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Herein is described how pixel coordinates should be encoded into
GKI metacode units and how this data is then converted back to pixel numbers
by the "lower level" code. For concreteness, the discussion is based on
a 512 x 512 display, where the pixels are numbered from 1 to 512 (one-based)
or 0 to 511 ( zero based). Only the X axis is discussed, the Y axis
being treated identically.
GKI metacode ranges from 0 through 32767, for a total of 32768
values. In NDC coordinates, the range is from 0.0 through 1.0.
These coordinates are show in the diagram following.
.sp
.nf
last GKI coordinate of pixel
63 127 191 255 319 32703 32767(!)
pixel | | | | | | |
extent-- |<-->||<-->||<-->||<-->||<-->|| ... |<-->||<--->|
| | | | | | |
the |-----|-----|-----|-----|-----| ... |-----|-----|
pixels | | | | | | ... | | |
|-----|-----|-----|-----|-----| ... |-----|-----|
num- (1-b) 1 2 3 4 5 511 512
bers (0-b) 0 1 2 3 4 510 511
| | | | | | ... | | |
GKI 0 64 128 192 256 320 32640 32704 32767(!)
| | | | | | ... | | |
NDC 0.0 1/512 2/512 3/512 4/512 5/512 511/512 1.0
.fi
.sp
The pixels are not points, but rather, in GKI/NDC space, have
"physical" extent. In NDC coordinates, the pixel boundaries are
easily calculated as (left boundary = zero-based pixel number / 512)
and (right boundary = 1-based pixel number / 512). In GKI coordinates,
each pixel spans 64 GKI units, with the left boundary given by
"zero-based pixel number times 64". The right boundary is then the
left boundary plus 64 and then actually references the next pixel.
That is, the left boundary is included in the pixel, while the right
boundary is not.
(Pixel 0 goes from 0 through 63, pixel one from 64 through 127, etc.)
This works for all pixels except the last one, which would have a
right boundary of 32768; in this special case, the right boundary
is defined to be 32767. As will be seen later on, this should cause
no difficulties.
Explicit reference to a particular pixel should, in GKI
coordinates, refer to the pixel's left (or for Y, lower) edge. Thus,
pixel 7 (one-based system) is, in GKI, 6*64 or 384.
Cell arrays are denoted by their lower-left and upper-right
corners, with the understanding that all pixels WITHIN this rectangle
are to be read/written. Thus, an array that covers (one-based)
(4,10) to (18, 29) implies that, in X, pixels 4 through 17 are referenced.
Therefore, the GKI coordinate range is from 3*64 up to 17*64, where
3*64 is the GKI coordinate for the left edge of pixel 4 and 17*64 is
the GKI coordinate for the right edge of pixel 17. (Remember, the
right edge of pixel 512 is 32767, not 32768.)
The (real) NDC coordinate that is then passed to the interface code
is determined by dividing the GKI coordinate by 32767. The interface
code will, ultimately, multiply by 32767 to give the GKI coordinates
passed to the lower level.
The lower level code translates the GKI coordinate values into
zero-based pixel numbers by multiplying by 512/32768 ( not 32767).
The (real) pixel numbers so determined are then truncated, and become
the ones to scroll to, zoom to, or put the cursor on. Therefore,
when refering to single pixels for such operations, use the left
boundary of the pixel as the desired GKI/NDC coordinate.
Pixel computation for cell arrays is somewhat more complicated.
The right boundary of a cell array can be the left boundary for
an adjacent cell array; if the simple truncation scheme were used, that
coordinate would be included in both cell array operations, which is not
acceptable (especially for hard copy devices where the resultant overplotting
would be, at best, objectionable). This problem gives rise to the following
algorithm. Left (and lower) positions are rounded up to the next pixel
boundary if the fractional position is greater than or equal 0.5. Right
(and upper) positions are rounded down to the next pixel boundary if the
fractional position is less than 0.5; since a fractional pixel value of 0.0
is less than 0.5, the right/upper pixel will be decreased even if it is
already on a boundary. The truncated values are then used as the
INCLUSIVE range of pixels to read or write. (If the positions lie
within the same pixel, that pixel becomes the X (or Y) range. If the
positions are in adjacent pixels, the right pixel operation is
not done if the left pixel moves into the same pixel as the right one.)
With this algorithm, the right edge of the display (NDC=1.0,
GKI=32767) becomes position 511.98, which is not rounded down as the
fractional part is >= 0.5, and, which, when truncated, turns into 511
which is what is desired as the (last) included pixel in the range.
For zoomed (image) displays, fractional pixel coordinates are
possible in the sense that, for a zoom of 4, pixels 16.0, 16.25,
16.50, and 16.75, all refer to the same datum. When setting the cursor,
the lower level code must distinguish all these cases, which have GKI
values (from a one-based coordinate system) 960, 976, 992, and 1008.
The lower level code will return these fractional pixel values when reading
the cursor, but the integral value is the real reference to the data
point. However, calls to the getcell and putcell routines should use
16 (aka 960) or the cell array rounding will interfere with what is
wanted. This does restrict getcell calls from starting/ending in the middle
of a zoomed (replicated) pixel, but makes the behavior of getcell
the same as putcell, which cannot write into the middle of a zoomed pixel.
In summary, users should reference individual pixels by
specifying their left (or lower) boundaries in GKI/NDC. For cursor
reference on zoomed displays, fractional (in the sense outlined above)
pixels may be referenced. Right (or upper) boundaries are used only
for cell arrays, and except for the very right-most, are determined by
the user in an operation similar to that for the left boundaries. GKI
coordinates that are a little too large (not more than 31 units for a
512 resolution device) will be rounded/truncated to the desired
coordinate. For cell array operations only, ones a little too small
will still address the correct pixel.
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