Cones, Planes and Cylinders, Oh my!

Another week in GEOS419 and we are moving to solidifying the project choice.  Further definition of the project let us know that the course was focused on vector analysis.  Since solar siting is largely based around continuous data such as DEMs it became unlikely to be a feasible project.

Week 3 in the course has been largely review again, this time we are looking at the basics of projections, developable surfaces, ellipsoids; basically transformations and distortions of area, distance, shape and direction.

Although a lot of this was review, the readings that we did, including Dent and Slocum, expanded on the ideas and theories around why distortion occurs and how it can be looked at with Tissot's Indicatrix, a method of using circles to describe and visually show how distortion occurs across the projection.  I have always found the ideas expressed in "Intro to GIS" classes interesting, in regard to spheres, ellipsoids, the geoid and how information from the surface of a 3d shape is projected onto one of the three classes of developable surfaces (a geometrical shape that can be flattened without distortion, ie a plane, cylinder or cone), so it was interesting to see some basic transformation formulas, even if the GIS package is going to handle all of it for us.

So the basic take home is that the earth is a 3d shape similar to a sphere, but more squished, an oblate spheroid is a good description.  The Earth can not have its shape described by a mathematical model as its surface has depressions and bulges.  There is an entire area of science known as geodesy that deals with the measurement of the earth and it's gravitational field.  The best description of the earth is the geoid, but it isn't a mathematical model, but it's shape can be approximated with ellipsoids placed in particular locations so they come close to the appropriate shape.

Before we get ahead of ourselves though, if we think of the Earth as a 3d model, it is impossible to take that 3d surface and translate it's surface onto 2d map without distorting some of the properties of the surface, as mentioned earlier: shape, area, direction and distance.  How we move things from the 3d surface to the 2d plane is of the utmost importance. We can minimize these distortions in one or two of these areas but not all four.  For example if we wanted to calculate areas how we project the information to the plane can't distort area, or our calculations will be incorrect.

Our basic theoretical steps to project from the surface of the Earth to a plane is as follows

1. Choose the size of a reference sphere (based on the map you want to create)
2. Choose a developable surface.
3. Place the surface against the globe or intersecting a globe
4. Project the surface features and graticule onto the developable surface
5. Unwrap the surface

In modern GIS it is important to understand what is happening, but it is handled by mathematical formulas now.

There is however one location on the surface of the 2d map where there is no distortion, and that is along the points where the developable surface touches the sphere, these can either be a point, a line, our multiple lines depending on the case of the surface, and are called standard lines or points.  As we move away from these lines our distortion either compresses (between the lines) or expands (outside the lines).  So depending on what we are mapping we can use that knowledge, especially on large scale maps to minimize the distortion.

On a small scale map there is always going to be areas that are far away from these places of no distortions, and so we are going to see areas of major distortion.  General world maps often preserve shape but wildly distort area, Greenland is a favorite example of this.  If we choose to preserve area on a world map we see a lot of distortion in shape.  

On a large scale map we can place our standard parallels close to the area we want mapped, and have nowhere on the map wildly distorted.  In general we should place the center of the map over the areas we want to map and the standard lines at about 1/6th of the extent of the map.  A popular projection known as TM (Transverse Mercator) has 3 projection types that are often used to map areas of a north/south extent, like a Canadian province.  Transverse Mercator is a cylindrical projection but instead of it being an upright cylinder, it is rotated 90 degrees.  Its 3 basic projections are:

• 3TM, designed for areas of an extent of 3 degrees or less
• UTM, designed for areas  of an extent of 6 degrees or less
• 10TM, designed for areas of an extent of 10 degrees of less

For areas with a large East/West extent it is good to use a conical projection and for a smaller scale map as we might see with one of these maps it is important to choose the projection that will preserve the properties you want.  Conical maps are generally used to map regions or countries, vs transverse Mercator which are used for larger scale maps.  Three conical projections to keep in mind are.

• Albers Equal Area Conic - Area
• Equidistant Conic - Distance
• Lambert Conformal Conic - Shape

Overall the distortion on a map can be controlled, but how we do it and what aspects we control are dictated, like so many things, by the purpose of the map and what we are trying to communicate.

Although this post is written entirely from memory without directly looking up anything, I would remis in not including a works cited, because these ideas came from the works of people before me, although I have removed the URLs as they will not work if you are not enrolled in the program.

Bibliography

• Dent, B. D., Torguson, J. S., & Hodler, T. W. (2009). Cartography: Thematic Map Design (6th ed.). McGraw-Hill.
• Price, M. (2019). Mastering ArcGIS Pro (1st ed.).
• SAIT. (n.d.-a). Map Scale and Generalization. Fall 2021 - GIS Data Analysis & Output (GEOS-419-O2A).
• SAIT. (n.d.-b). Selecting the Appropriate Map Projection. Fall 2021 - GIS Data Analysis & Output (GEOS-419-O2A). 
• Slocum, T. A., McMaster, R. B., Kessler, F., & Howrad, H. H. (2009). Thematic Cartography and Geovisualization (3rd ed.). Pearson Prentice Hall.