P3D Phantogram Phormula (Part 1) "Scoping the Situation"

[Bruce Springsteen <bsspringsteen@xxxxxxxxx>]

This will likely get me banished to T3D with John B, but the mob has
necessitating a posting.  Brevity may be the soul of wit, but in my
case it's more a laughing matter.  Here I go.

The requests for the following easily went over my P3D "threshold" -
more "Phanto Phanatics" than I imagined.  So let's cut to the numbers
(we need a *basic* familiarity with Cartesian coordinate systems in 3
dimensions - many or most of you will skim the preliminaries):

Imagine a sheet of paper in front of you on a table.  On that sheet of
paper is a small "+".  This + marks the intersection, or "origin", of
three imaginary lines in space, the X, Y, and Z axes of a
three-dimensional coordinate system.  The X axis runs through this
"origin" (called "O") from left to right on the paper surface.  To the
left of that origin, x has a negative value, to the right a positive
value.  Also on the paper, at right angles to the X axis, is the Z
axis.  Positive values of z are farther from you than O, negative
values nearer.  Also passing through O is the Y axis, emerging
straight up from the paper (where y values are positive) and straight
down (where y values are negative), at right angles to both of the
other axes.

Now we can define the position of any point in the surrounding space
by its distance from O in each of those 3 dimensions.  Any such
point's location is expressed as (x,y,z).  The origin O is at (0,0,0).
A point on the x axis, 10 cm to the left of O, would be at (-10,0,0).
If you also pushed that point away from you on the paper by 5 cm (in
the Z dimension), it would now be at (-10,0,5).  If you elevate that
point straight up from the page by 7 cm (in the Y dimension), its
location coordinates are now (-10,7,5). 

Next imagine a wire-frame cube, with edges 5 cm long, sitting on the
paper in front of you.  This cube is sitting with its square bottom
face centered on the origin mark "+", and its edges all parallel with
the X, Y and Z axes.  This cube has eight corners, four in a square on
the surface of the paper and four above those in a square 5 cm above
the first.  Let's name those corners.  Starting with the elevated
square, working clockwise from the far left corner we have points
A,B,C & D.  The bottom square, again going clockwise from the far left
corner, has points E,F,G & H.  

Each of those eight points has a location (x,y,z) in our system.
A = (-2.5, 5,  2.5) 
B = ( 2.5, 5,  2.5)
C = ( 2.5, 5, -2.5) 
D = (-2.5, 5, -2.5)
E = (-2.5, 0,  2.5)   (Notice that points E-H all have a y value of 0.
F = ( 2.5, 0,  2.5)    Points on the surface of the paper will always be
G = ( 2.5, 0, -2.5)    at 0 in the Y dimension, and will not move at all
H = (-2.5, 0, -2.5)    in the finished phantogram!)

Now comes the fun part.  Imagine a point exactly midway between your
two eyes, at about the top of your nose.  Let's call that point "CE"
for "center eye" or "Cyclops eye".  When you sit looking down at the
paper, your viewing position can be described as the (x,y,z) location
of that point.  If you sit with your nose centered on the X axis,
looking straight along the Z axis, with your eyes 32 cm above the
page, and 32 cm away from the origin O, your viewing point CE is at
the location (0, 32, -32).  Let's call that height (in the Y
dimension) above the page "h".  Let's call that distance (in the Z
dimension) from the origin "d".

Of course you are really looking at the cube not from CE, but from two
eyes to the left and right of CE, about 6.5 cm from each other (more
or less.) That distance between the eyes is the stereo base - call it
"b".  So now the "viewing position" of the cube in our scene can be
conveniently described in terms of CE's height and distance from
origin O, and by the distance between the two eyes, which are the
actual "stereo" points of view.  For the sake of this example then:

h = 32 cm  (ie: how high above the origin are you?)
d = 32 cm  (ie: how far from the origin are you?)
b = 6.5 cm (ie: how far apart are your eyes?)

To summarize, we now have:
A 3-dimensional coordinate system for locating points in space as
(x,y,z) from an origin "O".

A wire cube 5 x 5 x 5 centimeters sitting on a page squarely on the
origin "O".  We know the (x,y,z) locations of each of its eight corners.

A viewing position CE between our eyes that is a convenient height and
distance from the cube, as measured from O, with two actual eyes 6.5
cm apart.

Next installment:  "Making the Phantogram" 
    











 










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End of PHOTO-3D Digest 3204
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