Conversion of NGP-0 Coordinates to Device

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Specific Coordinates

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Conversion of NGP-0 coordinates to floating point PDP-10 coordinates

was discussed in RFC #387. In general, however, it is undesirable to

convert NGP coordinates to floating point coordinates because real

devices require integer addressing. To this end, a means is described

to convert NGP coordi- nates to integer coordinates in the range zero

to M, where M is the maximum address of the device screen on a machine

using 2's complement arithmetic. It would not, however, be difficult

to modify this algorithm to operate on machines using one's complement

or sign-magnitude arithmetic.

First consider the NGP coordinate format:

                   +--+-----------+
                   |  |   n       |
                   +--+-----------+
                    s ^  FRACTION
                    i
                    g
                    n

Where the sign occupies the most significant bit of the coordinate

followed by bits of numerical information (initial implementation of

NGP requires N=15). Negative numbers are represented by 2's

complement. Conversion to device coordinates is accomplished by:

                    D = S * f + S

Where D =>integer device coordinate

S =>scaling factor (typically M/2)
f =>NGP fractional coordinate

Let us rewrite this as:

                            n     n
                    D = S*(2 *f)/2 +S

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Now factor S into two terms:

I
S= Q * 2

Where Q is an odd integer and I is an integer.

When: I n n

                    D = Q * 2 *(2 *f)/2  +S

                             I-n   n
                      = Q * 2   *(2 *f)  +S
             n

The factor (2 *f) is represented in 2's complement form simply by

extending the sign bit of f into the upper portion of the computer

word, If Q = 1 (as it would be with many devices), it can be ignored.

If Q >< 1, we may console ourselves that an integer multiply is faster

on most machines than a floating point multiply. In fact, on a

PDP-10, this multiply can usually be performed with no access to

memory since Q is usually small.

I-n

We are now left with the 2 factor. This can be accomplished with an

arithmetic shift left by (I-n) or an arithmetic shift right by (n-I)

as is appropriate. The offset factor, S, may now be added using an

integer add.

The procedure for converting NGP coordinates to integer device

coordinates is then:

1 move coordinate to a register and extend sign

2. integer multiply by Q (if necessary)

3 arithmetic shift left by (I-n)

4. integer add S

This procedure would generally be much faster than:

1 move coordinate to register and extend sign

2. float fractional coordinate

3 floating point multiply

4. floating point add

5 conversion to fixed point

[ This RFC was put into machine readable form for entry ]

       [ into the online RFC archives by BBN Corp. under the   ]
       [ direction of Alex McKenzie.                      1/97 ]

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