Measurements and Scale

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Measurements and Scale

• Intro to Mapping Science

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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

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Linear and Areal

• Linear measured distance along a line
• Areal size of a portion of a surface

4
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.

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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

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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"

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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)

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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.

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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

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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.

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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.

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Why the metric system?

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

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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.

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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.

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SI Prefixes

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

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SI Prefixes

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

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SI Prefixes

• The metric system is incredibly easy to convert
  between different scales of measurement.
• Just move the decimal point.

.
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Defining areas

• Imperial
• Square feet
• Acres
• Square miles
• Metric
• Square meters
• Hectares
• Square kilometers

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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

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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.

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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.

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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.

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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

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Areal SI Prefixes
1 Meter

• 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 Decameter
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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.
• 124,000
• one inch on the map equals 24,000 inches (2,000
  feet) in the real world.

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Small and Large Scale

• Small scale maps are maps where the
  representative fraction is very small.
• example 11,000,000
• Features on small scale maps are small.
• Large scale maps are maps where the RF is
  relatively large.
• example 11,200 (one inch to 100 feet)
• Features on large scale maps appear large.

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Map Scale
things look large at large scale
Large
124,000
1500,000
13,000,000
Small
things look small at small scale
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Scale Specification Methods

• Representative fraction.
• Verbal or word statement.
• Graphic.

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Representative Fraction

• Ratio between distance on the map and distance on
  the Earth.
• 11 Very large scale (Steven Wright)
• 11,000 Large scale or 1/1,000
• 124,000 Medium scale
• 11,000,000 Small scale or 1/1,000,000
• Large scale versus small scale maps, a common
  point of confusion.

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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 11,000

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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 XX (Earth Distance /
  Map Distance)
• RF 1 (Earth Distance / Map Distance)OR RF
  1 X

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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 14,497,957

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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 doesnt! Use "x to y" instead.

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Comparing Verbal and RF

• 1 cm to 10m
• 11,000
• 1cm to 100m
• 110,000
• 1cm to 1km
• 1100,000
• 1cm to 20km
• 12,000,000

• 1 in to 600ft
• 17200
• 1in to 2,000ft
• 124,000
• 1in to 1mi
• 163,360
• 1in to 4mi
• 1253,440

metric's much easier to remember.
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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!

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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?

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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.

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Why do we worry about scale?
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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'.

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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.

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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 ?

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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'.

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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.

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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)

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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

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Generalization Algorithms

• Douglas-Poiker line simplification
• Lines and polygon boundaries
• Computer is never perfect
• Line linked to known geography
• Broken ring