> Taking the square root of the data de-emphasizes the densest areas, helping fix this. Actually, I’ll tell you a secret: I showed the 2.25th root of density, not the square root, because I’m way too detail obsessive.
This is an important detail. I used something similar before. Never assume that the best conversion from data to color is linear. The monitor is not linear, the eye is not linear, who knows if the printer is linear.
Not to be a wet noodle on the cerebral art explanation or ignore the technical discussion, both of which were interesting, but as an extremely practical matter a former coworker of mine had a life goal to row a boat around the perimeter of Lake Superior (a little bigger, and to the north) and this linearized map is kinda what he saw, which is both cool and would be an interesting way to document his photos and stories and it might have been a useful navigational tool for him, if zoomed in to human scale.
And no he didn't live in his boat and do it in one trip, it took him years (decades?) of his life. He'd do city B to city C on one vacation and city C to city D on some weekend etc.
Social media photo sharing sites are kind of "done" as a startup idea (or are they?) but I could see something like this, generalized to work on cars, planes, and bikes, being kinda popular to scroll and zoom thru someones trip photos as an unusual kind of map display. Maybe for navigation apps, scroll along a long line...
We have a sailboat on Lake Michigan. That map of Lake Michigan is, for the most part, how we experience it when we do a "sailing adventure" because we are going port-to-port along the coast.
A comment on the math: the article portrays it as quite verbose to say the least, which I think might indicate to some that it is "scary" or really complicated. I get the hunch that it need not be that way, if approached differently.
Affine transformations, conceptually, are quite elegant: https://en.wikipedia.org/wiki/Affine_transformation. Figuring out the values to use for a particular transformation is the key here. This is where the blog post gets really verbose in the math.
The complexity of the solution, as written, seems much higher than the inherent complexity of the problem. So my intuition suggests the mechanics of the solution (e.g. the amount of notation required, at least, if not the amount of computation) can be dramatically simplified if done a different way. I'd think that playing around with polar coordinates and various frames of reference would help find an elegant calculation.
those aren't affine transformations he's using, they're plane projective transformations (parallel lines aren't preserved). Yes, you can express what he means more simply, but as an 8x8 matrix problem - then just use a maths library to solve it. (see eg http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/EPSR...).
> I once briefly understood why we were doing what we were doing, but that knowledge has since vanished into the ether
Looks like generalized barycentric coordinate over the quadrilateral? Sort of like UV mapping, though it'd be good to hear from someone that knows more.
I've often thought about doing something like this for documenting my bike trips; a 'somewhat straightened' map would work great down the side of the page with the story. The straight line route maps that used to appear in AA map books are a bit dull, I wanted something more like http://w.faringdon.org/alljpg/hy%20Map%20London%20to%20St%20...
However I got sidetracked in the notion of how to linearize the map. What I want, is roughly: start with the route, smooth it so that the original route still wiggles on the final map (eg https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93..., then interpolate with splines)
Simply using straight line distance to the curve won't work, places can be the same distance from multiple points on the route (unless you divide space up as in the article).
So what I thought of instead was, treat the smoothed line as a charged wire, and map POIs as opposite charges. Electric field lines do not intersect, and if a POI 'charge' is allowed to fall towards the 'wire', the distance it travels and the place it lands give me coordinates. These calculations are relatively straightforward (see eg http://lampx.tugraz.at/~hadley/physikm/apps/linecharge.php), and I never need to figure out a closed form for the transformation, which I guess would be horrendous.
I'm not sure what you'd call this coordinate system; it bears some similarity with isobaric coordinate systems in meteorology though.
Ok, as I understand it, there is no canonical way to unfurl a lakeshore. Sure, a map of the lakeshore can be drawn differently. But there is no real parameterization of a shoreline. Its called the 'coastline of England' problem or some such. I.e. an approximate shoreline might be 200 miles. Draw every nook and cranny - 300 miles. Draw the accurate line around every grain of sand that's wet on one side and dry on the other - 20000 miles. What is the 'real shoreline'?
I love this. It's brilliant how enormous the cultural dependecies are in what we think of as completely uncontroversial geographic facts - like how "Upper" and "Lower" Egypt are the 'wrong way round' from a modern frame of reference - i.e. Upper Egypt was the south, because it was where the Nile flowed from.
"wrong way round' from a modern frame of reference"?
Isn't the modern use the same as the Egyptian one; that "upper" means "upstream", no matter the direction of river flow?
For example, in https://en.wikipedia.org/wiki/Ob_River you can see "The upper Ob valley, in the south, grows grapes, melons and watermelons, whereas the lower reaches of the Ob are Arctic tundra."
At http://www.protectingourwater.org/watersheds/map/upper_st_jo... you can see "The Upper St. Johns River Basin extends from the headwaters of the St. Johns River in Indian River and Okeechobee Counties to the confluence of the St. Johns and Econlockhatchee Rivers in Seminole County".
Up/Down have come to mean North/South in the way many people speak in English (US-centric? Not certain how other English speaking countries handle this). I think that's what GP post was referring to.
That it's actually referring to upriver/downriver in the case of Egypt is lost on those who aren't more closely familiar with the geography. It makes sense once explained, but it's non-obvious if you've become accustomed to up and down being synonymous with North and South.
Side note: There's a discussion about Common Core math and a comment by DanBC about how students may correctly fill out many math worksheets without developing any actual understanding just by coincidence (some idiosyncratic algorithm or method that's technically wrong, but happens to work for the worksheets). The same thing happens with language. If people only encounter an idiom in a limited context, they may come to misunderstand what the idiom actually expresses. In this case that it's not strictly the cardinal directions, but the direction of the rivers.
While I agree that many people use north=up/south=down, I'm not convinced that it's meaningful to talk about the 'modern frame of reference' in this way.
We have one major river in the US which flows south, the Mississippi. It's such a big influence on the local views that people in New Orleans may use "north" to mean "up river" even in a bend where the river flows from geographic south. The Mississippi is very influential in US history. Northern slavers used the threat to "sell down the river" to the South.
We also have the Hudson River, which is important for New York. Someone sent "up the river" was being sent to the Sing Sing Correctional Facility, thirty miles north of New York City.
So it's not surprising that those set a baseline for a lot of people.
But if you're from Portland, OR, then the river flows (locally) west. If you're from Jacksonville, FL, the river flows (locally) east, after coming from the south. If you're from Nebraska, the Platte flows mostly east. In Boston, the Charles flows mostly northeasterly. (And to double-check, the Upper Charles River Reservation is indeed 1) upstream, and 2) southwest of the mouth.)
To say that the modern frame of reference has changed, I would want to see evidence for change. Where is the "upper" part of a river system used for something other than upstream?
Otherwise, my belief is that eponeponepon is asserting a personal view that is not really backed by the evidence.
My point was that upper and lower seem to (anecdata) be disconnected from the context of rivers. Upper and Lower Egypt make sense once reminded of this, but for many (anecdata) it seems upper and lower mean north and south so their intuitive sense is that the naming is "wrong" or seems off.
Yes, I think I understand your point. Yes, there are many people who use "upper" and "lower" as rough equivalents to "north" and "south".
Here is but one of many examples - "Up and down can be confusing when one talks about Egypt. In ancient times the term Upper Egypt actually meant heading downwards into Africa, while Lower Egypt meant heading upwards on the map, to the Nile Delta." http://motherdaughterbookreviews.com/guest-post-the-magic-of...
My point, however, is that the phrase 'modern frame of reference' is meaningless. At best it's "personal sense of reference" or "a common mistake."
That is, our 'modern' use now seems to be identical to what it was when the terms "upper" and "lower" Egypt were coined.
Also, note that that book review is wrong. At least, according to Wikipedia, 'Lower Egypt was known as Ta-Mehu which means "land of papyrus"' and 'Upper Egypt was known as Ta Shemau which means "the land of reeds"', so it's not like modern English uses a direct translation of the Ancient Egyptian.
So, when was this earlier time that's supposed to be different from now?
This is an important detail. I used something similar before. Never assume that the best conversion from data to color is linear. The monitor is not linear, the eye is not linear, who knows if the printer is linear.