet• 640 acres in a square mile.

Metric Areal Measurements

• Based on the meter.• 1 square meter is defined as a square

having all sides measure 1 meter in length.• Employ the same SI prefixes to define areas

at different scales.• Be aware of changes in order of magnitude.

Metric Area Units

• Square meter– a square with 1 meter sides

• Hectare– 10,000 square meters– close to acre in size (1 ha = 2.471 ac)

• Square Kilometer– 1,000,000 square meters

Areal SI Prefixes• When changing

prefixes on linear measurements, you are only concerned with one direction.

• With area measurements, you change the order of magnitude in two directions.

1 Meter1 Decameter

1 square decameter

not 10 square meters -100 square meters!

Map Scale

• The scale of the map is the ratio of a distance measured on the map to that of the distance in the real world.

• A representative fraction shows that ratio, in the same units.

• 1:24,000 – one inch on the map equals 24,000 inches

(2,000 feet) in the real world.

Small and Large Scale

• Small scale maps are maps where the representative fraction is very small.– example: 1:1,000,000

• Features on small scale maps are small.• Large scale maps are maps where the RF is

relatively large.– example: 1:1,200 (one inch to 100 feet)

• Features on large scale maps appear large.

Map Scale

Small

Large

1:3,000,000

1:500,000

1:24,000

things look large at large scale

things look small at small scale

Scale Specification Methods

• Representative fraction.

• Verbal or word statement.

• Graphic.

Representative Fraction

• Ratio between distance on the map and distance on the Earth.– 1:1 [Very large scale] (Steven Wright)– 1:1,000 [Large scale] or 1/1,000– 1:24,000 [Medium scale]– 1:1,000,000 [Small scale] or 1/1,000,000

• Large scale versus small scale maps, a common point of confusion.

Representative Fraction• This method of specification is unit free. It

matters not whether you are talking about inches, centimeters, miles, or cubits--the idea is the same: this much on the map (an inch for instance) represents that much on the ground (1,000 inches).– In which case the RF = 1:1,000

Calculating Representative Fraction• Measure map distance between two points [map units].• Measure earth distance between the same two points [earth units].• Convert all measurements to the same metric [feet, meters].• Solve:

Map Distance = 1 . Earth Distance X

X = (Earth Distance / Map Distance)

• RF = 1 : (Earth Distance / Map Distance)OR RF = 1 : X

1

2

Distance from 1 to 2

Calculation of Representative Fraction• Map units = 5.25 inches• Earth units = 372.7 miles• Convert all measure to same metric [miles to

inches]:– Earth Units to Inches = miles * 5,280 * 12

= 372.7 * 5,280 * 12 Inches = 23,614,272

• Solve: 5.25 _ = 1_ 23,614,272 XX = 4,497,956.6

• RF = 1:4,497,957

Verbal Scale

• This many units on the map represent that many units on Earth.– One inch to 20 miles.– One inch to 400 miles.

• Unless you want to sound very ungeographical, avoid saying “One inch equals twenty miles” because clearly it doesn’t! Use "x to y" instead.

Comparing Verbal and RF

• 1 cm to 10m– 1:1,000

• 1cm to 100m– 1:10,000

• 1cm to 1km– 1:100,000

• 1cm to 20km– 1:2,000,000

• 1 in to 600ft– 1:7200

• 1in to 2,000ft– 1:24,000

• 1in to 1mi– 1:63,360

• 1in to 4mi– 1:253,440

metric's much easier to remember….

Graphic Scale

• Another map element, like your legend.• Draw line on map and divide into segments

so that each segment represents a certain distance on Earth.

• Use easily comprehendible units!

100 200 300 400 500

Miles

0

Why Use Graphic Scale

• Remains accurate when map is enlarged or reduced.

• Easy to transfer map units to Earth distance to answer, “How far is A from B?”

• Easy to plot specified Earth distance on map.– Where should the stops on our trip be if we

want to drive 400 miles each day?

Graphic Scale Construction• Determine the scale metric and the scale divisions:

– Metric: miles, feet, kilometers– Divisions: number of metric units (miles, feet, etc.) in

one division of the scale.• Measure an arbitrary map distance between two points

(inches), then determine Earth distance between same two points (miles).

• Convert all measurements to same metric.• ArcMap will generate these for you – if you have multiple

map frames on a layout - make sure the scale is linked to the desired map frame.

Why do we worry about scale?

Map scale in GIS• Map scale specifies the amount of reduction

between the real world and its graphic representation. Since a paper map is always the same size, its scale is fixed when it is printed, and cannot change.

• However, a map in a GIS can be shrunk or enlarged at will on the screen or on paper. You can zoom in until the screen displays a square meter or less, or zoom out until the screen displays all of NJ. This means that geographic data in a GIS doesn't really have a 'map scale'.

Display scale

• The display scale of a map is the scale at which it 'looks right'. Because a paper map is created at certain scale, its 'map scale' and 'display scale' are the same. The display scale influences two things about a map : – The amount of detail. – The size and placement of text and symbols.

Data accuracy and uncertainty

• Scale influences accuracy. • Data accuracy is an statement of how

closely a bit of data represents the real world.– What features have been omitted ? – What non-existent features are represented ? – How correct is their classification ? – How current is the data ? – How far away is a map feature from its actual

location in the world ?

Data resolution

• Resolution is the degree to which closely related entities can be discriminated.

• Usually, it is desirable to specify the resolution of a dataset as a minimum feature size. – For example, 'no lakes of less than 5 hectares

surface area should be captured'. In a GIS, this is the most important reason for having the same data represented at different 'scales'.

Data detail

• Data detail is a measure of how much information is stored for each feature. A GIS stores lines (eg, a lake shoreline) as a sequence of point locations, and draws it with the edges that join them. There is no limit to how many points can be stored, or how close together they may be.

• The amount of detail on line features should be limited just like data density. It does not make sense to store points at intervals which are shorter than the accuracy of their locations.

Generalization• If your end result is a cartographic product,

you must ensure that your features are represented appropriately for the scale at which the map is drawn

• You will have to simplify features– Show dual carriageway as single line– Smooth outline of lakes, coastlines– Change feature type (points instead of polys)

Generalization

• GIS data may preserve data beyond what you need or want

• ArcGIS can differentiate between incredibly small values– State Plane (feet) default is 0.003937 inches

• Software may have difficulties displaying overly detailed data at smaller scales

Generalization Algorithms

• Douglas-Poiker line simplification

• Lines and polygon boundaries

• Computer is never perfect– Line linked to known

geography– Broken ring