Transformations - Transformations and Projections in Computer Graphics - page 19

91 sine values, from sin 0 ◦ =0tosin90 ◦ = 1, multiplies each by 2 14 =16 , 384, rounds
them, and stores them in a table as 16-bit integers ranging from 0 to 16,384.
d2r=Pi/180;
Table[Round[N[16384*Sin[i*d2r]]], {i,0,90}]

θ sin θ θ sin θ θ sin θ θ sin θ θ sin θ

0 0 1 286 2 572 3 857 4 1143

5 1428 6 1713 7 1997 8 2280 9 2563

10 2845 11 3126 12 3406 13 3686 14 3964

15 4240 16 4516 17 4790 18 5063 19 5334

20 5604 21 5872 22 6138 23 6402 24 6664

25 6924 26 7182 27 7438 28 7692 29 7943

30 8192 31 8438 32 8682 33 8923 34 9162

35 9397 36 9630 37 9860 38 10087 39 10311

40 10531 41 10749 42 10963 43 11174 44 11381

45 11585 46 11786 47 11982 48 12176 49 12365

50 12551 51 12733 52 12911 53 13085 54 13255

55 13421 56 13583 57 13741 58 13894 59 14044

60 14189 61 14330 62 14466 63 14598 64 14726

65 14849 66 14968 67 15082 68 15191 69 15296

70 15396 71 15491 72 15582 73 15668 74 15749

75 15826 76 15897 77 15964 78 16026 79 16083

80 16135 81 16182 82 16225 83 16262 84 16294

85 16322 86 16344 87 16362 88 16374 89 16382

90 16384

Table 1.10: Sine Values as 16-Bit Integers.

The 91 values are listed in Table 1.10, but notice that they are only approximations
of the true sine values. (Even floating-point sine values are, in general, just approxi-
mations, but normally better than our integers.) This means that the use of this table
for many successive rotations of a point may place it farther and farther away from its
true position. When we perform many successive rotations of an object that consists
of many points, placing points away from where they should be generally results in a
deformation of the object.
We assume that the points are represented by coordinates that are 16-bit integers.
Calculating the rotated coordinates ( x ∗ ,y ∗ )ofapoint( x, y ) can now be done, for
example, by

x ∗ =rshift( x

Table( θ ) , 14) ,
y ∗ =rshift( x × Table( θ ) , 14) + rshift( y × Table(90 − θ ) , 14) .

× Table(90 − θ ) , 14) − rshift( y ×

Notice how the required cosine values are obtained from the end of the table. This
method works because the table has 91 entries. Multiplying a 16-bit integer coordinate