GB,
"...you simply lay the rectangular grid on top of the piece of pie..."
Whenever someone uses the word "simply", I know I'm in trouble. I'm fairly dense. This is not sinking in.
Let's say you have a point at the intersection of the Northing Axis and the Central Parallel. Any points due west or east along that Central Parallel will have no change in latitude(Y), right? On the Grid Projection, those points will not "curve" up? The Grid Northing will remain the same for all points?:-S
Dave
Here's my drawing...
Dave,
"As you go east or west along a line of latitude from the central meridian the northings increase."
You've just pulled the rug out from under my feet. Grid East should be perpendicular to Grid North. If this is not so, then I'll have to admit defeat and never move to a state with a Lambert Projection.
Dave
Here's my drawing...
Yes, the grid is perpendicular but lines of latitude are curved.
As you travel east or west on the grid line you fall south the same if you backsight geodetic north, turn 90 degrees right or left. The grid line is tangent to the line of latitude at the central meridian.
Dave -
To avoid any misunderstanding, let's make sure we are limiting the discussion here to only the northern hemisphere. I don't think anyone wants to cross the equator with a conic projection.
The central parallel or central latitude is usually only discussed in the context of a single-parallel Lambert projection (such as used in France), but that does not matter.
As you go east or west of the central meridian maintaining constant latitude the northing increases, like in the US maps we have seen forever.
In the image above, the US-Canada border is pretty much 49 degrees north from the southeast corner of Manitoba to a few miles west of Point Roberts, Washington. The northing of the latitude-defined part of the border is lowest at the central meridian.
GB
Here's my drawing...
Don't give up David. You'll get it. It'll be worth it once it clicks.
Remember that two 90° turns from the central meridian will not be pointed north again. You'll be pointed parallel to the meridian which is the same as grid north. Convergence of meridians causes this.
GB,
OK, so the term Central Parallel is only used when it's a Single-Parallel Lambert. I was wrongly using the term to describe the halfway parallel between the two Secant Standard Parallels.
Aren't angles maintained on a Lambert Conformal Projection? If East-West is not perpendicular to North-South on a State Plane Grid System, then I'm going to have to admit defeat on this one.
I'll dive in and re-read all my links this weekend. Maybe something will click.
Thanks for making the attempt.
Dave
Here's my drawing...
Thanks for trying, Dave. I need to go read and think awhile. I will re-read and consider your words.
Dave
Here's my drawing...
Shawn,
I hardly ever give up. The only two things I ever gave up on as completely unknowable are Einstein's Theory of Relativity, and Women. 😉
I've got a long day tomorrow. I'll pick it back up this weekend.
Dave
Dave -
Study the conformality section starting on page 17 of the Stem document and note the limitations conveyed by the use of terms like "infinitesimal lines", "very short lines", and "small areas".
In the US map above, consider the northeast corner of Arizona. The north line is a parallel of latitude and the east line is a meridian. They cross at right angles in the geodetic sense and they are mapped at right angles as well, if only for the infinitesimal line segments immediately adjacent to the intersection.
You'll get it sooner or later, keep on it!
GB
Here's my drawing...
😛
The same issues occur in transverse Mercator so you're stuck!
We're talking about two sets of lines on a map.
1. There are the grid lines of easting and northing. They are straight and rectangular and parallel to the paper/map edges. It's a 2D Cartesian grid.
2. The graticule lines. These are lines of equal latitude or equal longitude that have been projected into the map's coordinate system [for display]. They can be straight or curved and are sometimes parallel to an edge of the paper/map.
In any map projection that has a convergence angle, grid north does not always equal geodetic or true north. That will be true in conic projections, and cylindrical projections that aren't in the normal aspect (where cylinder is upright).
A Mercator map is a cylindrical projection in normal aspect. Longitude and latitude lines are straight and parallel to the paper/map edge. The pole can't be displayed (Y = infinity) but data is stretched east-west to the same width as the equator or the standard parallel. Spacing between equal latitude invervals increases as you move away from the equator (or standard parallel) in order to maintain conformality.
A longitude or latitude value will have the same easting or northing value anywhere in the map.
A transverse Mercator map uses a 90 degree rotated cylinder. The longitude lines must converge to the poles so when displayed they are tilted towards the central meridian and slightly curved. Latitude lines are also curved slightly.
For conic projections in a normal aspect, longitude lines are tilted towards the central meridian but straight. Latitude lines are curved.
In these latter cases, for any point not on the central meridian, a longitude or latitude value will not project to the same easting or northing value.
Melita
Let me take a stab. Much of what i say has been said already in some manner, but i wish to tie some things together and emphasize clearing up some confusing issues:
Using very large areas is always good to illustrate concepts because they exaggerate and make obvious what would otherwise be very subtle. On very small areas, and assuming we're close to the points/lines of contact (standard parallels, central meridians, etc), the projection just won't matter, except to us surveyors.
Properties common to all projections
On the map, the scale factor will vary from place to place and may also vary (at a given place) depending on which direction one is facing.
On the map, the meridian convergence -- the difference between true (geodetic) north and grid (map) north -- will vary from place to place.
A geodesic line -- a true shortest route on the ellipsoid -- will not generally be projected as a straight line on the map.
A geodetic angle -- what we surveyors measure (on the ellipsoid) -- will not generally be projected onto the map without distortion.
Properties common to all conformal projections
On the map, the scale factor will still vary from place to place but will be constant (at a given place) no matter which direction one is facing.
A geodetic angle -- what we measure between two directions -- will always be projected onto the map without distortion.
That is why we surveyors like conformal projections: angles are preserved and distances can be converted independently of direction. There are still meridian convergence angles though.
Issues I have with some discussions of projections
Conformal projections preserve "shape". What is shape? On a plane, we know that two polygons are similar -- have the same shape -- if corresponding vertex angles are equal and if corresponding sides are in equal proportion. But we cannot measure shape as a single quantity. On an ellipsoid, a polygon's sides are geodesics. The same polygon on a map -- after a conformal projection -- may have non-straight sides, will have have corner angles preserved, and will have distorted side lengths. Is the shape preserved? No more than it is preserved on any other type of map projection, if the shape is small enough and close enough to the "contact lines".
Some projections preserve (or some distort) directions (in this context, meaning bearings or azimuths, not circle readings). What are directions? As surveyors we know that they are angles measured between north and some other direction. And we know that there are generally two different norths on a map, true and grid. So unless we disambiguate between true and grid, it is meaningless to consider what a projection does to "directions".
I've needed this thread for quite a while, now.
Thanks, all!
I don't know about you, but I'm claiming PDH's just from the read and refreshment of damaged brain cells on this one.
Pablo B-)
Chowderhead Gets It
Melita,
I re-read every post on the thread. Carefully. Slowly. I even moved my lips as I was reading. Everyone here was trying to tell me the same thing, but it didn't connect until you wrote:
"We're talking about two sets of lines on a map."
If I start walking East, I will diverge from the Latitude I began at. (Except if I start at a point on the Equator.)
Thank you.
Dave
Martin,
"On the map, the scale factor will still vary from place to place but will be constant (at a given place) no matter which direction one is facing."
I'm struggling with that one. Since the scale factor applies to line length, wouldn't it matter greatly?
Dave
GB,
"...'infinitesimal lines', 'very short lines', and 'small areas'."
Thank you. I will re-attack page 17 tomorrow.
Dave
>
> "On the map, the scale factor will still vary from place to place but will be constant (at a given place) no matter which direction one is facing."
>
> I'm struggling with that one. Since the scale factor applies to line length, wouldn't it matter greatly?
Yes, scale factor matters. Sorry, I didn't mean to suggest that it didn't matter.
Martin,
No, I mean doesn't the direction matter? If the scale is different at any point on the projection, I'd think that direction was critical to determining line length. How the scale is applied to a line whose scale varies is a whole 'nuther incomprehensible noodle-buster. :-S
Dave
> I mean doesn't the direction matter? If the scale is different at any point on the projection, I'd think that direction was critical to determining line length. How the scale is applied to a line whose scale varies ...
Ah yes, of course, good question. So, if the line is very long and if the scale factor is significantly different at the other end (in that case direction is important), then i believe the average of the two scale factors would be used. Who's done this, anyone?
Hey, what we're doing here is more of a thought experiment -- just what your friend Einstein used to do. In practice, a single site-wide average scale factor is plenty good enough (for the women we hang out with). 😉
It's been awhile since I needed to do it, but as I recall:
k' = (k1 + k2 + 4((k1+k2)/2))/6
k1 = Scale Factor @ Pt.1
k2 = Scale Factor @ Pt.2
Loyal
