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ch4.mp
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ch4.mp
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% CH4.mp
% MetaPost input file with chapter four pictures.
verbatimtex
%&latex
\documentclass{book}
\usepackage{bookjh}
\usepackage{linalgjh}
\begin{document}
etex
input jh
ahangle:=40;
input arrow
input jh3d
%========================CRAMER'S RULE=======================
% Parallelogram picture
beginfig(1);
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v;
z0=(0w,0v);
z1=(2w,1v); z2=(1w,3v); % endpoints of the two vectors
z3=(6w,8v);
z4=2*z1; z5=2*z2; % solution is x=2, y=2
pickup pencircle scaled line_width_light; %inside parallegram
draw z0--z1--(z1+z2)--z2--cycle;
pickup pencircle scaled line_width_light; %outside parallegram
draw z0--z4--z3--z5--cycle;
drawvec_dark_with_border(z0,z1); drawvec_dark_with_border(z0,z2);
drawvec_dark(z0,z1); drawvec_dark(z0,z2);
label.lrt(btex {\tiny $\colvec{2 \\ 1}$} etex,z1);
label.lrt(btex {\tiny $x_2\cdot\colvec{2 \\ 1}$} etex,z4);
label.lft(btex {\tiny $\colvec{1 \\ 3}$} etex,z2);
label.lft(btex {\tiny $x_1\cdot\colvec{1 \\ 3}$} etex,z5);
label.rt(btex {\tiny $\colvec{6 \\ 8}$} etex,z3);
endfig;
% Parallelogram picture with shaded box
beginfig(2);
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v;
z0=(0w,0v);
z1=(2w,1v); z2=(1w,3v); % endpoints of the two vectors
z3=(6w,8v);
z4=2*z1; z5=2*z2; % solution is x=2, y=2
path inside_pgram;
inside_pgram=z0--z1--(z1+z2)--z2--cycle;
pickup pencircle scaled line_width_light; %inside parallegram
draw inside_pgram;
pickup pencircle scaled line_width_light; %outside parallegram
draw z0--z4--z3--z5--cycle;
drawvec_dark_with_border(z0,z1); drawvec_dark_with_border(z0,z2);
fill inside_pgram withcolor shading_color;
drawvec_dark(z0,z1); drawvec_dark(z0,z2);
label.lrt(btex {\tiny $\colvec{2 \\ 1}$} etex,z1);
label.lft(btex {\tiny $\colvec{1 \\ 3}$} etex,z2);
%label.rt(btex {\tiny $\colvec{6 \\ 8}=x_1\cdot\colvec{1 \\ 3}+x_2\cdot\colvec{2 \\ 1}$} etex,z3);
endfig;
% Parallelogram picture with shaded box dilated by x
beginfig(3);
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v;
z0=(0w,0v);
z1=(2w,1v); z2=(1w,3v); % endpoints of the two vectors
z3=(6w,8v);
z4=2*z1; z5=2*z2; % solution is x=2, y=2
path inside_pgram;
inside_pgram=z0--z5--(z1+z5)--z1--cycle;
pickup pencircle scaled line_width_light; %inside parallegram
draw inside_pgram;
pickup pencircle scaled line_width_light; %outside parallegram
draw z0--z4--z3--z5--cycle;
drawvec_dark_with_border(z0,z1); drawvec_dark_with_border(z0,z2);
fill inside_pgram withcolor shading_color;
drawvec_dark(z0,z1); drawvec_dark(z0,z2);
label.lrt(btex {\tiny $\colvec{2 \\ 1}$} etex,z1);
label.lft(btex {\tiny $x_1\cdot\colvec{1 \\ 3}$} etex,z5);
%label.rt(btex {\tiny $x_1\cdot\colvec{1 \\ 3}+x_2\cdot\colvec{2 \\ 1}$} etex,z3);
endfig;
% Parallelogram picture with shaded box dilated by x and shifted by y
beginfig(4);
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=10pt; v:=u; w:=v;
z0=(0w,0v);
z1=(2w,1v); z2=(1w,3v); % endpoints of the two vectors
z3=(6w,8v);
z4=2*z1; z5=2*z2; % solution is x=2, y=2
path inside_pgram;
inside_pgram=z0--z3--(z3+z1)--z1--cycle;
pickup pencircle scaled line_width_light; %inside parallegram
draw inside_pgram;
pickup pencircle scaled line_width_light; %outside parallegram
draw z0--z4--z3--z5--cycle;
drawvec_dark_with_border(z0,z1); drawvec_dark_with_border(z0,z2);
fill inside_pgram withcolor shading_color;
pickup pencircle scaled line_width_light; % far edge shown inside shaded
z6=whatever[z4,z3]=whatever[z1,z3+z1];
draw z6--z3 withcolor medgray;
drawvec_dark(z0,z1); drawvec_dark(z0,z2);
label.lrt(btex {\tiny $\colvec{2 \\ 1}$} etex,z1);
label.ulft(btex {\tiny $\colvec{6 \\ 8}$} etex,z3);
%label.rt(btex {\tiny $x\cdot\colvec{1 \\ 3}+y\cdot\colvec{2 \\ 1}$} etex,z3);
endfig;
%============= TOPIC: projective plane ============================
%
%
beginfig(5) % introductory central projection
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.2in; v:=u; w:=v;
drawing_scale:=30pt;
save lpn; lpn:=new_vect; vect_def(lpn,0,4,0);
% Locate the observer and describe where he's looking
set_point(Obs,15,4,6);
Obs_phi:=90; Obs_dist:=10;
point_of_view_abs(lpn,Obs_phi);
% where to locate cneter of middle figure, and right-hand figure
save rt_mid, mid_mid, left_mid; numeric rt_mid, mid_mid, left_mid;
rt_mid = 15; mid_mid = .7rt_mid; left_mid = 0;
% draw the line to be projected to the plane, in the middle
save l_back, l_front; % two ends of line
l_back:=new_vect; l_front:=new_vect;
vect_def(l_back,-2,mid_mid,-.5); vect_def(l_front,3,mid_mid,.25*3);
project_point(80,l_back); project_point(81,l_front);
free_vect(l_front); free_vect(l_back);
save fig_mid; path fig_mid;
fig_mid = z80--z81;
pickup pencircle scaled line_width_light;
draw fig_mid;
% show the plane on the right in proportion
save p_ufront, p_lfront, p_lback, p_uback; % four corners of plane
p_ufront:=new_vect; p_lfront:=new_vect; p_lback:=new_vect; p_uback:=new_vect;
vect_def(p_ufront,3,rt_mid,2); vect_def(p_lfront,3,rt_mid,-2);
vect_def(p_lback,-3,rt_mid,-2); vect_def(p_uback,-3,rt_mid,2);
project_point(91,p_ufront); project_point(92,p_lfront);
project_point(93,p_lback); project_point(94,p_uback);
free_vect(p_uback); free_vect(p_lback);
free_vect(p_lfront); free_vect(p_ufront);
save fig_rt; path fig_rt;
fig_rt = z91--z92--z93--z94--cycle;
pickup pencircle scaled line_width_light;
draw fig_rt;
% points on line and plane (pts a, b, c)
save mid_pta, mid_ptb, mid_ptc, rt_pta, rt_ptb, rt_ptc;
mid_pta:=new_vect; mid_ptb:=new_vect; mid_ptc:=new_vect;
rt_pta:=new_vect; rt_ptb:=new_vect; rt_ptc:=new_vect;
vect_def(mid_pta,0,mid_mid,0); vect_def(mid_ptb,1,mid_mid,.25*1);
vect_def(mid_ptc,2,mid_mid,.25*2);
vect_def(rt_pta,0,rt_mid,0); vect_def(rt_ptb,.5,rt_mid,.25*.5);
vect_def(rt_ptc,1,rt_mid,.25);
project_point(70,mid_pta); project_point(71,mid_ptb);
project_point(72,mid_ptc);
project_point(73,rt_pta); project_point(74,rt_ptb);
project_point(75,rt_ptc);
free_vect(rt_ptc); free_vect(rt_ptb); free_vect(rt_pta);
free_vect(mid_ptc); free_vect(mid_ptb); free_vect(mid_pta);
% make line on plane with similar triangles
save proportion; numeric proportion;
proportion = length(z73-(0,0))/length(z70-(0,0));
z76 = proportion[(0,0),z71];
z77 = proportion[(0,0),z72];
pickup pencircle scaled line_width_light;
draw (1.8[z77,z73])--(1.3[z73,z77]);
draw (0,0)--z73 withcolor lightgray;
drawpoint(z70); drawpoint(z73);
label.top(btex {\tiny $A$} etex,z73);
draw (0,0)--z76 withcolor lightgray;
drawpoint(z71); drawpoint(z76);
label.top(btex {\tiny $B$} etex,z76);
draw (0,0)--z77 withcolor lightgray;
drawpoint(z72); drawpoint(z77);
label.top(btex {\tiny $C$} etex,z77);
save stickfig_hgt; numeric stickfig_hgt; stickfig_hgt = 1.5u;
pickup pencircle scaled line_width_light;
drawstickfigure(stickfig_hgt) shifted (left_mid,-.9stickfig_hgt);
%
free_vect(lpn);
endfig;
beginfig(6) % projection from source to image; by a movie projector
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.18in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,4,0);
% Locate the observer and describe where he's looking
set_point(Obs,15,4,6);
Obs_phi:=90; Obs_dist:=10;
point_of_view_abs(lpn,Obs_phi);
% where to locate cneter of middle figure, and right-hand figure
save rt_mid, mid_mid, left_mid; numeric rt_mid, mid_mid, left_mid;
rt_mid = 15; mid_mid = .7rt_mid; left_mid = 0;
% draw the line to be projected, in the middle
% in contrast to the prior picture, it now forms part of the Source plane
save s_ufront, s_lfront, s_lback, s_uback; % four corners of plane
s_ufront:=new_vect; s_lfront:=new_vect; s_lback:=new_vect; s_uback:=new_vect;
vect_def(s_ufront,2.5,mid_mid,2); vect_def(s_lfront,2.5,mid_mid,-2);
vect_def(s_lback,-2.5,mid_mid,-2); vect_def(s_uback,-2.5,mid_mid,2);
project_point(85,s_ufront); project_point(86,s_lfront);
project_point(87,s_lback); project_point(88,s_uback);
free_vect(s_uback); free_vect(s_lback);
free_vect(s_lfront); free_vect(s_ufront);
save src_plane; path src_plane;
src_plane = z85--z86--z87--z88--cycle;
pickup pencircle scaled line_width_light;
draw src_plane;
save l_back, l_front; % two ends of line
l_back:=new_vect; l_front:=new_vect;
vect_def(l_back,-2,mid_mid,-.5); vect_def(l_front,2,mid_mid,.25*2);
project_point(80,l_back); project_point(81,l_front);
free_vect(l_front); free_vect(l_back);
save fig_mid; path fig_mid;
fig_mid = z80--z81;
pickup pencircle scaled line_width_light;
draw fig_mid;
% show the plane on the right in proportion
save p_ufront, p_lfront, p_lback, p_uback; % four corners of plane
p_ufront:=new_vect; p_lfront:=new_vect; p_lback:=new_vect; p_uback:=new_vect;
vect_def(p_ufront,3,rt_mid,2); vect_def(p_lfront,3,rt_mid,-2);
vect_def(p_lback,-3,rt_mid,-2); vect_def(p_uback,-3,rt_mid,2);
project_point(91,p_ufront); project_point(92,p_lfront);
project_point(93,p_lback); project_point(94,p_uback);
free_vect(p_uback); free_vect(p_lback);
free_vect(p_lfront); free_vect(p_ufront);
save fig_rt; path fig_rt;
fig_rt = z91--z92--z93--z94--cycle;
pickup pencircle scaled line_width_light;
draw fig_rt;
% points on line and plane (pts a, b, c)
save mid_pta, mid_ptb, mid_ptc, rt_pta, rt_ptb, rt_ptc;
mid_pta:=new_vect; mid_ptb:=new_vect; mid_ptc:=new_vect;
rt_pta:=new_vect; rt_ptb:=new_vect; rt_ptc:=new_vect;
vect_def(mid_pta,0,mid_mid,0); vect_def(mid_ptb,1,mid_mid,.25*1);
vect_def(mid_ptc,2,mid_mid,.25*2);
vect_def(rt_pta,0,rt_mid,0); vect_def(rt_ptb,.5,rt_mid,.25*.5);
vect_def(rt_ptc,1,rt_mid,.25);
project_point(70,mid_pta); project_point(71,mid_ptb);
project_point(72,mid_ptc);
project_point(73,rt_pta); project_point(74,rt_ptb);
project_point(75,rt_ptc);
free_vect(rt_ptc); free_vect(rt_ptb); free_vect(rt_pta);
free_vect(mid_ptc); free_vect(mid_ptb); free_vect(mid_pta);
% make line on plane with similar triangles
save proportion; numeric proportion;
proportion = length(z73-(0,0))/length(z70-(0,0));
z76 = proportion[(0,0),z71];
z77 = proportion[(0,0),z72];
pickup pencircle scaled line_width_light;
draw (1.8[z77,z73])--(1.3[z73,z77]);
drawpoint(z70); drawpoint(z73);
% make the movie projector
save proj_width, proj_hgt; numeric proj_width, proj_hgt;
proj_width = w; proj_hgt = 1.5v;
save reel; path reel; reel = fullcircle scaled .85proj_hgt;
save projector; path projector;
projector = (.3w,.2v)--(0w,.1v)--(0w,.5proj_hgt)--(-proj_width,.5proj_hgt)
--(-proj_width,-.5proj_hgt)--(0w,-.5proj_hgt)
--(0w,-.1v)--(.3w,-.2v);
pickup pencircle scaled line_width_light;
draw projector;
draw reel shifted (-1.2proj_width,.55proj_hgt);
draw reel shifted (-1.2proj_width,-.55proj_hgt);
save number_beams; numeric number_beams; number_beams = 7;
for i = 1 upto number_beams:
draw ((.5w,0v)--(1.5w+uniformdeviate .35w,0v)) rotated (((i-1)/(number_beams-1))[-25,25]) withcolor lightgray;
endfor
% labels
x10 = 0w; y92-y10 = 10pt;
label(btex {\small \textit{projector $P$}} etex,z10);
x11 = .5[x85,x87]; y11 = y10;
label(btex {\small \textit{source $S$}} etex,z11);
x12 = .5[x91,x93]; y12 = y10;
label(btex {\small \textit{image $I$}} etex,z12);
%
free_vect(lpn);
endfig;
beginfig(7) % projection from P->I->S; by an artist
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.18in; v:=u; w:=v; %was .2
drawing_scale:=25pt; %was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,4,0);
% Locate the observer and describe where he's looking
set_point(Obs,15,4,6);
Obs_phi:=90; Obs_dist:=10;
point_of_view_abs(lpn,Obs_phi);
% where to locate cneter of middle figure, and right-hand figure
save rt_mid, mid_mid, left_mid; numeric rt_mid, mid_mid, left_mid;
rt_mid = 15; mid_mid = .7rt_mid; left_mid = 0;
% draw the line to be projected, in the middle
% in contrast to the prior picture, it now forms part of the Source plane
save s_ufront, s_lfront, s_lback, s_uback; % four corners of plane
s_ufront:=new_vect; s_lfront:=new_vect; s_lback:=new_vect; s_uback:=new_vect;
vect_def(s_ufront,2.5,mid_mid,2); vect_def(s_lfront,2.5,mid_mid,-2);
vect_def(s_lback,-2.5,mid_mid,-2); vect_def(s_uback,-2.5,mid_mid,2);
project_point(85,s_ufront); project_point(86,s_lfront);
project_point(87,s_lback); project_point(88,s_uback);
free_vect(s_uback); free_vect(s_lback);
free_vect(s_lfront); free_vect(s_ufront);
save src_plane; path src_plane;
src_plane = z85--z86--z87--z88--cycle;
pickup pencircle scaled line_width_light;
draw src_plane;
save l_back, l_front; % two ends of line
l_back:=new_vect; l_front:=new_vect;
vect_def(l_back,-2,mid_mid,-.5); vect_def(l_front,2,mid_mid,.25*2);
project_point(80,l_back); project_point(81,l_front);
free_vect(l_front); free_vect(l_back);
save fig_mid; path fig_mid;
fig_mid = z80--z81;
pickup pencircle scaled line_width_light;
draw fig_mid;
% show the plane on the right in proportion
save p_ufront, p_lfront, p_lback, p_uback; % four corners of plane
p_ufront:=new_vect; p_lfront:=new_vect; p_lback:=new_vect; p_uback:=new_vect;
vect_def(p_ufront,3,rt_mid,2); vect_def(p_lfront,3,rt_mid,-2);
vect_def(p_lback,-3,rt_mid,-2); vect_def(p_uback,-3,rt_mid,2);
project_point(91,p_ufront); project_point(92,p_lfront);
project_point(93,p_lback); project_point(94,p_uback);
free_vect(p_uback); free_vect(p_lback);
free_vect(p_lfront); free_vect(p_ufront);
save fig_rt; path fig_rt;
fig_rt = z91--z92--z93--z94--cycle;
pickup pencircle scaled line_width_light;
draw fig_rt;
% points on line and plane (pts a, b, c)
save mid_pta, mid_ptb, mid_ptc, rt_pta, rt_ptb, rt_ptc;
mid_pta:=new_vect; mid_ptb:=new_vect; mid_ptc:=new_vect;
rt_pta:=new_vect; rt_ptb:=new_vect; rt_ptc:=new_vect;
vect_def(mid_pta,0,mid_mid,0); vect_def(mid_ptb,1,mid_mid,.25*1);
vect_def(mid_ptc,2,mid_mid,.25*2);
vect_def(rt_pta,0,rt_mid,0); vect_def(rt_ptb,.5,rt_mid,.25*.5);
vect_def(rt_ptc,1,rt_mid,.25);
project_point(70,mid_pta); project_point(71,mid_ptb);
project_point(72,mid_ptc);
project_point(73,rt_pta); project_point(74,rt_ptb);
project_point(75,rt_ptc);
free_vect(rt_ptc); free_vect(rt_ptb); free_vect(rt_pta);
free_vect(mid_ptc); free_vect(mid_ptb); free_vect(mid_pta);
% make line on plane with similar triangles
save proportion; numeric proportion;
proportion = length(z73-(0,0))/length(z70-(0,0));
z76 = proportion[(0,0),z71];
z77 = proportion[(0,0),z72];
pickup pencircle scaled line_width_light;
draw (1.8[z77,z73])--(1.3[z73,z77]);
drawpoint(z70); drawpoint(z73);
% little painter person
save stickfig_hgt; numeric stickfig_hgt; stickfig_hgt = 1.5u;
pickup pencircle scaled line_width_light;
drawstickfigure(stickfig_hgt) shifted (left_mid,-.9stickfig_hgt);
% beret
pickup pencircle scaled line_width_light;
save beret; path beret; beret=fullcircle scaled .2u xscaled 1.65 rotated 20;
fill beret shifted (-.05w,.06v); draw beret shifted (-.05w,.06v);
drawpoint((-0.05w-.03w,.06v+.2u-.09v));
% paintbrush
pickup pencircle xscaled .9line_width_dark yscaled line_width_light;
draw (.4w,-.6v)--(.45v,-.25v);
% labels
x10 = 0w; y92-y10 = 10pt;
label(btex {\small \textit{painter $P$}} etex,z10);
x11 = .5[x85,x87]; y11 = y10;
label(btex {\small \textit{image $I$}} etex,z11);
x12 = .5[x91,x93]; y12 = y10;
label(btex {\small \textit{source $S$}} etex,z12);
%
free_vect(lpn);
endfig;
beginfig(8) % projection from source to pinhole to image; by eclipse
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.18in; v:=u; w:=v; %was .2in
drawing_scale:=25pt; %was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,4,0);
% Locate the observer and describe where he's looking
set_point(Obs,15,4,6);
Obs_phi:=90; Obs_dist:=10;
point_of_view_abs(lpn,Obs_phi);
% where to locate cneter of middle figure, and right-hand figure
save rt_mid, mid_mid, left_mid; numeric rt_mid, mid_mid, left_mid;
rt_mid = 15; mid_mid = .7rt_mid; left_mid = 0;
% draw the line to be projected, in the middle
% in contrast to the prior picture, it now forms part of the Source plane
save s_ufront, s_lfront, s_lback, s_uback; % four corners of plane
s_ufront:=new_vect; s_lfront:=new_vect; s_lback:=new_vect; s_uback:=new_vect;
vect_def(s_ufront,2.5,mid_mid,2); vect_def(s_lfront,2.5,mid_mid,-2);
vect_def(s_lback,-2.5,mid_mid,-2); vect_def(s_uback,-2.5,mid_mid,2);
project_point(85,s_ufront); project_point(86,s_lfront);
project_point(87,s_lback); project_point(88,s_uback);
free_vect(s_uback); free_vect(s_lback);
free_vect(s_lfront); free_vect(s_ufront);
save src_plane; path src_plane;
src_plane = z85--z86--z87--z88--cycle;
pickup pencircle scaled line_width_light;
draw src_plane;
save l_back, l_front; % two ends of line
l_back:=new_vect; l_front:=new_vect;
vect_def(l_back,-2,mid_mid,-.5); vect_def(l_front,2,mid_mid,.25*2);
project_point(80,l_back); project_point(81,l_front);
free_vect(l_front); free_vect(l_back);
save fig_mid; path fig_mid;
fig_mid = z80--z81;
pickup pencircle scaled line_width_light;
%draw fig_mid;
% show the plane on the right in proportion
save p_ufront, p_lfront, p_lback, p_uback; % four corners of plane
p_ufront:=new_vect; p_lfront:=new_vect; p_lback:=new_vect; p_uback:=new_vect;
vect_def(p_ufront,3,rt_mid,2); vect_def(p_lfront,3,rt_mid,-2);
vect_def(p_lback,-3,rt_mid,-2); vect_def(p_uback,-3,rt_mid,2);
project_point(91,p_ufront); project_point(92,p_lfront);
project_point(93,p_lback); project_point(94,p_uback);
free_vect(p_uback); free_vect(p_lback);
free_vect(p_lfront); free_vect(p_ufront);
save fig_rt; path fig_rt;
fig_rt = z91--z92--z93--z94--cycle;
pickup pencircle scaled line_width_light;
draw fig_rt;
% points on line and plane (pts a, b, c)
save mid_pta, mid_ptb, mid_ptc, rt_pta, rt_ptb, rt_ptc;
mid_pta:=new_vect; mid_ptb:=new_vect; mid_ptc:=new_vect;
rt_pta:=new_vect; rt_ptb:=new_vect; rt_ptc:=new_vect;
vect_def(mid_pta,0,mid_mid,0); vect_def(mid_ptb,1,mid_mid,.25*1);
vect_def(mid_ptc,2,mid_mid,.25*2);
vect_def(rt_pta,0,rt_mid,0); vect_def(rt_ptb,.5,rt_mid,.25*.5);
vect_def(rt_ptc,1,rt_mid,.25);
project_point(70,mid_pta); project_point(71,mid_ptb);
project_point(72,mid_ptc);
project_point(73,rt_pta); project_point(74,rt_ptb);
project_point(75,rt_ptc);
free_vect(rt_ptc); free_vect(rt_ptb); free_vect(rt_pta);
free_vect(mid_ptc); free_vect(mid_ptb); free_vect(mid_pta);
% make line on plane with similar triangles
save proportion; numeric proportion;
proportion = length(z73-(0,0))/length(z70-(0,0));
z76 = proportion[(0,0),z71];
z77 = proportion[(0,0),z72];
pickup pencircle scaled line_width_light;
%draw (1.8[z77,z73])--(1.3[z73,z77]);
drawpoint(z70); % drawpoint(z73);
% sun and moon
save sunsize; numeric sunsize; sunsize = 1.2u;
save sun, moon; path sun, moon;
sun = fullcircle scaled sunsize;
moon = fullcircle scaled sunsize shifted (.2w,0v);
draw sun shifted (0,0);
fill moon withcolor lightgray; draw moon;
% image
save sunimagesize; numeric sunimagesize; sunimagesize = .2u;
save sunimage, moonimage, image; path sunimage, moonimage, image;
sunimage = fullcircle scaled sunimagesize;
moonimage = fullcircle scaled sunimagesize shifted (.1w,0v); % reversed
draw sunimage shifted z73;
fill moonimage shifted z73 withcolor lightgray; draw moonimage shifted z73;
% labels
x10 = 0w; y92-y10 = 10pt;
label(btex {\small \textit{source $S$}} etex,z10);
x11 = .5[x85,x87]; y11 = y10;
label(btex {\small \textit{pinhole $P$}} etex,z11);
x12 = .5[x91,x93]; y12 = y10;
label(btex {\small \textit{image $I$}} etex,z12);
%
free_vect(lpn);
endfig;
%
% In these plane-to-plane projections, I've eliminated perspective
% as confusing.
beginfig(9) % projection with rail tracks on S
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.17in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,0,0);
% Locate the observer and describe where he's looking
set_point(Obs,8,8,1);
Obs_phi:=90; Obs_dist:=5;
point_of_view_abs(lpn,Obs_phi);
% Source plane at an angle with vertical
save splane_angle; numeric splane_angle; splane_angle=35;
save xmin, xmax, ymin, ymax, zmin, zmax;
numeric xmin, xmax, ymin, ymax, zmin, zmax;
-xmin=6; xmax=4; -ymin=ymax=1.25;
zmin=-xmax*sind(splane_angle); zmax=-xmin*sind(splane_angle);
save s_lower, s_midb, s_midf, s_upper; % center line of source plane
s_lower:=new_vect; s_midb:=new_vect; s_midf:=new_vect; s_upper:=new_vect;
vect_def(s_lower,xmin,0,zmax); vect_def(s_midb,0,ymin,0);
vect_def(s_midf,0,ymax,0); vect_def(s_upper,xmax,0,zmin);
project_point(80,s_lower); project_point(81,s_midb);
project_point(82,s_midf); project_point(83,s_upper);
free_vect(s_upper); free_vect(s_midf);
free_vect(s_midb); free_vect(s_lower);
z84-z80=z80-z85; z81-z84=z82-z85; % lower end
z86-z83=z83-z87; z86-z81=z87-z82; % upper end
save splane_lower, splane_upper; path splane_lower, splane_upper;
splane_lower = z81--z84--z85--z82;
splane_upper = z81--z86--z87--z82;
z88=.5[z81,z82];
% Image plane yz-plane
save i_upper, i_lower; % center line of image plane
i_upper:=new_vect; i_lower:=new_vect;
vect_def(i_upper,0,0,zmax); vect_def(i_lower,0,0,zmin);
project_point(90,i_upper); project_point(91,i_lower);
free_vect(i_lower); free_vect(i_upper);
z94-z91=z91-z95; z81-z94=z82-z95; % upper end
z92-z90=z90-z93; z92-z81=z93-z82; % lower end
save iplane_lower, iplane_upper; path iplane_lower, iplane_upper;
iplane_upper = z81--z92--z93--z82;
iplane_lower = z81--z94--z95--z82;
% projection point
save proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
numeric proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
proj_pt_x=2; proj_pt_zslant=proj_pt_x*sind(splane_angle);
proj_pt_zstart=1.5; proj_pt_z=proj_pt_zstart-proj_pt_zslant;
save proj_point; proj_point:=new_vect;
vect_def(proj_point,proj_pt_x,0,proj_pt_z); project_point(40,proj_point);
free_vect(proj_point);
x41=x40; z41=whatever[z80,z83]; % intersect with S plane below P pt
x42=x40; y42=y90-(y88-y41); % top of vert line through P
x43=x40; y43=y91-(y88-y41); % bot of vert line through P
x44=x80; % left of slant line through P
x45=x88; % intersect slant line and I plane; vanishing point
x46=x83; % right of slant line through P
z40-z44=z41-z80;
z45=whatever[z40,z44];
z46-z45=z83-z88;
%z45=whatever[z46,z44];
% The in-plane lines
save line_space; numeric line_space; line_space=.3;
z50=line_space[z80,z84]; z51=line_space[z80,z85];
z52=line_space[z83,z86]; z53=line_space[z83,z87];
z54=.05[z50,z52]; z55=.95[z50,z52]; % rear (i.e., leftward) of the tracks
z56=.05[z51,z53]; z57=.95[z51,z53]; % front of the rail tracks
% draw the planes and lines
pickup pencircle scaled line_width_light;
fill iplane_lower--cycle withcolor white; draw iplane_lower;
draw z41--z43 withcolor shading_color; % bot of vert line thru P
fill splane_lower--cycle withcolor white; draw splane_lower;
fill splane_upper--cycle withcolor white; draw splane_upper;
draw z54--z55; % rail track
draw z56--z57 dashed evenly scaled .75; % other
draw z42--z41 withcolor shading_color; % top of vert line thru P
draw z44--z45 withcolor shading_color; % right of slant line
fill iplane_upper--cycle withcolor white; draw iplane_upper;
draw z45--z46 withcolor shading_color; % left of slant line
drawpoint(z40);
% labels
label.rt(btex {\small $S$} etex,z85);
label.rt(btex {\small $I$} etex,z95);
label.ulft(btex {\small $P$} etex,z40);
%
free_vect(lpn);
endfig;
beginfig(10) % projection with rail tracks on P->S->I
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.17in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,0,0);
% Locate the observer and describe where he's looking
set_point(Obs,8,8,1);
Obs_phi:=90; Obs_dist:=5;
point_of_view_abs(lpn,Obs_phi);
% Source plane at an angle with vertical
save splane_angle; numeric splane_angle; splane_angle=35;
save xmin, xmax, ymin, ymax, zmin, zmax;
numeric xmin, xmax, ymin, ymax, zmin, zmax;
-xmin=6; xmax=4; -ymin=ymax=1.25;
zmin=-xmax*sind(splane_angle); zmax=-xmin*sind(splane_angle);
save s_lower, s_midb, s_midf, s_upper; % center line of source plane
s_lower:=new_vect; s_midb:=new_vect; s_midf:=new_vect; s_upper:=new_vect;
vect_def(s_lower,xmin,0,zmax); vect_def(s_midb,0,ymin,0);
vect_def(s_midf,0,ymax,0); vect_def(s_upper,xmax,0,zmin);
project_point(80,s_lower); project_point(81,s_midb);
project_point(82,s_midf); project_point(83,s_upper);
free_vect(s_upper); free_vect(s_midf);
free_vect(s_midb); free_vect(s_lower);
z84-z80=z80-z85; z81-z84=z82-z85; % lower end
z86-z83=z83-z87; z86-z81=z87-z82; % upper end
save splane_lower, splane_upper; path splane_lower, splane_upper;
splane_lower = z81--z84--z85--z82;
splane_upper = z81--z86--z87--z82;
z88=.5[z81,z82];
% Image plane yz-plane
save i_upper, i_lower; % center line of image plane
i_upper:=new_vect; i_lower:=new_vect;
vect_def(i_upper,0,0,zmax); vect_def(i_lower,0,0,zmin);
project_point(90,i_upper); project_point(91,i_lower);
free_vect(i_lower); free_vect(i_upper);
z94-z91=z91-z95; z81-z94=z82-z95; % upper end
z92-z90=z90-z93; z92-z81=z93-z82; % lower end
save iplane_lower, iplane_upper; path iplane_lower, iplane_upper;
iplane_upper = z81--z92--z93--z82;
iplane_lower = z81--z94--z95--z82;
% projection point
save proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
numeric proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
proj_pt_x=2; proj_pt_zslant=proj_pt_x*sind(splane_angle);
proj_pt_zstart=1.5; proj_pt_z=proj_pt_zstart-proj_pt_zslant;
save proj_point; proj_point:=new_vect;
vect_def(proj_point,proj_pt_x,0,proj_pt_z); project_point(40,proj_point);
free_vect(proj_point);
x41=x40; z41=whatever[z80,z83]; % intersect with S plane below P pt
x42=x40; y42=y90-(y88-y41); % top of vert line through P
x43=x40; y43=y91-(y88-y41); % bot of vert line through P
x44=x80; % left of slant line through P
x45=x88; % intersect slant line and I plane; vanishing point
x46=x83; % right of slant line through P
z40-z44=z41-z80;
z45=whatever[z40,z44];
z46-z45=z83-z88;
%z45=whatever[z46,z44];
% The in-plane lines
save line_space; numeric line_space; line_space=.3;
z50=line_space[z80,z84]; z51=line_space[z80,z85];
z52=line_space[z83,z86]; z53=line_space[z83,z87];
z54=.05[z50,z52]; z55=.95[z50,z52]; % rear (i.e., leftward) of the tracks
z56=.05[z51,z53]; z57=.95[z51,z53]; % front of the rail tracks
% these lines need to end next to intersection of vert line thru P
z20=whatever[z81,z84]; % line from z20 to z21 is cutoff line for tracks
z21=whatever[z82,z85];
z21-z20=z85-z84; z41=whatever[z21,z20];
z22=whatever[z20,z21]=whatever[z54,z55]; % cutoff for solid track
z23=whatever[z20,z21]=whatever[z56,z57]; % cutoff for dashed track
% the lines for this picture lie below S
z10=whatever[z81,z82]=whatever[z54,z55]; % intersect solid track and I plane
z11=whatever[z45,z10]=whatever[z94,z95]; % crosses bottom of I plane
z12=whatever[z81,z82]=whatever[z56,z57]; % intersect dashed track and I plane
z13=whatever[z45,z12]=whatever[z94,z95]; % crosses bottom of I plane
% draw the planes and lines
pickup pencircle scaled line_width_light;
fill iplane_lower--cycle withcolor white; draw iplane_lower;
draw z41--z43 withcolor shading_color; % bot of vert line thru P
draw z10--z11; % projection of solid track
draw z12--z13 dashed evenly scaled .75; % projection of dashed track
fill splane_lower--cycle withcolor white; draw splane_lower;
fill splane_upper--cycle withcolor white; draw splane_upper;
draw z10--z22; % rail track
draw z12--z23 dashed evenly scaled .75; % other
draw z42--z41 withcolor shading_color; % top of vert line thru P
draw z44--z45 withcolor shading_color; % right of slant line
fill iplane_upper--cycle withcolor white; draw iplane_upper;
draw z45--z46 withcolor shading_color; % left of slant line
drawpoint(z40);
% labels
label.rt(btex {\small $S$} etex,z85);
label.rt(btex {\small $I$} etex,z95);
label.ulft(btex {\small $P$} etex,z40);
%
free_vect(lpn);
endfig;
beginfig(11) % projection with rail tracks S->I->P
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.17in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,0,0);
% Locate the observer and describe where he's looking
set_point(Obs,8,8,1);
Obs_phi:=90; Obs_dist:=5;
point_of_view_abs(lpn,Obs_phi);
% Source plane at an angle with vertical
save splane_angle; numeric splane_angle; splane_angle=35;
save xmin, xmax, ymin, ymax, zmin, zmax;
numeric xmin, xmax, ymin, ymax, zmin, zmax;
-xmin=6; xmax=4; -ymin=ymax=1.25;
zmin=-xmax*sind(splane_angle); zmax=-xmin*sind(splane_angle);
save s_lower, s_midb, s_midf, s_upper; % center line of source plane
s_lower:=new_vect; s_midb:=new_vect; s_midf:=new_vect; s_upper:=new_vect;
vect_def(s_lower,xmin,0,zmax); vect_def(s_midb,0,ymin,0);
vect_def(s_midf,0,ymax,0); vect_def(s_upper,xmax,0,zmin);
project_point(80,s_lower); project_point(81,s_midb);
project_point(82,s_midf); project_point(83,s_upper);
free_vect(s_upper); free_vect(s_midf);
free_vect(s_midb); free_vect(s_lower);
z84-z80=z80-z85; z81-z84=z82-z85; % lower end
z86-z83=z83-z87; z86-z81=z87-z82; % upper end
save splane_lower, splane_upper; path splane_lower, splane_upper;
splane_lower = z81--z84--z85--z82;
splane_upper = z81--z86--z87--z82;
z88=.5[z81,z82];
% Image plane yz-plane
save i_upper, i_lower; % center line of image plane
i_upper:=new_vect; i_lower:=new_vect;
vect_def(i_upper,0,0,zmax); vect_def(i_lower,0,0,zmin);
project_point(90,i_upper); project_point(91,i_lower);
free_vect(i_lower); free_vect(i_upper);
z94-z91=z91-z95; z81-z94=z82-z95; % upper end
z92-z90=z90-z93; z92-z81=z93-z82; % lower end
save iplane_lower, iplane_upper; path iplane_lower, iplane_upper;
iplane_upper = z81--z92--z93--z82;
iplane_lower = z81--z94--z95--z82;
% projection point
save proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
numeric proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
proj_pt_x=2; proj_pt_zslant=proj_pt_x*sind(splane_angle);
proj_pt_zstart=1.5; proj_pt_z=proj_pt_zstart-proj_pt_zslant;
save proj_point; proj_point:=new_vect;
vect_def(proj_point,proj_pt_x,0,proj_pt_z); project_point(40,proj_point);
free_vect(proj_point);
x41=x40; z41=whatever[z80,z83]; % intersect with S plane below P pt
x42=x40; y42=y90-(y88-y41); % top of vert line through P
x43=x40; y43=y91-(y88-y41); % bot of vert line through P
x44=x80; % left of slant line through P
x45=x88; % intersect slant line and I plane; vanishing point
x46=x83; % right of slant line through P
z40-z44=z41-z80;
z45=whatever[z40,z44];
z46-z45=z83-z88;
%z45=whatever[z46,z44];
% The in-plane lines
save line_space; numeric line_space; line_space=.3;
z50=line_space[z80,z84]; z51=line_space[z80,z85];
z52=line_space[z83,z86]; z53=line_space[z83,z87];
z54=.05[z50,z52]; z55=.95[z50,z52]; % rear (i.e., leftward) of the tracks
z56=.05[z51,z53]; z57=.95[z51,z53]; % front of the rail tracks
% the lines for this picture lie above S and below the vanishing point
z10=whatever[z81,z82]=whatever[z54,z55]; % intersect solid track and I plane
z12=whatever[z81,z82]=whatever[z56,z57]; % intersect dashed track and I plane
% draw the planes and lines
pickup pencircle scaled line_width_light;
fill iplane_lower--cycle withcolor white; draw iplane_lower;
draw z41--z43 withcolor shading_color; % bot of vert line thru P
fill splane_lower--cycle withcolor white; draw splane_lower;
fill splane_upper--cycle withcolor white; draw splane_upper;
draw z54--z10; % rail track
draw z56--z12 dashed evenly scaled .75; % other
draw z42--z41 withcolor shading_color; % top of vert line thru P
draw z44--z45 withcolor shading_color; % right of slant line
fill iplane_upper--cycle withcolor white; draw iplane_upper;
draw z10--z45; % image of solid track
draw z12--z45 dashed evenly scaled .75; % image of dashed track
draw z45--z46 withcolor shading_color; % left of slant line
drawpoint(z40);
% labels
label.rt(btex {\small $S$} etex,z85);
label.rt(btex {\small $I$} etex,z95);
label.ulft(btex {\small $P$} etex,z40);
%
free_vect(lpn);
endfig;
beginfig(12) % projection with rail tracks on S
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.18in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,0,0);
% Locate the observer and describe where he's looking
set_point(Obs,8,8,1);
Obs_phi:=90; Obs_dist:=5;
point_of_view_abs(lpn,Obs_phi);
% Source plane at an angle with vertical
save splane_angle; numeric splane_angle; splane_angle=35;
save xmin, xmax, ymin, ymax, zmin, zmax;
numeric xmin, xmax, ymin, ymax, zmin, zmax;
-xmin=6; xmax=4; -ymin=ymax=1.25;
zmin=-xmax*sind(splane_angle); zmax=-xmin*sind(splane_angle);
save s_lower, s_midb, s_midf, s_upper; % center line of source plane
s_lower:=new_vect; s_midb:=new_vect; s_midf:=new_vect; s_upper:=new_vect;
vect_def(s_lower,xmin,0,zmax); vect_def(s_midb,0,ymin,0);
vect_def(s_midf,0,ymax,0); vect_def(s_upper,xmax,0,zmin);
project_point(80,s_lower); project_point(81,s_midb);
project_point(82,s_midf); project_point(83,s_upper);
free_vect(s_upper); free_vect(s_midf);
free_vect(s_midb); free_vect(s_lower);
z84-z80=z80-z85; z81-z84=z82-z85; % lower end
z86-z83=z83-z87; z86-z81=z87-z82; % upper end
save splane_lower, splane_upper; path splane_lower, splane_upper;
splane_lower = z81--z84--z85--z82;
splane_upper = z81--z86--z87--z82;
z88=.5[z81,z82];
% Image plane yz-plane
save i_upper, i_lower; % center line of image plane
i_upper:=new_vect; i_lower:=new_vect;
vect_def(i_upper,0,0,zmax); vect_def(i_lower,0,0,zmin);
project_point(90,i_upper); project_point(91,i_lower);
free_vect(i_lower); free_vect(i_upper);
z94-z91=z91-z95; z81-z94=z82-z95; % upper end
z92-z90=z90-z93; z92-z81=z93-z82; % lower end
save iplane_lower, iplane_upper; path iplane_lower, iplane_upper;
iplane_upper = z81--z92--z93--z82;
iplane_lower = z81--z94--z95--z82;
% projection point
save proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
numeric proj_pt_x, proj_pt_z, proj_pt_zstart, proj_pt_zslant;
proj_pt_x=2; proj_pt_zslant=proj_pt_x*sind(splane_angle);
proj_pt_zstart=1.5; proj_pt_z=proj_pt_zstart-proj_pt_zslant;
save proj_point; proj_point:=new_vect;
vect_def(proj_point,proj_pt_x,0,proj_pt_z); project_point(40,proj_point);
free_vect(proj_point);
x41=x40; z41=whatever[z80,z83]; % intersect with S plane below P pt
x42=x40; y42=y90-(y88-y41); % top of vert line through P
x43=x40; y43=y91-(y88-y41); % bot of vert line through P
x44=x80; % left of slant line through P
x45=x88; % intersect slant line and I plane; vanishing point
x46=x83; % right of slant line through P
z40-z44=z41-z80;
z45=whatever[z40,z44];
z46-z45=z83-z88;
%z45=whatever[z46,z44];
% The in-plane lines
save line_space; numeric line_space; line_space=.3;
z50=line_space[z80,z84]; z51=line_space[z80,z85];
z52=line_space[z83,z86]; z53=line_space[z83,z87];
z54=.05[z50,z52]; z55=.95[z50,z52]; % rear (i.e., leftward) of the tracks
z56=.05[z51,z53]; z57=.95[z51,z53]; % front of the rail tracks
% these tracks need to end next to intersection of vert line thru P
z20=whatever[z81,z84]; % line from z20 to z21 is cutoff line for tracks
z21=whatever[z82,z85];
z21-z20=z85-z84; z41=whatever[z21,z20];
z22=whatever[z20,z21]=whatever[z54,z55]; % cutoff for solid track
z23=whatever[z20,z21]=whatever[z56,z57]; % cutoff for dashed track
% the lines for this picture lie below S
z10=whatever[z81,z82]=whatever[z54,z55]; % intersect solid track and I plane
z11=whatever[z45,z10]=whatever[z92,z93]; % crosses top of I plane
z12=whatever[z81,z82]=whatever[z56,z57]; % intersect dashed track and I plane
z13=whatever[z45,z12]=whatever[z92,z93]; % crosses top of I plane
% draw the planes and lines
pickup pencircle scaled line_width_light;
fill iplane_lower--cycle withcolor white; draw iplane_lower;
draw z41--z43 withcolor shading_color; % bot of vert line thru P
fill splane_lower--cycle withcolor white; draw splane_lower;
fill splane_upper--cycle withcolor white; draw splane_upper;
draw z55--z22; % rail track
draw z57--z23 dashed evenly scaled .75; % other
draw z42--z41 withcolor shading_color; % top of vert line thru P
draw z44--z45 withcolor shading_color; % right of slant line
fill iplane_upper--cycle withcolor white; draw iplane_upper;
draw z45--z11; % image of solid track
draw z45--z13 dashed evenly scaled .75; % image of dashed track
draw z45--z46 withcolor shading_color; % left of slant line
drawpoint(z40);
% labels
label.rt(btex {\small $S$} etex,z85);
label.rt(btex {\small $I$} etex,z95);
label.ulft(btex {\small $P$} etex,z40);
%
free_vect(lpn);
endfig;
beginfig(13) % projection, like with rail tracks, on S, showing infinities
numeric u; %scaling factor
numeric v; %vertical scaling factor
numeric w; %horizontal scaling factor
u:=.18in; v:=u; w:=v; % was .2in
drawing_scale:=25pt; % was 30pt
save lpn; lpn:=new_vect; vect_def(lpn,0,0,0);