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In our previous article, we discussed how State
Planes came about. In
years past, a surveyor who is to define the boundary of a plot of land
for the purpose of recording ownership at the county courthouse would
often start at aoint whose geographic coordinates (i.e. latitude and
longitude) were not only precisely known, but also published.
From that point, the surveyor would use the principles of plane
geometry and trigonometry to measure and calculate the exact position
of all points on the boundary of the plot of land.
Since plane geometry was used, in essence what the surveyor did
was to approximate the surface of the round earth with a flat plane
tangent to the earth at the “known point”.
Using plane geometry was, and still is, very
important. It enabled
surveyors to complete their tasks without an inordinate amount of
difficult mathematics. Thus,
the results of their work were less susceptible to error and
affordable by the customer. (It
was not until the availability of the digital computer that actually
doing ellipsoidal calculations was to become a realistic alternative.)
Finally, last time we noticed that having a different flat
plane for different surveys made it difficult to merge data acquired
in different surveys into a cohesive map.
So the State Plane concept was invented.
That is, a single plane established and adopted by the powers
that be, and all property boundaries (and surveying for engineering
projects as well) could use the same plane.
Now that we’re all comfortable with that
idea, I have to be the bearer of sad tidings.
The actual “plane” of State Planes is not actually a plane.
That’s right! Just when you thought the world of surveying and coordinate
systems actually made some sense, we find out that State Plane
coordinates are not actually based on a plane.
(I’m sure the more skeptical Casual Cartographers knew
something like this was coming.)
What are they based on? That’s
the topic of this and the next couple of articles.
Imagine, if you will, that the earth were a big
wire mesh globe. Let’s
give our imaginary globe a mesh density of one wire per 10 degrees of
longitude and 10 degrees of latitude.
That is, our imaginary globe would appear to be constructed
using 36 meridians and 18 parallels of latitude, all glued together.
(You might choose to weld your wires together, but mine are
glued with Crazy Glue.)
Imagine further, that at the center of this
globe, a incredibly powerful, but very small, flash bulb was
installed. A scientist
would call this flash bulb a point source of light.
Mine has a remote control, of course, so I can flash the bulb
without having to get up out of my chair.
Figure 1
Now, imagine further that we had a flat piece
of photographic film, specially formulated so that it was sensitive
only the light emitted by our incredibly powerful, but also very small
flash bulb. If we placed
our flat photographic film against our wire mesh globe, at a “known
point”, and initiated a flash from our imaginary flash bulb, the
resulting image would be similar to that shown in Figure 1.
This image represents what the surveyor used to do (and often
times still does).
The image of our wire mesh globe produced clean
crisp lines in the center of the image.
This is where the photographic film actually touched, or was
very close to touching, the wire mesh globe.
On the image, as distance from this point increases, the images
of our wire mesh meridians and parallels tend to get fuzzy as in
Figure 2. That is, the
image loses focus as the distance from the “known point”
increases.
Figure 2
This trip into imaginary land graphically
illustrates a basic point. If
we approximate the surface of the earth with a flat plane tangent to
the earth at a “known point”, that approximation will be pretty
good in the immediately vicinity of that point.
As the distance from this point increases, the validity of the
approximation deteriorates. At
a certain distance, depending upon the application of course, the
approximation becomes invalid. That
is, the approximation becomes incapable of producing results that meet
the accuracy criteria of our application.
In terms of our imaginary scenario, at point where the distance
between the photographic film and our wire mesh globe reaches a
certain value, the image becomes unusable for our purposes.
Figure 3
In order to increase the size of the region for
which the approximation is valid, we need to devise a means by which
we can get more of the photographic film closer to the wire mesh
globe. That is, reduce as mush as possible the distance between the
wire mesh globe and the photographic film.
Perhaps you have already figured this out.
One way to do this is to wrap the photographic film around the
wire mesh globe so that it takes on the shape of a cylinder.
See figure 3. Having
done this, a flash from our point source of light produces an image on
our special imaginary photographic film similar to that shown in
Figure 4. Where the film
actually touched the imaginary wire mesh globe, the lines are clean
and crisp, indicating a high degree of validity for our approximation.
Since we wrapped our film around the globe such that the film
was touched at the equator, the clean crisp line on our image
represents the equator.
Figure 4
North and south of the equator, the lines tend
to get fuzzy. The further
north (or south) you go, the fuzzier they get.
Of course, there cannot be any image of the north pole, or the
south pole, as there was no photographic film adjacent to either of
these features of our wire mesh globe.
Again, our imaginary cartography system illustrates the effect
of using a cylinder to improve the effectiveness of approximating the
round surface of the earth. That is, using this technique, we can use the resulting
approximation for the entire extent of the equator, and regions close
to the equator. A huge
increase in the size of the region in which our approximation is
valid.
Does this mean we have to learn “cylindrical
geometry” in order to be a surveyor?
Fortunately not. This
is true as the shape represented by the cylinder can easily be
flattened out. That
is, by simply unrolling our photographic file, or cutting along the
dotted line so to speak, the surface of a cylinder can be turned into
a flat plane. For those
of use who are rather clumsy, take heart.
Mathematically this is very easy to do.
(In fact, we did it right here two months ago when we
introduced the Platte Carree projection!)
So we have two concepts that are very important
to the cartographer, casual or otherwise.
First, we have the concept of a developable surface.
The cylinder is said to be a developable surface as it can be
made to lie perfectly flat. You have to cut it appropriately, but nevertheless the result
will lie perfectly flat. Once
the surface is made into a perfectly flat surface, all the techniques
of plane geometry can be applied.
Second, we see that we can use the cylindrical
shape to significantly improve on the original idea of using a flat
plane to approximate the surface of the earth.
That is, instead of simply having one point that is completely
true, using the cylinder approach made the entire equator true.
In other words, our approximation now produces precise results
anywhere along the entire equator.
Depending upon the accuracy required by the application of our
mapping technique, a wide region on either side of the equator now
produces acceptable results.
What good is this, you ask? Colorado (or any other state for that matter) is nowhere near
the equator? This has
been a waste of my time! Well,
give me another minute or so.
Is the cylinder the only shape that is
developable? What other
shapes could be of value in improving the approximation?
I suspect that most Casual Cartographers already know of one. The cone. Like
the cylinder, if you “cut along the dotted line”, you can get a
cone lie perfectly flat. The
cone is another developable surface.
How can we use the cone to our benefit.
Perhaps even to map Colorado?
Figure 5
Let’s return to our imaginary wise mesh
globe. This time, roll up
a piece of your imaginary photographic film so that it takes the shape
of a cone, i.e. like a party hat.
Place the party hat on our wire mesh globe (see figure 5) and
flash the light. Figure 6
provides a rendition of what the image of on your photographic film
could look like. As
before, where the photographic film actually came in contact with the
wire mesh globe, the images of our wire mesh meridians and parallels
are in focus. They are
clean and crisp suggesting that in that region, the approximation of
using a cone, rather than a flat plane, to approximate the surface of
the earth is valid. Again,
as the distance from the clean crisp line increases, the lines get
fuzzy, again indicating that the validity of the approximation
deteriorates as the distance from the “line of contact” increases.
Take a moment, and compare the results of using
the cylinder and the cone. Notice
that the cylinder approach limited the validity of our approximation
to the region close to the equator.
(Until next month anyway.)
Notice also, that the affect of using the cone idea enabled us
to move the clean crisp line, i.e. the region in which our
approximation was valid, to a different parallel.
In fact, depending upon whether we rolled our photographic film
into a short fat cone, or a tall skinny cone, we could move the region
of validity to any desired parallel. By creating a cone such that when placed on our wire mesh
globe, the point of contact was at 39 degrees north latitude, the
result would do reasonably well to map Colorado.
Thus, use of the cone concept greatly increases our ability to
get an approximation whose region of validity is where we want it.
That is, someplace other than the equator.
Note that just because we rolled the
photographic film into, say, a very short fat cone, or a very skinny
tall cone, does not affect the ability to “cut along the dotted
line” and make it lie perfectly flat.
See Figure 6. Thus, the cone idea enables us to select the
region of our wire mesh globe for which the resulting approximation is
valid.
Figure 6
I suspect there are a few Casual Cartographers
out there who have figured out that, in reality, the cylinder could
also be considered to be a very very very tall cone.
That is a cone that is infinitely tall.
Perhaps not so obvious, is that if you were to try to make the
absolutely shortest and fattest cone possible, you would in essence be
making a flat plane. Which
gets use right back to where we started all of this.
So, we should understand that the cylinder and the flat plane
are really just very special cases of the cone.
However, since us mere mortals haven’t yet figured out how to
divide by zero, or where infinity really is, we end up treating these
special cases separately.
So, in order to develop a State Plane, an
ordinary flat plane is not used directly, the region of validity is
just too small for an entire state.
The region in which the flat plane approximation is valid is
just too small to be used as a State Plane.
However, all is not lost.
By using the cylinder or cone ideas, the size of the region in
which the flat plane approximation is valid can be greatly enlarged.
Thus, State Planes are actually based on either a cylinder or a
cone; either of which can be flattened out to actually produce the
“plane” upon which a State Plane is actually based.
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