Transformations - Transformations and Projections in Computer Graphics - page 24

t=Table[ArcTan[2.^{-i}], {i,0,15}]; (* arctans in radians *)
d=1; x=2.1; y=0.34; z=46. Degree;
Do[{Print[i,", ",x,", ",y,", ",z,", ",d],
xn=x+y d 2^{-i}, yn=y-x d 2^{-i},
zn=z-d t[[i+1]], d=Sign[zn], x=xn, y=yn, z=zn}, {i,0,14}]
Print[0.60725x,", ",0.60725y]

Figure 1.13: Mathematica Code for CORDIC Rotations.

Exercise 1.30: Instead of using the complex CORDIC method, wouldn't it be simpler
to perform a rotation by a direct use of Equation (1.17)? After all, this only requires
the calculation of one sine and one cosine values.

1.2.5 Similarities

A similarity is a transformation that scales all distances by a fixed factor. It is easy to
show that a similarity is produced by the special transformation matrix
⎛

⎞

ac 0
−

⎝ ⎠ .

ca 0
mn 1

To show this, we observe that translations preserve distances, so we can ignore

the translation part of the matrix above and restrict ourselves to the matrix ac

−ca .It

transforms a point P =( x, y )tothepoint P ∗ =( x ∗ ,y ∗ )=( ax

− cy, cx + ay ). Given

the two transformations P 1 → P 1 and P 2 → P 2 , it is straightforward to illustrate the

relation
distance 2 ( P 1 P 2 )= (Δ x ∗ ) 2 +(Δ y ∗ ) 2
=[( ax 2 −

cy 1 )] 2 +[( cx 2 + ay 2 ) ( cx 1 + ay 1 )] 2

cy 2 ) − ( ax 1 − −

c Δ y ) 2 +( c Δ x + a Δ y ) 2

=( a Δ x −

= a 2 Δ x 2 2 a Δ xc Δ y + c 2 Δ y 2 + c 2 Δ x 2 +2 c Δ xa Δ y + a 2 Δ y 2

=( a 2 + c 2 )(Δ x 2 +Δ y 2 )
=( a 2 + c 2 ) distance 2 ( P 1 P 2 ) ,

−

implying that distan ce ( P 1 P 2 )= √ a 2 + c 2 distance ( P 1 P 2 ). Thus, all distances are

scaled by a factor of √ a 2 + c 2 .
In general, a similarity is a transformation of the form P ∗ =( x ∗ ,y ∗ )=( ax − cy +

( cx + ay )+ n ), where the ratio of expansion (or shrinking) is k = √ a 2 + c 2 .If k is
positive, the similarity is called direct ;if k is negative, the similarity is opposite .

m, ±

Exercise 1.31: Discuss the case k =0.