Transformations - Transformations and Projections in Computer Graphics - page 22

bring this vector to its correct size, it should be multiplied by the scale factor

m

K m = cos θ i .

i =0

(Notice that cos( d i θ i )=cos θ i since cosine is an even function.) This is discouraging
because it suggests that m multiplications are needed just to calculate the scale factor
K m . However, the first 16 scale factors are listed in Table 1.11 and even a quick glance
shows that they converge to the number 0.60725 ... . Reference [Vachss 87] shows that
K m can be obtained simply by using the m most significant bits of this number and
ignoring the rest.
Using the identity sin 2 θ +cos 2 θ = 1 and the definition tan θ i =2 −i ,weget

1
1+tan 2 θ i

1
√ 1+2 − 2 i ,

cos θ i = =

which is why the scale factors of Table 1.11 were so easily calculated to a high precision
by the code
N[Table[Product[(2^(-2i)+1)^(-1/2),{i,0,n}],{n,0,16}],17]//TableForm .

y

P * =( x * , y * )

θ 3

θ

θ 2

θ 1

P =( x , y )

θ 0

x

Figure 1.12: CORDIC Rotation.

Exercise 1.29: Suggest another way to calculate K m .
Any practical CORDIC implementation (see [Jarvis 90] for a C program) should
have the following two features.