Steven D'Aprano wrote:
On Tue, 14 Apr 2009 14:42:32 -0700, norseman wrote:
Grids are uniform! Same size, non-changing across whole backdrop. There
is nothing in uniform that says X==Y. Units along axis need not be same.
Corners don't even have to be 90degrees. (Spherical) But they must
measure as same size cells across the board. Just like any grid paper.
Except for log paper, log-log paper, normal-probability paper, and other
non-uniform grids.
http://en.wikipedia.org/wiki/File:Log_paper.svg
Or Penrose tilings:
http://en.wikipedia.org/wiki/Penrose_tiles
For those reading this that just said "AH-Hah!", Spherical (Lat./Long.)
is not measured in Cartesian (distance like feet or meter) but in angles
(like 7 and 1/2 minute USGS Quads). 7.5minutes of Longitude at the
equator does not have the same arc length as 7.5minutes at the poles.
But both are 7.5minutes and thus form a (polar) grid. ENOUGH OF THIS -
sorry for being long winded.
Whoever wrote Tk was not crazy. Just didn't use a dictionary.
Nor should they. "Grid" has technical meanings (note plural) that are not
well-suited to a dictionary definition. I'm amused that Wiktionary gives
one definition for grid as a rectangular array of uniformly sized squares
or rectangles, and illustrated it with a curvilinear grid of non-uniform
pieces!
http://en.wiktionary.org/wiki/grid
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Log grids are actually same units apart. Find a picture of a Slide Rule.
Circular... are polar and again lines increase by same unit.
Again - nothing dictates X==Y as the only choice. Each axis may be
different from any other. Relief models usually hold X & Y and
exaggerate the Z by many multiples of the other scale(s). Even the Grand
Canyon when displayed on a small table without an exaggerated Z is very
unimpressive.
Not all dimensional objects can be depicted in planar mode without
distortion. But grids are defined as same unit spacing.
Google up some geodetics and geo-referencing and check the "maps" of the
world in different planar projections.
Trying to display an object of n Dimensions on an object of less than n
dimensions is the nightmare of every mapmaker.
If that's not clear, try: get an orange and peal the skin off in one
contiguous piece and place it flat on the table.
Actually, to be fully correct, to display any n dimensional object on
any non-n dimensional object will result in distortion.
The location of a point on the Earth being depicted on paper is put
through a (compound) formula to properly convert grid locations.
Same in reverse, but the algorithm is different.
ps. I used to have a "ball" made of pentagons. It had no holes.
Homework assignment: generate the math to properly display tiny objects
glued to all surfaces of that "ball", maintaining proper spatial
relationships, on paper. Hint: define the grid for the paper first. You
will also need one for the "ball".
Steve
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