# PROGRAM: Vanishing Points
# AUTHOR: Andrejs Treibergs
# DATE: 6-23-01
> with(plots):with(linalg);
Warning, the protected names norm and trace have been redefined and
unprotected


  [BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp,

        Wronskian, addcol, addrow, adj, adjoint, angle, augment,

        backsub, band, basis, bezout, blockmatrix, charmat, charpoly,

        cholesky, col, coldim, colspace, colspan, companion, concat,

        cond, copyinto, crossprod, curl, definite, delcols, delrows,

        det, diag, diverge, dotprod, eigenvals, eigenvalues,

        eigenvectors, eigenvects, entermatrix, equal, exponential,

        extend, ffgausselim, fibonacci, forwardsub, frobenius,

        gausselim, gaussjord, geneqns, genmatrix, grad, hadamard,

        hermite, hessian, hilbert, htranspose, ihermite, indexfunc,

        innerprod, intbasis, inverse, ismith, issimilar, iszero,

        jacobian, jordan, kernel, laplacian, leastsqrs, linsolve,

        matadd, matrix, minor, minpoly, mulcol, mulrow, multiply,

        norm, normalize, nullspace, orthog, permanent, pivot,

        potential, randmatrix, randvector, rank, ratform, row, rowdim,

        rowspace, rowspan, rref, scalarmul, singularvals, smith,

        stackmatrix, submatrix, subvector, sumbasis, swapcol, swaprow,

        sylvester, toeplitz, trace, transpose, vandermonde, vecpotent,

        vectdim, vector, wronskian]

# A cube with long lines to infinity.
> bob:=[[0,0,0],[0,0,1],[1000,0,1],[0,1,1],[0,1,0],[1000,0,0],[0,0,0],[0
> ,1000,0],[1,0,0],[1,0,1],[0,1000,0],[0,0,1]]:spacecurve(bob,view=[-1..
> 10,-1..10,-1..10]);

> spacecurve(bob,view=[-1..10,-1..10,-1..2],scaling=constrained# 
> ,axes=normal);

# Rotation function.
> rogue:=lis->[cos(theta)*lis[1]-sin(theta)*lis[2],sin(theta)*lis[1]+cos
> (theta)*lis[2],lis[3]];
> theta:=0.5;hlis:=map(rogue,bob);
# Perpendicular projection of cube.
# 
# 
> ep:=[0.0,-3.5,1.90];
> cp:=[0.0,0.0,0.5];lis:='lis';
> T:=lis->[lis[1]-ep[1],lis[2]-ep[2],lis[3]-ep[3]]:
> dp:=T(cp):
> r1:=sqrt(dp[1]^2+dp[2]^2):
> s1:=dp[1]/r1:c1:=dp[2]/r1:
> R:=lis->[c1*lis[1]-s1*lis[2],c1*lis[2]+s1*lis[1],lis[3]]:
> rdp:=R(dp):
> r2:=sqrt(dp[1]^2+dp[2]^2+dp[3]^2):
> s2:=rdp[3]/r2:c2:=rdp[2]/r2:
> S:=lis->[lis[1],-s2*lis[2]+c2*lis[3],c2*lis[2]+s2*lis[3]]:
> S(rdp):
> SRT:=lis->S(R(T(lis))):
> GPp:=lis->[op(1,SRT(lis)),op(2,SRT(lis))]:GPpP:=lis->op(3,SRT(lis)):
> delta=10.1;GPp(lis);GPpP(lis);

                        ep := [0., -3.5, 1.90]


                          cp := [0., 0., .5]


                              lis := lis


                             delta = 10.1


  [1.000000000 lis[1],

        .3713906764 lis[2] - .464238346 + .9284766909 lis[3]]


        .9284766909 lis[2] + 3.955310703 - .3713906764 lis[3]

> nn:=nops(hlis); kpro:=map(GPp,hlis): 
> display([plot(kpro,color=`navy`),
> plot({[GPp([0,0,-3]),GPp([0,0,10])],[GPp([0,-3,0]),GPp([0,10,0])],[GPp
> ([-3,0,0]),GPp([10,0,0])]},color=gray),plot([[0,0]],style=point,symbol
> =circle)], scaling=constrained,title=`Parallel Projection in Eye to
> Center direction`,axes=none,view=[-21..21,-21..21]);
# 

                               nn := 12


# Perspective view of cube.
# 
> GPx:=lis->[op(1,GPp(lis))/GPpP(lis),op(2,GPp(lis))/GPpP(lis)]:
> colo:=[`magenta`,`red`,`coral`,`yellow`,`green`,`cyan`,`blue`,`navy`,`
> violet`,`maroon`]:
> nn:=nops(hlis); kpro:=map(GPx,hlis):  display([plot(kpro,color=blue),
> plot({[GPx([0,0,-3]),GPx([0,0,10])], [GPx([0,-3,0]),GPx([0,10,0])],
> [GPx([-3,0,0]),GPx([10,0,0])]},color=black),
> plot([GPx([-1111,1111,0]),GPx([1011,1111,0])]),plot([[0,0]],style=poin
> t,symbol=circle)], scaling=constrained,title=`Perspective Projection
> in Eye to Center direction`,axes=none,view=[-01.1..1.1,-0.7..0.7]);

                               nn := 12


# Construction of vanishing points.
> d:=array(1..10): e:=array(1..10): convert(e,list):
> for ii from 1 to 10 do;
> theta:=(ii+6)/17.45;d[ii]:=map(rogue,bob);e[ii]:=
> [[[-2,-1],[5,-1]],[[-tan(theta),-1],[0,-2],[cot(theta),-1]],[(sin(s)+c
> os(2.0*theta))/sin(2.0*theta),-1-cos(s)/sin(2.0*theta),s=-Pi/2..Pi/2]]
> ;od:
> leftwig:=t->[[0,0,0], [-1000.0*tan(t/2.0),1000.0,0], [1,0,0],
> [2,0,0],[-1000.0*tan(t/2.0),1000.0,0],[3,0,0]]:
> theta:=.5:leflin:=leftwig(theta): rightwig:=t->[[-1,0,0],
> [1000.0*tan(Pi/4.0 - t/2.0),1000.0,0], [-2,0,0],[-3,0,0],
> [1000.0*tan(Pi/4.0 - t/2.0),1000.0,0],[0,0,0]]:
> riglin:=rightwig(theta):
> 
> kproj:=[seq(map(GPx,d[j]),j=1..10)]: ovie:=[seq(plot(
> [kproj[j],[GPx([-1111,1111,0]),GPx([1011,1111,0])],
> op(e[j]),[GPx([0,0,-300]),GPx([0,0,2.2])],
> [GPx([-4,0,0]),GPx([4,0,0])]],scaling=constrained,title=`Perspective
> Projection in Eye to Center
> direction`,axes=none,view=[-2..2,-3..1]),j=1..10)]:
> display(ovie,scaling=constrained,insequence=true);

# 3 x 3 x 1 box.
> # 
> boe:=[[0,0,0],[0,0,1],[2,0,1],[2,0,0],[1,0,0],[1,0,1],[1000,0,1],[0,3,
> 1],[0,3,0],[1000,0,0],[0,0,0],[0,2,0],[0,2,1],[0,1,1],[0,1,0],[0,1000,
> 0],[3,0,0],[3,0,1],[0,1000,0],[0,0,1]]:
> spacecurve(boe,view=[-1..10,-1..10,-1..2],scaling=constrained,axes=nor
> mal);

# Parallel projection of 3x3x1 box wiith measuring lines.
> theta:=0.5:hlie:=map(rogue,boe):leflin:
> nn:=nops(hlie); kpro:=map(GPp,hlie): kprm:=map(GPp,leflin):
> kprr:=map(GPp,riglin):display([plot(kpro,color=`violet`),
> plot(kprm,color=`blue`), plot(kprr,color=`green`),
> plot({[GPp([0,0,-3]),GPp([0,0,10])],[GPp([0,-3,0]),GPp([0,10,0])],[GPp
> ([-5,0,0]),GPp([10,0,0])]},color=gray),plot([[0,0]],style=point,symbol
> =circle)], scaling=constrained,title=`Parallel Projection in Eye to
> Center direction`,axes=none,view=[-11..11,-3..11]);

                               nn := 20


# Perspective drawing with 3x3x1 bos with measuring lines.
> nn:=nops(hlie); kpro:=map(GPx,hlie):kprm:=map(GPx,leflin):
> kprr:=map(GPx,riglin):  display([plot(kpro,color=`violet`),
> plot(kprm,color=`blue`), plot(kprr,color=`green`),
> plot({[GPx([0,0,-3]),GPx([0,0,10])], [GPx([0,-3,0]),GPx([0,10,0])],
> [GPx([-5,0,0]),GPx([10,0,0])]},color=brown),
> plot([GPx([-1111,1111,0]),GPx([1011,1111,0])]),plot([[0,0]],style=poin
> t,symbol=circle)], scaling=constrained,title=`Perspective Projection
> in Eye to Center direction`,axes=none,view=[-01.1..1.1,-0.7..0.7]);

                               nn := 20


# Perspective drawing of 3x3x1 box with the construction of vanishing
# points animation.
> lef:=array(1..10):convert(lef,list):rig:=array(1..10):convert(rig,list
> ):
> for ii from 1 to 10 do;
> theta:=(ii+6)/17.45;d[ii]:=map(rogue,boe);e[ii]:=
> [[[-2,-1],[5,-1]],[[-tan(theta),-1],[0,-2],[cot(theta),-1]],[(sin(s)+c
> os(2.0*theta))/sin(2.0*theta),-1-cos(s)/sin(2.0*theta),s=-Pi/2..Pi/2]]
> ; od:
> kproj:=[seq(map(GPx,d[j]),j=1..10)]: ovie:=[seq(plot(
> [kproj[j],[GPx([-1111,1111,0]),GPx([1011,1111,0])],
> op(e[j]),[GPx([0,0,-300]),GPx([0,0,2.2])],
> [GPx([-4,0,0]),GPx([4,0,0])]],scaling=constrained,title=`Perspective
> Projection in Eye to Center
> direction`,axes=none,view=[-2..2,-3..1]),j=1..10)]:
> display(ovie,scaling=constrained,insequence=true);

# Perspective drawing of 3x3x1 box with the construction of vanishing
# points and measuring points animation.
# 
> for ii from 1 to 10 do; theta:=(ii+6)/17.45;d[ii]:=map(rogue,boe);
> e[ii]:=
> [[cos(s)*sec(theta)-tan(theta),-1-sec(theta)*sin(s),s=0..Pi/2-theta],
> [cot(theta)-csc(theta)*cos(s),-1-csc(theta)*sin(s),s=0..theta],
> [(sin(s)+cos(2.0*theta))/sin(2.0*theta),-1-cos(s)/sin(2.0*theta),s=-Pi
> /2..Pi/2], [[-2,-1],[5,-1]],
> [[-tan(theta),-1],[0,-2],[cot(theta),-1]], 
> [[cot(theta)-csc(theta),-1],[0,-2],[sec(theta)-tan(theta),-1]]];
> lef[ii]:=map(GPx,leftwig(theta)):
> rig[ii]:=map(GPx,rightwig(theta));od:
> kproj:=[seq(map(GPx,d[j]),j=1..10)]: ovie:=[seq(plot(
> [kproj[j],[GPx([-1111,1111,0]),GPx([1011,1111,0])],
> op(e[j]),rig[j],lef[j],[GPx([0,0,-300]),GPx([0,0,2.2])],
> [GPx([-4,0,0]),GPx([4,0,0])]],scaling=constrained,title=`Constructing
> Vanishing Points and Measuring
> Points`,axes=none,v