Short Mathematica and Matlab code snippets for "Transformations and Projections" (excluding the codes in chapter 3).

------ Chapter 1 ------

Page 23

d2r=Pi/180;
Table[Round[N[16384*Sin[i*d2r]]], {i,0,90}]

Page 28

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]

Page 50

tm=Sqrt[x^2+y^2];
a={{x/tm,-y/tm,0},{y/tm,x/tm,0},{0,0,1}};
b={{z,0,Sqrt[1-z^2]},{0,1,0},{-Sqrt[1-z^2],0,z}};
c={{Cos[t],-Sin[t],0},{Sin[t],Cos[t],0},{0,0,1}};
FullSimplify[a.b.c.Transpose[b].Transpose[a] /. x^2+y^2->1-z^2]

Page 51 (This is Matlab code)

n=3;
A=[.5774,-.5774,-.5774; .5774,.7886,-.2115; .5774,-.2115,.7886]
% Rotation from 1,1,1 to x-axis
Q=eye(n);
for j=1:n-1,
  for i=n:-1:j+1,
    T=eye(n);
    D=sqrt(A(j,j)^2+A(i,j)^2);
    cos=A(j,j)/D; sin=A(i,j)/D;
    T(j,j)=cos; T(j,i)=sin; T(i,j)=-sin; T(i,i)=cos; T
    A=T*A;
	Q=Q*T';
  end;
end;
Q
A

Page 51 (This is Matlab code)

T1=[0.7071,0,0.7071; 0,1,0; -0.7071,0,0.7071];
T2=[0.8165,0.5774,0; -0.5774,0.8165,0; 0,0,1];
T3=[1,0,0; 0,0.9660,0.2587; 0,-0.2587,0.9660];
p=[1;1;1];
a=T1*p
b=T2*a
c=T3*b

------ Chapter 4 ------

Page 149

(* hemispherical fisheye projection *)
Clear[k, n, P, Q, L]
k=10; n=50;
scal[q_]:=(k Tan[ArcTan[q/k]/2])/q;
P=Table[{Random[Real,{-10.,10.}], Random[Real,{-10., 10.}]},{n}];
Q=Table[Sqrt[P[[i]].P[[i]]], {i, n}];
L=Table[Line[{P[[i]], scal[Q[[i]]] P[[i]]}], {i, n}];
Show[Graphics[L], Graphics[Circle[{0, 0}, 10]],
 Graphics[Point[{0, 0}]],  AspectRatio -> 1]

Page 151

k = 10;
angl = {22.5, 45., 67.5, 89.};
k Tan[angl Degree]
k Tan[angl/2 Degree]


Page 157

k = 10; n = 50; scal[q_] := (k Tan[ArcTan[q/k]/2])/q;
P = Table[{Random[Real, {-10.,10.}], Random[Real, {-10.,10.}]}, {n}];
x = -5; y = 5; (* Location of viewer *)
Pt = P - Table[{x, y}, {n}];
Q = Table[Sqrt[Pt[[i]].Pt[[i]]], {i, n}];
L = Table[Line[{P[[i]]+{x, y}, (scal[Q[[i]]] P[[i]])+{x, y}}], {i, n}];
Show[Graphics[L], Graphics[Circle[{0, 0}, k]],
 Graphics[{AbsolutePointSize[5], Point[{0, 0}]}],
 Graphics[{AbsolutePointSize[5], Point[{x, y}]}],
 AspectRatio -> Automatic, PlotRange -> All]

Page 195

k=10.;
Table[k z/(z+k), {z,0,100,5}]
Table[%[[i+1]]-%[[i]], {i,1,20}]
Table[Point[{%%[[i]],0}], {i,1,21}];
Show[Graphics[%]]

Page 196

l=20.; r=0.1;
Table[l(1-(z r/(z+l))), {z,0,100,5}]
Table[%[[i]]-%[[i+1]], {i,1,20}]
Table[Line[{{i, 17}, {i, %%[[i]]}}], {i,1,21}]
Show[Graphics[%]]



----------- Answers to exercises ---------

Page 251

t14=2^14;
Print["(x*=",(8192-(2 14189.))/t14,",y*=",(14189.+(2 8192))/t14,")"]
Print["(x*=",Cos[60 Degree]-2. Sin[60 Degree],
 ",y*=",Sin[60 Degree]+2. Cos[60 Degree], ")"]

Pages 251-252

t14=2^14;
Print["(x*=",(2845.-(2 16135.))/t14,",y*=",(16135.+(2 2845.))/t14,
  ")"]
Print["(x*=",Cos[80 Degree]-2. Sin[80 Degree],
 ",y*=",Sin[80 Degree]+2. Cos[80 Degree], ")"]

Pages 267-268

(* exercise for hemispherical fisheye projection *)
k=1;
scal[q_]:=(k Tan[ArcTan[q/k]/2])/q;
{scal[1.],scal[10.], scal[100.], scal[1000.], scal[10000.]}