zProblem

[norseman]

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
> 
> 
> 
========================================
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