Now, we have a solid idea about how

to build the 3D spatial reference framework with datum.

The next issue is how to convert 3D

to 2D spatial reference framework; 2D plane coordinate system,

on which most spatial data applications are developed.

The process is called map projection.

There are numerous map projections.

Among them, we'll focus on

conformal map projections in which the shape of space objects is preserved.

And they are considered standard map projection for large scale maps.

For conformal projections, there are three main types depend on the projection surface,

which are cylindrical, conical and planar.

Though the mathematical formulations are very

complicated and rooted from differential geometry,

which is a branch of mathematics,

the concept is rather simple in which the source

of the light is located at the center of the Earth,

spatial objects are projected from the surface of the Earth

to the projection surface: cylinder, cone or plane.

The shape on the projection surface is the outcome of the conform projection.

Now, you are looking at examples of cylindrical projection.

If projection surface is upright and meets the Earth on a single line,

we call it normal or standard projection.

If the projection surface cuts through the Earth,

consequently, it meets on two lines,

we call it secant projection.

When projection surface is inclined, it is called oblique projection.

When projection surface laid down,

it is called transverse projection.

The parameters to define the projections are tangent lines,

the original projection, and scale factor if it is secant projection.

Similarly to a cylindrical projection,

conical projection are also categorized into normal,

secant, oblique, which depends on how conical projection surface meets the Earth.

The parameters of the projections are again

the tangent line between projection surface in the Earth and projection center.

You are looking at different types of planar projections again which

depends on the way how projection surface and the Earth are related.

Let's take a look at a popular prediction, transverse Mercator shortly,

TM projection, which is

transverse cylindrical projection presented in the previous slide.

It is widely used for national and international mapping system,

particularly with Universal Transverse Mercator, UTM.

For example, the US topographic maps are based on UTM.

TM basically inherits mathematical foundation and many traits from normal Mercator,

the cylindrical projection, except for the orientations.

In TM, the axis of the cylinder lies on the equatorial plane,

and the tangent line is located on any selected meridian.

So it is called central meridian.

The figures shows a secant version in which the scale is reduced at the center, at the central meridian.

Because cylinder slices through the Earth,

the reduced scale at the central meridian in secant projection is called scale factor.

Now, let me ask a question,

where do you think is the most accurate TM projection?

In other words, where do you expect no distortion in the projection surface?

Of course, on the line where the cylinder and earth meet each other.

The scale becomes drastically larger,

while it is farther from the center meridian in any projection.

So TM is suitable for the areas with north and south long extent.

As briefly discussed, shapes,

size and location of mapping area are important for selection of map projection.

If the extent is north-south long,

transverse Mercator would be a good choice.

If east-west long and large,

conical projection would be a reasonable choice.

If the area is circular and relatively small,

panel projection would work out.

Basically, the shape, size,

location of the given area are collectively considered for deciding how

to locate the tangent lines between the Earth and projection surface.

The figure would substantiate the argument.

The state of Alaska has a long tail to the south-eastern direction.

For the area, the tangent line is inclined

so that the selection should be an oblique projection.

For example, an oblique cylindrical projection in this case.

So far, we have discussed only conformal mapping generally for a large scale map.

However, for a world wide mapping at a small scale,

conformal mapping would not be a solution.

In that case, equal area projection should be used such as Goode's homolosine,

mollweide, Lambert cylindrical equal area projection and so on.

This issue is related to the subject and proposal of the map,

which should be considered in the first place.

Now, for checking your understanding of spatial reference framework,

let's take a look at a few examples.

The first one is Wisconsin transverse Mercator.

The map projection is a secant transverse cylindrical

because the state of Wisconsin is rather north-south long.

Datum is North American Datum, NAD83 based on GRS80 ellipsoid.

The central meridian of WTM is minus 90 degree,

penetrating the center of Wisconsin.

And 520,000 meters is added to east-west in order to make x coordinate a positive value

on the other hand, a large value over 4 million meters were subtracted

from y-axis because the latitude of the origin is an equator.

So y coordinate could be unreasonably large values without minus false northing.

Finally, the scale factor at the central meridian is 0.996.

Can you visualize the map projection along with the given datum?

If you can say yes, now,

you understand the concept of spatial reference framework very well.

Let's take a look at another example.

Texas statewide mapping system.

This time, the map projection is a secant

conical because state of Texas is rather east-west long and large.

The datum is North American datum 27 based on Clarke 1866 ellipsoid.

The central meridian and the latitude of origin is minus 100 degree,

and 31 degree 10 minutes.

Because it's a secant conical prediction,

it has two standard parallel.

On each projection surface,

slice through the Earth.

They are 27 degree 25 minutes and 34 degrees 55 minutes in latitude.

Also, it has false easting and false northing on x-axis and y-axis to make the coordinate

have a positive and reasonable values.

Now, the last issue in spatial reference framework,

which is coordinate transformation.

In spatial data applications, you come up with these problem very

often when you combine two or more data sets and analyze them together.

In that case, you have to make map projections of each dataset unified for

aligning spatial data on top of each other.

There are mainly four major methods for coordinate transformation,

from coordinate system A to coordinate system B.

The first approach is the direct transformation.

Without consideration of map projection,

coordinate system A can be transformed to B with a heuristic polynomial function,

generally, third order polynomials.

It works well, In case that the given area is relatively small.

However, if it requires a mathematically thorough transformation with high accuracy,

we should consider datum side,

meaning that this approach wouldn't work out very well.

So that's the second case of coordinate transformation.

In each coordinate system A transformed backward to datum A,

and transformed forward to coordinate system B.

This would work when the two coordinate systems A and B share their datum,

meaning that they have the same datum.

For example, on some coordinate system A is

Lambert conformal conic based on NAD83,

and coordinate system B is WTM on NAD83.

They share the same datum so that the method shown on

the figure would give you a mathematical rigid solution.

Now, my question is,

what if two coordinate systems have different datum?

The third method should be applied to the case.

Coordinate system A transformed to backward to datum A. Datum A approximate

to datum B by regression or other 3D to 3D transformation.

And then datum B is projected to coordinate system B.

The example could be a transformation within Lambert conformal conic on

NAD83 and Texas statewide mapping system on NAD27, two different datums.

In the case, two map projections have different datums so that the third method would work out.

Theoretically, there is a more rigid transformation method.

The fourth method is to take another series of

steps to geocentric coordinate systems, meaning x, y,

z in 3D Cartesian coordinate which is only applied to geodetic coordinate transformation,

which requires the most accurate outcomes in transformation.

As mentioned, the third method approximates the datum-to-datum transformation.

So it inevitably comes up with some error,

while the fourth method would include only minimum error in coordinate transformation.

Isn't it somewhat complicated?

Yes, it certainly is.

However, one good news is that, in reality,

the fourth method is never used in GIS applications,

An even better news is that most GIS softwares have function of

so-called on-the-fly projection, with which

users don't have to worry about coordinate transformation,

as long as coordinate systems of your datasets are well-defined in metadata.

GIS softwares automatically align spatial data on top of each other,

even though they are based on different map projection.