02-mapping-data.Rmd

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# Mapping Data {#mapping-data}

You probably already accept that the Earth is "round" and not "flat".
You have probably held and touched a globe at some point in your life.
But have you ever wondered how we describe location and measure
something as large as the Earth? In this chapter, we will explore
fundamental concepts for how we measure the Earth and orient ourselves
with coordinate systems.

::: {.box-content .learning-objectives-content}
::: {.box-title .learning-objectives-top}
#### Learning Objectives {.unnumbered}
:::

1.  Understand the models of Earth's figure and shape
2.  Describe different vertical datums and how they are used to
    reference height
3.  Understand the difference between cartesian, celestial, geographic,
    and projected coordinate systems
4.  Recognize the differences among major types of map projections
5.  Explore how projected coordinate systems distort and represent the
    world around us
:::

### Key Terms {.unnumbered}

Antipode, Great Circle, Small Circle, Geodesy, Vertical Datum,
Horizontal Datum, Deflection of the Vertical, Ellipsoid, Spheroid,
Geoid, Elevation, Orthometric Height, Geoid Height, Geodetic Height,
Coordinate System, Celestial Coordinate System, Cartesian Coordinate
System, Geographic Coordinate System, Projected Coordinate System, Map
Projection, Tissot's Indicatrix

## Introduction to geodesy

**Geodesy** is the fascinating science of measuring the shape,
orientation, and gravity of Earth. Naturally, some of the questions that
come to mind when thinking about such a grand topic are *I thought the
shape of Earth is a sphere?* and *How do we orient ourselves on Earth?*
and *What does gravity have to do with mapping location?*

All of these questions stem from need to represent **location**. For our
purposes, location is the position of something relative to something
else. In order to actually describe a location on Earth, we first need
to know the size and shape of Earth. Some of the first estimations of
Earth's size and shape were made by Eratosthenes, a Greek mathematician
from the second and third centuries B.C. Eratosthenes was responsible
for many concepts we use in our everyday lives:

-   Conceiving the first spherical model of Earth
-   The first accurate measure of Earth's circumference
-   Calculating the tilt of Earth's axis
-   Calculating the distance of Earth to the Sun
-   Invention of the leap day

Eratosthenes accurately calculated the circumference of Earth by
noticing how the Sun shone directly down the bottom of a well in Syene
(modern Egypt) at noon on the summer solstice. He later made a second
observation at Alexandria at noon on the summer solstice with a pole and
noticed a shadow. He measured the angle of the shadow and inferred the
circumference of Earth, which was already known to be spherical. The
calculation goes something like this:

[insert circumference calculation here]

Pretty simple, right? Turns out, Eratosthenes was off by only 75 km or
less than 0.2% in his calculation! The actual North-South circumference
of Earth is about 40,075 km. His calculation worked because the Sun's
rays are nearly parallel when they strike Earth. So if you observe the
Sun at the same time in two locations on Earth on the North-South axis,
you will notice the Sun has a different elevation above the horizon,
which means different lengths of shadows will be cast on the ground.
This is also a way to prove that the Earth is in fact round because a
flat Earth would have equally-sized shadows everywhere at any given time
of day.

## Models of Earth

Here is a simple thought experiment to consider. Suppose you are trying
to measure your own height. You probably have not given much thought
about how to technically do this because it seems intuitive: place a
measuring tape at the bottom of your feet and mark the measurement at
the top of your head. If we break this down, there are some important
rules to follow (Figure 2.2):

1.  The measuring tape must originate somewhere. In other words, we need
    to define a reference point or surface of zero height (i.e., the
    ground).
2.  The measuring tape must be a straight line and originate at a
    90-degree angle, perpendicular to the ground.
3.  The measurement must terminate at a point along an imaginary line
    that is tangential to your head, and yes, that line must be
    perpendicular to the measuring tape and also parallel with the
    ground.

![Figure 2.2 Diagram for measuring height above a
datum](images/02-height.png){.center} <br/>

Whenever you measure your height, the ground is easy to define. It is
whatever point you are standing on. This starting point it also known as
a **datum**. A datum is simply a reference point, set of points, or a
surface from which distances can be measured. It does not matter if you
are below sea level, atop Mount Everest, or on the 30th floor of a
skyscraper. You will always get an accurate and repeatable measure of
your height using a datum that is defined directly below your feet. But
what about measuring the height of terrain on Earth? Whenever we measure
the height of Earth's terrain above some reference surface, we are
measuring **elevation**.

The same rules above apply when we measure elevation. In order for
elevation measurements to be comparable across the world, we need to
define a reference surface, a datum, for the entire planet. There are
actually several ways that we can model the shape of Earth in order to
produce a datum. Models of Earth's shape are often referred to as either
vertical datums (the plural of datum) if you are referencing elevation
or horizontal datums if you are referencing location. A **vertical
datum** is a 3D surface model that is used to reference heights or
elevations for the Earth. A simple question like *How high is Mount
Logan in Yukon, Canada?* is complicated by the need for a reference
surface and the fact that Earth's shape is irregular. In this section,
we will review three types of vertical datums:

-   Geodetic - based on geometry
-   Tidal - based on sea level
-   Gravimetric - based on gravity

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#### Remember this? {.unnumbered}
:::

<p id="box-text">

Models are abstractions of reality and help us understand and
communicate complex ideas.

</p>
:::

### Geodetic vertical datums

A **geodetic vertical datum** is one that describes the Earth's shape in
the simplest possible terms using standard geometry. Despite what a
globe might lead you to believe, the Earth is not perfectly spherical,
but it is close to being spherical. In fact, the radius of Earth varies
by no more than 22 km or 0.35%, hardly anything you would ever notice if
you were holding it in your hand. That small difference is, however,
significant enough to lead to mapping inaccuracies at the local level if
a spherical model of Earth was adopted (Figure 2.3). Instead, we
frequently describe Earth's shape as an oblate ellipsoid, which is
essentially a sphere that has been flattened, and we define this
ellipsoid with a semimajor and semiminor axis. Sometimes you will see
the term *spheroid* used, which just means "sphere-like" and is
interchangeable with the term *ellipsoid*.

![Figure 2.3 Spherical geodetic
datum](images/02-geodetic-datum.png){.center} <br/>

There are many different ellipsoids that have been defined and are
currently in use as datums. The most commonly used ellipsoid is called
the World Geodetic System of 1984 or usually abbreviated as WGS 1984 or
WGS 84. In fact, there are hundreds of ellipsoids that have been defined
over recent centuries to model the shape of the Earth. The reason for so
many other ellipsoids is due in part to technological advances that have
improved the accuracy and precision of surveying as well as estimation
of the ellipsoidal parameters. Many of these ellipsoids are not
*geocentric*, that is, not originating from the center of mass of Earth.
These datums are known as **regional datums**, which still describe the
dimensions that approximate the shape of Earth, but are instead oriented
so that the surface of the ellipsoid is congruent with a particular
regional surface of Earth. For example, the European Datum 1950, the
South American Datum 1969, the North American Datum 1983, and the
Australian Geodetic Datum 1966 conform well to their respective
continents, even better than WGS 1984 in most cases, but poorly anywhere
else in the world.

![Figure 2.4 Sphere versus
ellipsoid](images/02-sphere-ellipsoid.png){.center} <br/>

Figure 2.4 greatly exaggerates the flattening of the ellipsoid to
illustrate the above points. In reality, the sphere is flattened using a
flattening factor calculated as $f=(CA-CG)/CA$ and defined exactly as
$f=298.257223560$ for WGS 1984. Thus, the semiminor axis (i.e.,
rotational axis) for the WGS 1984 ellipsoid (meters) is

$$
CG=CA-(CA×\frac{1}{f})=6378137-(6378137×\frac{1}{298.257223560})=6356752.3
$$ where $G$ is the North Pole and $A$ is a point on the Equator. The
sphere, of course, is much simpler where $radius=CB=CA=6378137$.

### Tidal Vertical Datums

A **tidal vertical datum** is likely one that you are familiar with. The
premise of a tidal vertical datum is to use mean sea level as a
reference surface, above which are positive elevations and below are
negative elevations. This has a lot of advantages, like it is intuitive
and oceans cover more than 70% of the planet's surface so much of
Earth's land mass is near an ocean. However, the disadvantages are that
sea level changes over time with tides and also with climate change. The
not-so-obvious problem with a tidal vertical datum is that the sea level
is actually not constant around the planet not only due to tides, but
also temperature, air pressure, and gravity. In other words, mean sea
level measured at a gauge station in Halifax on the Atlantic Ocean will
not be the same distance from the center of Earth as mean sea level
measured at Victoria on the Pacific Ocean (Figure 2.5). The primary
challenge with a tidal vertical datum is extending it away from the
coastline through a network of survey points using a process known as
levelling, and even still, it is only meaningful during the epoch in
which the mean sea level was measured at a number of tidal gauge
stations.

![Figure 2.5 Conceptual tidal datum for
Canada](images/02-tidal-datum.png){.center} <br/>

### Gravimetric Vertical Datums

The **geoid** is a physical approximation of the figure of Earth. The
shape represents Earth's surface with calmed oceans in the absence of
other influences such as winds and tides. It is computed using gravity
measurements of Earth's surface and is best thought of as the surface or
shape that the oceans would take under the influence of Earth's gravity
and rotation alone. In other words, the geoid represents the shape Earth
would take if the oceans covered the entire planet. More specifically,
the geoid is a **gravimetric** model of Earth's shape that is defined as
an equipotential surface from a constant gravity potential value. Due to
the distribution of mass on Earth, gravity is not constant across the
planet's surface. As a result, the surface of Earth's oceans is not
smooth like a sphere, but instead undulates depending on where gravity
forces water to remain at rest. You can think of Earth's gravitational
field as a series of parallel lines extending outwards from the center
of mass of Earth into space. Any of these lines that you choose is an
equipotential surface where the force of gravity is constant. Keep in
mind that the force of gravity is stronger nearer the center of mass of
Earth and weaker as you move away from it. Thus, the geoid is an
arbitrary equipotential gravity surface that is chosen to roughly
coincide with present-day mean sea level.

When you measure the height of something relative to a gravimetric
vertical datum like the geoid, you must level your instrument. Levelling
forms a vertical line that is orthogonal or perpendicular to the geoid,
known as a **plumb line**. It is incredibly easy to visualize a plumb
line. Simply tie a rock to the end of a string and hold the string with
your outstretched arm. The length of the straightened string traces a
plumb line to the center of mass of Earth, wherever you are. Because
gravity changes with location on Earth and all plumb lines are
converging on a singular point, plumb lines are never parallel. This
phenomenon has important implications for comparing observations on the
the ground with a geodetic model of Earth like an ellipsoid. In other
words, the plumb line that you traced with your string is pointing to
the center of mass of the geoid, but the center of the ellipsoid is
often in a slightly different direction. This difference is known as the
**deflection of the vertical** and is measured as the angular difference
between the centre of the geoid and the centre of a reference ellipsoid.
Like other measurements of geodetic location (i.e., latitude and
longitude), the deflection of the vertical is comprised of two angles:
$ξ$ (xi) representing the north-south angular difference and $η$ (eta)
representing the east-west angular difference.

::: {.box-content .case-study-content}
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#### Case Study {.unnumbered}
:::

#### Case study title max of forty characters {#box-text .unnumbered}

<p id="box-text">

The Canadian Geodetic Vertical Datum of 2013 (CGVD2013) is the current
gravimetric vertical datum used in Canada to reference heights. It is
defined with a potential gravity value of 62,636,856.0 $m^2$ $s^-2$. The
pervious vertical datum in Canada - the Canadian Geodetic Vertical Datum
of 1928 (CGVD28) - was actually a tidal vertical datum that corresponded
to mean sea level measured at Yarmouth, Halifax, Pointe-au-Père,
Vancouver and Prince-Rupert, and a height in Rouses Point in New York.
It turns out that Halifax referenced to CGVD2013 is 64 centimeters
*below* Halifax referenced to CGVD28!

</p>

<p id="box-text">



It should be evident by now that the reference surface that you choose
as a vertical datum will determine the measured elevation of Earth's
terrain. Additionally, We frequently need to convert elevations between
geodetic and gravimetric vertical datums. For example, when you use a
Global Positioning Service (GPS) receiver, you are provided with an
elevation that is relative to the WGS 1984 ellipsoid (more on that in
[Chapter 4]()). The difference in height between an ellipsoid and the
geoid is referred to as **geoid height (N)** while the difference in
height between an ellipsoid and Earth's surface is referred to as
**geodetic or ellipsoidal height (h)**. The difference in height between
the geoid and the Earth's surface is called **orthometric height (H)**
(Figure 2.6), and is given as:

$$
H = h - N
$$

![Figure 2.6 Orthometric Height (H) is the ellipsoidal height (h) less
the geoid height (N)](images/02-H-h-N.png){.center} <br/>

To illustrate the concept of a gravimetric datum, suppose we constructed
a large, straight tunnel through the physical Earth that was tangential
to the ellipsoid. If we allowed the oceans to flow freely through this
tunnel, your experiences might convince you that water would flow from
one end to the other. But in fact, this tunnel is so large, that the
gravity field is changing. So the water would actually come to rest at
the surface of the geoid or gravimetric model, as shown in Figure 2.7
below.

![Figure 2.7 Thought experiment showing where water would be at rest
within a tunnel through the geoid due to the equipotential force of
gravity (g)](images/02-geoid-tunnel.png){.center} <br/>

## Referencing Location

### Cartesian Coordinate Systems

Now that we have explored how to reference heights to vertical datums,
we will now turn to considering how to reference location on Earth.
Before we we jump to the three dimensional case of Earth, consider how
you would map your room and identify your location within that room.
Assuming you are in a rectangular room, you could easily pick a corner
and first measure the distances between the four corners of the room,
giving you the dimensions. You could then proceed to measure your
distance to any two walls and quite easily define your position within
the room relative to the first corner that you picked. This is an
example of a **coordinate system** that provides reference for the
relative locations of anything contained within the extent of the
coordinate system (i.e., the four corners of your room).

Fundamentally, a coordinate system is defined by a common unit of
measurement (e.g., meters, feets, degrees), an orientation defining the
direction that measurements positive or negative, and an **origin**,
which is an arbitrary point where the measurements begin at zero. On a
one dimensional line, you can define any other location on the line as a
measured distance from the origin at $[0]$. On two dimensional maps,
like your room, we have two axes that are perpendicular from which we
can define any location with measured distances from the origin $[0,0]$.
We can also extend this to the three dimensional Earth, which simply
requires another axis to define the origin at $[0,0,0]$. All of these
cases are referred to as **Cartesian coordinate systems**, so-named
after French philosopher Descartes (also known for the phrase, "I think,
therefore I am") who first described the two dimensional case in 1637 in
*La Géometrie*.

### Celestial Coordinate Systems

You might be wondering, *How do we reference locations on Earth?* Before
the technological era of geoids, astronomical observations provided the
basis for the earliest coordinate systems. The ancient civilizations of
Greece, Egypt, and Babylon all recognized the use of celestial
coordinate systems for defining the location of stars, planets, and
other celestial bodies in the sky. Recall that Eratosthenes, a Greek
mathematician living about 2300 years ago, had already worked out the
spherical shape of Earth. A celestial coordinate system simply extends
the spherical shape of the Earth outward into space to locate objects
using angular measurements. It is a *geocentric* coordinate system, that
is based on an origin at the center of Earth with an orientation
following the rotational axis of Earth (i.e., spinning around the
semiminor, North-South axis). This is also known as an equatorial
coordinate system, because it is oriented relative to the Equator of
Earth. The equatorial coordinate system is the reason why you can
navigate by Polaris, the North Star, which would appear nearly directly
overhead if you were standing at the North Pole. Even though Earth is
not perfectly spherical, astronomical observations have been reliably
used for millennia to transit the irregular oceans and terrain of Earth.

### Geographic Coordinate Systems

Geocentric coordinate systems are essential for global navigation and
mapping. In the modern era, we use **geographic coordinate systems**
(sometimes abbreviated GCS) that are oriented to the rotational axis of
Earth, much like the equatorial coordinate system. The origin of a
geographic coordinate system is the center of Earth and the units of
measurement are degrees of **longitude** and degrees of **latitude**.
Degrees of latitude (denoted by lambda, $λ$) measure the angle from the
equitorial plane North (+) or South (-), while degrees of longitude
(denoted by phi, $φ$) measure the angle from the polar plane East (+) or
West (-). Thus, geographic coordinate systems use angular units of
measurement. Any combination of latitude and longitude gives coordinates
$[λ,φ]$ on a sphere or ellipsoid. Positive values of latitude put you in
the Northern Hemisphere, while negative values of longitude put you in
the Western Hemisphere. Constant lines of latitude, known as
**parallels** because they are always parallel to one another, and lines
of longitude, known as **meridians**, form a grid that fits over a
sphere or ellipsoid called a **graticule**.

For most of your life, you have probably believed that there is a
singular combination of capital-L Latitude and Longitude values that
absolutely define some location on Earth in perpetuity. This is perhaps
the most profound geographic lie that we were taught as young school
children. In fact, there are as many "types" of latitude as there are
geographic coordinate systems. By now, you should be able to recognize
the difference between *geocentric latitude* that is referenced to a
sphere and *geodetic latitude* that is referenced to an ellipsoid. The
main difference is that geocentric latitude is the angle relative to the
centre of the sphere at the equatorial plane and geodetic latitude is
the angle relative to the equatorial plane (i.e., not necessarily the
centre). Figure 2.8 illustrates how the same angle can put you in two
very different places depending on the geographic coordinate system you
are using.

![Figure 2.8 Geocentric versus geodetic
latitude](images/02-geocentric-geodetic-latitude.png){.center} <br/>

$CS$ represents the line connecting the centre of a spherical geographic
coordinate system to the surface of the sphere at a geocentric latitude
$φ′=45°$ and $DE$ represents the line connecting the equatorial plane to
the surface of the ellipsoid at a geodetic latitude $φ=45°$. Notably,
$CS$ is parallel to $DE$ and $CS$ intersects with the surface of the
sphere at a 90° angle and $DE$ intersects the surface of the ellipsoid
at a 90° angle.

A coordinate system that is referenced to an ellipsoid is known as a
**horizontal datum**. For example, the World Geodetic System of 1984
(WGS 1984) is a *geodetic* datum that has a geographic coordinate system
referenced to it. So the origin of WGS 1984 is also the origin of the
GCS from which latitude and longitude measurements are derived. So if
you measured your latitude and longitude using the WGS 1984 horizontal
datum, you would be placed somewhere on the ellipsoid also defined by
WGS 1984. It is important to note, however, that the same exact
*geocentric latitude* and longitude measures would place you somewhere
else on a sphere. Other factors such as plate tectonics, glaciation, and
ocean tides cause the Earth's surface to be in constant motion
underneath any fixed horizontal datum. For example, Europe has drifted
about 60 meters from North America since Eratosthenes first calculated
the circumference of Earth. Therefore, any "type" of latitude or
longitude is only useful during a particular epoch of time.

If you mapped the Earth as an ellipsoid in a three dimensional Cartesian
coordinate system, you could describe location using three coordinates
$[x,y,z]$, with $[0,0,0]$ being the center of Earth. However, we do not
often express mapped coordinates of Earth using Cartesian coordinates.
Instead of referring to the North Pole at $[0,0,6356752.3]$ meters (the
polar radius of Earth) we usually refer to it with a single coordinate,
90° N. Why? The North and South Poles are the only points on Earth that
can be defined with a single coordinate because they coincide with the
orientation of a geographic coordinate system. Imagine placing a
protractor at a 90° angle relative to a table. If you rotate it along
that perpendicular axis, one arm of the protector would spin in a 360°
circle, but the other arm of the protractor would always point up or
down at 90° in the same direction. For all other locations on Earth, a
pair of $[λ,φ]$ coordinates are needed to define location. With
space-based global navigation systems it is more common to combine
coordinates in both horizontal and vertical datums together. For
example, $[λ,φ]$ expresses your location relative to the horizontal
datum while $[λ,φ,h]$ expresses your ellipsoidal height (h) at a
location and $[λ,φ,H]$ expresses your orthometric height (H) at a
location.

### Projected Coordinate Systems

Despite everything covered so far in this chapter, we very rarely see or
display geographic data in 3-dimensions. In fact, most geographic data
you will likely encounter will be either 1- or 2-dimensional (more on
that in Chapter 3). Geographic coordinate systems are incredibly
important for understanding how geographic data are fundamentally
"attached" to the Earth. However, geographic coordinate systems are not
suitable for creating maps, those 2-dimensional spatial models that can
be easily displayed on your computer screen or printed on a sheet of
paper. Instead, cartographers rely on **projected coordinate systems**
that flatten a 3-dimensional geographic coordinate system to a
2-dimensional map. Really, these projected coordinate systems involve
transformations called **map projections** that convert 3-dimensional
coordinate space into 2-dimensional coordinate space, which means the
map units are linear such as meters. Whenever we move from a geographic
to a projected coordinate system, we lose information, and distortion
results.

Cartographers have wrestled with how to project Earth onto a printed
page for millennia. The fundamental mathematics for map projections were
first comprehensively described by Claudius Ptolemy around 150 C.E.
Ptolemy's work *Geography* was one of the earliest treatises on
cartography and map making that included an atlas of regional maps of
Europe, Africa, and Asia. Ptolemy's work built on and came several
centuries following Eratosthenes and earlier Greek geocentric
observations by Plato and Aristotle. Ptolemy observed that a globe was
the best way to represent the intervals and proportions of Earth's
surface without distortion. However, globes are not very useful for
looking at regions in detail and you can only see part of a globe at any
given time. Thus, a mathematical language is needed to translate a
geographic coordinate system to a planar or projected coordinate system.
*Geography* was lost to antiquity before it was rediscovered, copied,
and translated centuries later, first by Muslim cartographers in the 9th
century C.E. and later by Italian cartographers in the 15th century C.E.
during the Renaissance, which gave rise to the many types of map
projections that we see today.

Because all map projections result in distortion from the loss of the
third spatial dimension, it is useful to think about map projections in
terms of what they preserve. There are four main characteristics that
can be distorted or preserved, which give rise to the primary types of
map projections that are in use for environmental management
applications:

-   **Conformal** projections preserve shape and angles
-   **Equal-area or equivalent** projections preserve area
-   **Azimuthal** projections preserve direction
-   **Equidistant** projections preserve scale and distances

Some map projections can preserve several of these characteristics at
once, but only a globe can simultaneously preserve area, direction,
distance, and shape. Any map projection will have inherent trade-offs
representing these characteristics accurately. It is beyond the scope of
this textbook to discuss all map projections. Instead, we will focus on
several key examples of map projections that are commonly used for
environmental management applications. For a more comprehensive
discussion of map projections generally, the reader is referred to *Map
Projections: A Working Manual* by John Snyder (1987). In the next
section, we look at how map projection distortation can be measured.

### Measuring map projection distortion

**Tissot's Indicatrix** is often used to visualize distortion from map
projections, named after Nicholas Auguste Tissot. The metric is
relatively simple: Tissot's Indicatrix is a perfect circle on the
surface of a 3-dimensional globe, but will form an ellipse whenever
projected to a 2-dimensional coordinate system. For this reason,
Tissot's Indicatrix is sometimes referred to as Tissot's Ellipse. Since
ellipses can vary along two axes, Tissot's Indicatrix can represent
areal, angular, and linear map distortions both longitudinally and
latitudinally at any location in the map. This is very handy, because we
can place Tissot's Indiciatrices (the plural of indicatrix) at different
locations and examine how distortion changes throughout the map
projection.

So how do we use this tool? The quotient between a line projected onto a
map $a$ and the same line on a globe $a'$ is $\frac{a}{a'}=1$ when there
is no distortion on that axis of the ellipse. This quotient is also
called a scale factor because it is showing how the map scale is
modified locally by a map projection. Figure 2.9 below shows an example
of a reference indicatrix that is a perfect circle on a globe with axes
$a$ and $b$.

![Figure 2.9 Reference Tissot
Indicatrix](images/02-reference-indicatrix.png){.center} <br/>

Let us assume that $a = b = 1$. Then, the reference indicatrix has the
following properties: $a=b$, $a×b=1$, and $Area=πab=π^2$. If
$\frac{a'}{a'}>1$, then we can conclude that the the projection is
*expanding* the distance along the $a$ line. If $\frac{a'}{a'}<1$, then
we can conclude that the projection is *compressing* the distance along
the line $a$. For example, suppose $a=2$ and $b=0.5$, then we have
modified the scale of the indicatrix along both axes, but the areal
solution is the same as the reference indicatrix shown as the red dashed
circle in Figure 2.10 below.

![Figure 2.10 Equivalent Tissot
Indicatrix](images/02-equivalent-indicatrix.png){.center} <br/>

The indicatrix above in Figure 2.X is an example of an *equivalent*
indicatrix, which has the following properties: $a>b$, $a×b=1$, and
$Area=πab=π^2$.

If $\frac{a}{a'}×\frac{b}{b'}>1$, then we can conclude that the
projection is *inflating* the area. If $\frac{a}{a'}×\frac{b}{b'}<1$,
then we can conclude that the projection is *deflating* the area.
Consequently, whenever the quotients of both axes are equivalent (i.e.,
$\frac{a}{a'}=\frac{b}{b'}$), Tissot's Indicatrix forms a perfect circle
and the ellipse is conformal with angles true to the globe. For example,
suppose $a=b=2$, then we have modified the scale of the indicatrix along
both axes, but with the same factor. This results in a *conformal*
indicatrix that is not equivalent, shown in Figure 2.11 below.

![Figure 2.11 Conformal Tissot
Indicatrix](images/02-conformal-indicatrix.png){.center} <br/>

This conformal indicatrix has the following properties: $a=b$, but
$a×b≠1$, and therefore $Area=πab=4π^2$. As you can see, whenever $a×b=1$
the indicatrix is equivalent (equal-area) and whenever $a=b$ the
indicatrix is conformal.

### Map projections for environmental management

#### Mercator

Mercator is a cylindrical map projection that represents meridians and
parallels as straight lines. The cylindrical surface is oriented such
that the rotational axis of Earth runs through the openings of the
cylinder and the Equator represents the tangent where the cylinder meets
the Earth's surface. Scale along the tangent is true because the
translation from the spherical Earth to the cylindrical surface is
one-to-one at the tangent, so this is also the location on the
projection where there is no distortion. This has the effect of
accurately representing the shape and angles on the map (i.e.,
conformal), but greatly distorts area as you move away from the Equator.
In fact, the North and South Poles are represented as 2-dimensional
lines at the top and bottom edges of the map instead of 1-dimensional
points. Although scale and area change as you move North or South along
a meridian, scale and area are equivalent along any parallel, but not
necessarily true to a globe. In other words, you can compare area or
scale anywhere along a parallel, but only at the Equator is the area and
scale true to the globe.

The Mercator map projection is perhaps the most pervasive and reproduced
projection around us (Figure 2.12). Because angles are preserved, you
can easily and accurately navigate long distances across Earth, and this
was exactly the purpose that Gerardus Mercator envisioned when he first
identified the projection for sea-faring Europeans in 1569. You may also
recognize the Mercator projection from web mapping applications like
Google Maps, which use it because it ensures that North-South roads
intersect at right angles with East-West roads.

![Figure 2.12 Mercator map projection with Tissot's Indicatrices in
red](images/02-Mercator-map.png){.center} <br/>

#### Universal Transverse Mercator (UTM)

Universal Transverse Mercator (UTM) is very similar to a Mercator
projection except that the cylinder is rotated or transverse by 90° so
that the opening of the cylinder is perpendicular to the rotational axis
of Earth. This has the effect of moving the tangent from the Equator to
any Meridian. In fact, you can rotate the cylinder at any angle you want
where 0° is a true Mercator, 90° is a transverse Mercator, and any other
angle is considered an oblique Mercator. UTM is actually a system of 60
different transverse Mercator projections that are defined to represent
6° Longitudinal intervals of Earth's surface (60 zones x 6° = 360°).
Each projection is defined as a zone, which is also divided into North
and South zones depending whether you are in the Northern or Southern
Hemisphere. Canada spans 15 UTM zones from Zone 7 North in the Yukon to
Zone 22 North covering Newfoundland. Figure 2.13 below shows a map of
UTM Zone 13 over Saskatchewan.

![Figure 2.13 Universal Transverse Mercator Zone 13 map projection with
Tissot's Indicatrices in red](images/02-UTM-13N-map.png){.center} <br/>

Besides defining the orientation of the cylinder, we can also specify
its size or diameter. When the diameter of the cylinder is equivalent to
the diameter of Earth at the Equator, a single tangent line is formed.
If the diameter of the cylinder is smaller than Earth's diameter at the
Equator, then the cylinder has two lines that contact Earth's surface,
known as **secants**. The purpose for having two secants is that the
projection distortion can be more evenly distributed across the map. In
the case of UTM, a scale factor of 0.9996 is applied to shrink the
transverse cylinder slightly, forming two secants that are 360 km apart
East-West. In between the two secants lies the **central meridian**,
which is used to define the origin for the projected coordinate system.
Notice in Figure 2.X how Tissot's Indicatrices are near perfect circles
along the central meridian of the map. It is important to realize that
the secants are parallel to each other and the central meridian, which
means the secants are not meridians on Earth and form what are called
**small circles**, a line that does not divide Earth into two equal
portions.

[Figure 2.14. Close-up of a single UTM zone showing the central meridian
and secants]

UTM uses a unique coordinate system that deserves some explanation. Just
like with latitude and longitude coordinates $[λ,φ]$, an arbitrary
origin needs to be defined so that we know where we are in relation to
the origin on the map. For projected coordinate systems like UTM that
use linear units of measure such as meters, the origin is defined as the
intersection of the central meridian and the Equator. Simple enough,
right? There is one catch: UTM does not use any negative coordinates by
convention. Thus, the origin of the coordinate system for each zone must
be moved so that coordinates West of the central meridian and South of
the Equator are positive. To do this, a constant value is added to all
East-West coordinates (known as **Eastings**) and all North-South
coordinates (known as **Northings**) to create what are known as **False
Eastings** and **False Northings**, respectively. A value of 500,000 m
is added to all Eastings so that the western limit of the zone is
located at 0 m, the central meridian is located at 500,000 m, and the
eastern limit of the zone is located at 1,000,000 m. A value of
10,000,000 m is added to all Northings in the Southern Hemisphere so
that the Equator is at 10,000,000 m and 0 m is near the South Pole for
all southern UTM zones. You might recognize the importance of the
10,000,000 m value because this represents approximately one-quarter of
the Earth's North-South circumference.

[Figure 2.15. Showing a UTM zone with the central meridian and annotated
by false eastings/northings]

#### Sinusoidal

Sinusoidal is a *pseudocylindrical* map projection, so-named because
these projections approximate a true cylindrical projection except that
Meridians are curved instead of straight like with Mercator or UTM.
Because Meridians are curved, Sinusoidal maps represent the North and
South Poles as single points instead of lines as is the case with
Mercator. Thus, Sinusoidal maps are not conformal and distort shape, but
they are in fact equal-area (Figure 2.16). Equal-area map projections
like Sinusoidal are important for accurately accounting for land cover
and other global mapping efforts. Therefore, it is common to find global
datasets distributed in a Sinusoidal map projection.

![Figure 2.16 Sinusoidal map projection with Tissot's Indicatrices in
red](images/02-Sinusoidal-map.png){.center} <br/>

#### Albers

Albers is another example of an equivalent map projection, but unlike
Sinusoidal, an Albers projection uses a cone as a projection surface
instead of a pseudocylinder. Like the cylindrical case, the cone can be
sized and oriented in any way, but the cone of an Albers projection is
typically oriented so that the vertex of the cone aligns with the
rotational axis of Earth. The base of the cone has a diameter that
usually results in two secants, known as **standard parallels** on an
Albers map. Thus, Albers projections tend to distort latitudinally as
you move North-South away from the standard parallels, but even more so
as you move toward the base of the cone (Figure 2.17). Besides being an
equal-area projection, Albers is a good choice for mapping regions
because shape and scale are mostly preserved near the standard
parallels. For example, the province of British Columbia has adopted a
modified Albers projection that situates the standard parallels at 50° N
and 58.5° N, which are near the northern and southern latitudinal limits
of the province. This narrow band of latitude between the standard
parallels ensures that there is relatively little distortion in shape
and scale within the province, which is comparable to UTM. However,
British Columbia is a longitudinally wide province, spanning 6 of
Canada's 15 total UTM zones, so Albers has a distinct advantage of being
able to show the entire province with little distortion. For the same
reasons, you will often find Canada-wide data distributed in an Albers
projection.

![Figure 2.17 BC Environment Albers map projection with Tissot's
Indicatrices in red](images/02-BC-Albers-map.png){.center} <br/>

#### Azimuthal

Azimuthal projections use a flat circular plane to project the Earth onto a map. This plane is usually oriented so that it is tangent at a single point, usually the North or South Pole. In the case of a polar azimuthal projection, meridians radiate outward as straight lines from the pole to the edge of the circular plane and the parallels are represented as concentric circles. As a result, distortion increases as you move away from the centre point of the map with the outer edge of the plane representing an **antipode** or an opposite point on Earth. The primary benefit of azimuthal projections is that they preserve direction and distance between the centre point and any other point in the map (Figure 2.18). The shortest geographic distance between the centre point and any other point creates a line known as a **great circle**, which divides the Earth into two equal portions. So another benefit of an Azimuthal project is that great circles can be mapped as straight lines. Azimuthal projections are commonly used when distance and direction are important, such as weather RADAR stations or air traffic control towers. It is important to realize that the centre point for an azimuthal projection can be any point on Earth and the equidistant property can be exploited for a number of applications.

![Figure 2.18 North Pole Azimuthal map projection with Tissot's Indicatrices in red](images/02-Azimuthal-map.png){.center}

## Summary

In this chapter, you learned about the science and technology of geodesy that goes into mapping data. We described the different models of Earth's shape and the advantages and disadvantages that each model has. More generally, models of Earth represent vertical datums to which heights are referenced. When we think of "where" something is on Earth, we must use horizontal datums to reference location. There are two types of horizontal datums that are very important to geomatics: (1) a geographic coordinate system uses lines of latitude and longitude to define locations on Earth's 3-dimensional surface while (2) a projected coordinate system flattens Earth into a manageable 2-dimensional coordinate space. In the case of projected coordinate systems, we have many choices when deciding which map projection to use when we map data, each with its own uses and distortions. The next time that you look at a map, ask yourself with your new found appreciation of geodesy, how is this information being misrepresented to me?

### Reflection Questions {.unnumbered}

1. Describe the process of measuring the height of something on Earth.
2. Explain the difference between a geographic coordinate system and a projected coordinate system.
3. Name as many projected coordinate systems as you can.
4.

### Practice Questions {.unnumbered}

1. Choose a continent to roughly draw and then draw a grid of latitude and longitude lines in 3 different projected coordinate systems on a piece of paper. Describe the distortions that you expect and point to where they appear on your continent.
2. x`x`

## Recommended Readings {.unnumbered}

Ensure all inline citations are properly referenced here.

```{r include=FALSE}
knitr::write_bib(c(
  .packages(), 'bookdown', 'knitr', 'rmarkdown', 'htmlwidgets', 'webshot', 'DT',
  'miniUI', 'tufte', 'servr', 'citr', 'rticles'
), 'packages.bib')
```