Reading James E. Stem's State Plane Coordinate System of 1983, I suddenly looked up from the page and wondered thusly:
OK, I can see how the Transverse Mercator Projection works. When you "unwrap" the cylinder, the Central Meridian is a straight, North-South line. Meridian lines diverge from "True North" the farther away from the Central Meridian you get. Cool. Got it. No problemo.
BUT.......
How does a Lambert Conformal Conic Projection (upside-down snow cone) work? When you "unwrap" the snow cone holder, the Standard Parallels are not straight lines. The projection would have to be "rubber-sheeted" to make the Standard "Lines" straight. Without perpendicular lines, no grid. And with rubber-sheeting, you lose the conformance of "True" North.
Dave
It's been a while since I studied this, and my understanding is crude, but I believe part of the math with a Lambert conic is that the radial distance from the tip of the cone to the POINT in the projection is calculated. The arc along the projection from the central meridian to the point is figured with that radius. This arc distance is then figured as a departure (perpendicular to North) from the central meridian. I know there is more to it than this as the math is more involved, but I think this is a proper general visual of what is taking place.
See http://geodesyattamucc.pbworks.com/w/file/fetch/63366627/Lec5_2013_SPCS_lecture.pdf
Take a look at the dimensions of the cone and the dimensions of the zone.
HTH,
DMM
Shawn,
I was hoping you'd weigh in on this one.
The only way I can visualize the rubber-sheeting is to imagine "tipping" the projection plane until the Standard Parallels are "straight". No tipping would be needed at the Equator, but more and more tipping would be required the farther you get from the Equator. Or something. I wish I could wrap my head around it.
In Stem's article, he mentioned that several people wanted to make SPCS83 strictly Transverse Mercator Projections and eliminate the Lamberts. I can see some merit there. Unfortunately, the proposed 2-degree spacing of the secants just didn't work.
Dave
GOM/DMM,
Thanks for the link. Heading there now. I'll report back.
Dave
Hi Dave,
I'm not sure I understand your questions. Does this link help any:
In a state plane zone using Lambert conformal conic, the standard parallels are closely spaced, relatively. So they're going to be a lot 'straighter' than what's implied by a world view. Just as in a TM-based SPCS zone, the farther you get from the central meridian, the larger the convergence angle will be.
The standard lines in a TM zone aren't straight either. They don't follow a meridian, but instead roughly parallel the central meridian.
The latitude and longitude lines are perpendicular because these projections are conformal, but they do not form a rectangular grid. The easting, northing lines are the rectangular grid.
Melita
Melita,
Thanks for the link.
In order to make a State Plane Grid with a Lambert Conformal Projection, the area above the Central Parallel has to be "expanded" and the area below the Central Parallel has to be "compressed". Straightening out the Central Parallel adds the same diverging-meridians distortion that the TM Projection suffers from. What conforms in a Lambert Conformal Projection?
Dave
GOM/DMM,
This is reasonable and understandable:
Here's where my grasp suffers:
The Central Parallel MUST be perpendicular to the Northing Axis in order to define a (rectangular) State Plane Grid. This is probably the key to my understanding, but it's beyond my mental grasp:
Dave
As you can see from the point n,e, it is not really East from the perpendicular to the central meridian (if it were truly east, it would be substantially more northerly). This is where the distortion comes in. basically you take the arc distance from the point to the central meridian with the radius being the tip of the pointy cone to the point. Then from that radius, along the central meridian, you turn 90 (on the flattened cone) the arc distance. The curved lines you're seeing are True East (so to speak). The Grid doesn't follow True East, it follows perpendicularly from the central meridian.
Here's an image showing the rectangular grid (which determines northing and easting) in addition to the standard parallels, does this help?
This diagram is vastly larger than a practical projection zone would be, so there is a lot of exaggeration. But the exaggeration makes it easy to grasp the concept of convergence of meridians.
To address your conformality question below, the adjective conformal as applied to projections means that the relative angles between lines are correct and that the scale factor at any specific point is the same in every direction (but of course the scale factor changes throughout the map area).
GB
Shawn,
Do the projected meridian lines that converge at the point of the dunce cap coincide with the meridian lines that hug the globe and converge at the North Pole? If they are the same, that would explain a lot.
Dave
Here's my drawing...
Flatten the cone on the table and break down the triangles as shown...
Calculate the latitude and departure of A by calculating the orange triangle. The origin is SW of the bottom of the page so that has to be taken into account when calculating northings and eastings.
In the projections world, conformal always means local shape (aka angle) is maintained. A right angle on the ellipsoid will be a right angle in the map.
Yes, they're the same, but the length of a meridian will differ because the line has been projected to the cone. Because of conformality, the longitude and latitude lines at that point meet at right angles. Angles are conserved, distance, direction, and area are not.
GB,
The meridians won't converge if the Standard Parallels projected to Grid are perpendicular to the Northing Axis. So it wouldn't look like a huge slice of pie anymore. It baffles me how to go about arm wrestling that piece of pie into a rectangular grid.
"...the scale factor at any specific point is the same in every direction..."
I read that in Stem's White Paper and didn't understand it. The scale factor would be 1.0 for the length of two points on a Standard Parallel, but would be entirely different if one of the points was anywhere else.
Dave
Melita,
Thank you! That makes sense that the angles are conserved, but not distance, direction and area.
Dave
Melita,
"...conformal always means local shape (aka angle) is maintained."
I wondered how shape could be conserved in a projection. Thanks for tying the two together. I understand now.
Dave
Here's my drawing...
Dave,
Nice drawing. To which point's coords would the departure and latitude be added to?
Would (n,e) be adjusted to the intersection of the red lines?
Dave
Dave -
The standard parallels projected to grid are perpendicular to the northing axis only for an infinitesimal distance where they cross the central meridian. Did you find something in the Stem document that indicates otherwise?
You don't need to wrestle the "piece of pie" into a rectangular grid - you simply lay the rectangular grid on top of the piece of pie as in the diagram above and as shown in Dave Karoly's diagram below.
In a Lambert projection, any change in latitude results in a change of scale factor.
The significance of the scale factor being the same in every direction is that a (tiny) circle on the ellipsoid is mapped as a (tiny) circle. If the scale factor varied with direction a circle on the ellipsoid would look like an ellipse or a close approximation on the map.
GB
Here's my drawing...
The northing is calculated at the dashed line above your red line.
As you go east or west along a line of latitude from the central meridian the northings increase.
If you travel north along a meridian line east of the central meridian eastings decrease. If you travel north along a meridian line west of the central meridian eastings increase.
