The Wild World of Geodesy!
PLEASE interrupt if you have questions!
Go to http://lyzidiamond.com /geodesy to follow along.
(Use left and right arrows to navigate. Links have a purple background.)
What is geodesy?
Wikipedia says: the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space.
:-/
Let's break it down.
We can all agree: the Earth is three-dimensional.
Yes. :)
But the Earth is not a sphere. It is its own fancy shape.
We measure and describe that shape in a few ways.
First: the geoid.
"Lumpy Earth"
The geoid is the shape of the Earth without land masses, tides, and wind. It assumes that all locations have the same gravitational potential.
Gauss described it as "the mathematical figure of the Earth."
But the lack of uniformity makes it hard to use as a reference for measurement.
Enter the reference ellipsoid.
or

"oblate ellipsoid."
The reference ellipsoid is mathematically defined: you can read more about that here.
The important part is that the short (polar) axis is roughly aligned with the rotation axis of the Earth.
Why is it important to have a mathematically defined shape for the Earth?
Because we, as cartographers are engaged in measuring the Earth. We need to know what goes where.
Hooray! Now we get to talk about coordinate systems and datums!
So we now know that the reference ellipsoid for Earth is a significant approximation of the geoid, which is itself a significant approximation of the real shape of the Earth.
But the reference ellipsoid is super helpful to us when it comes to knowing where things are in relation to each other.
It enables the creation of a geodetic datum: a coordinate system and set of reference points that are used to locate places on the Earth.
A datum provides starting points for measurement.
Horizontal datums measure positions on the Earth's surface. Vertical datums measure elevations.
Let's walk through an example.
You and I are standing just to the left of this hill.
You are walking up the hill. I am walking next to the hill.
We each take fifteen steps forward.
If you look at the photo, I am closer to the right side of the photo than you are.
We tend to think about measuring space in a Cartesian fashion.

Cartesian coordinate system
But remember: the Earth is three-dimensional and lumpy, even in the reference ellipsoid.
If you looked at our relative positions on and next to the hill from above, it will seem as though I traveled further.
But I didn't.
A datum takes this into account and makes a coordinate system that (roughly) reflects it.

Not perfect, but at least not flat.
Notice in that image where it says Local Datum?
There are many different datums that refer to many different reference ellipsoids.
For example, a datum may define a location for the Equator, but that location may be different for a different datum.
Why so many?
Because, again, of the irregular shape of the Earth.
Some ellipsoids are better for certain areas.
The National Geodetic Survey manages the US's national datums: NAD83 and NAVD88.
One widely-used datum is WGS84, or the World Geodetic System of 1984.
WGS84 is a vertical and horizontal datum.
This is the datum that is used by the Global Positioning System (GPS).
Going back to the hill example...
If I used my GPS to get to a point on the other side of the hill, it would give me different directions if I was going over the hill than if I was going beside the hill.
Using WGS84, it knows the relationship between distance on the earth and relative location.
But even if there wasn't a hill, this would still be necessary. Why?
Because the Earth is three-dimensional.
Cartesian distance becomes flawed on a curved surface.
And our job as cartographers is to take this three-dimensional thing and represent it on a two-dimensional plane.
Because a map fits in our pockets much more easily than a globe.
And imagine how big that globe would have to be to be useful in a city!
Which brings in another important part of geodesy:
PROJECTIONS!
But first...
While datums allow us to understand locations in reference to each other on a three-dimensional plane ...
... projections are all about how we show those locations on maps.
Gauss figured out that a sphere is one of the only shapes that cannot be dismantled into a plane.
Think about it like peeling an orange.

Omnomnom.
It is impossible to take that orange peel and lay it flat on the table.
So to display the Earth on a page, we need to stretch it.
Map projections simply set the rules about how the map is to be stretched or distorted.
Map projections distort in four ways:
Size.
Shape.
Position.
Direction.
Hi, Mercator!
The Mercator projection was defined by cartographer Gerardus Mercator in 1569.
It became popular because of its strength in nautical navigation.
Every projection has distortion, but Mercator is a particularly acute offender.
Most notably, Mercator distorts relative size of land masses towards the poles.

Stretchy!
Another common projection is the Gall-Peters projection.
Gall-Peters preserves relative size of land masses but distorts shape.
There are an infinite number of map projections.
Which projections do you like best? Why?
MerCATor!
To create a projection, you need to select an Earth shape (like the reference ellipsoid) and then determine a transformation of points to a Cartesian plane.
Those transformations relate to a projection surface.

Transverse Mercator
Projection surfaces include cylinders, cones, planes, spheres, and ellipsoids.
Think about it like you're wrapping the globe with one of these surfaces, then unrolling it onto a plane.
In addition to the projection surface, you also have to decide on aspect.
Aspect is how the surface is oriented on the globe.
In addition to classifying projections by surface, we can classify them by distortion.
Azimuthal projections preserve relative direction.
Conformal projections preserve local shape.
Equal-area projections preserve area.
Equidistant projections preserve distance.
I am going to stop here because my head is about to explode with geometry.
In the next installment of geodesy at #maptime, we will talk about Spherical Mercator and web mapping.
We'll talk about great circles and rhumb lines and the common problems in geodesy.
Someday...
But for now:
What did we learn today?
We learned about the science of geodesy!
We learned about geoids and reference ellipsoids!
We learned about datums and coordinate systems!