## Saturday, December 24, 2011

### My Differential Geometry Images

So I am writing some notes on differential geometry, and using metapost for tricky diagrams. Here are a few of the tricky diagrams I have.

The first three images are:

The third image is a tad bit bigger:

Note the fonts in this picture are messed up, since I had to use postscript fonts. It looks far more beautiful in TeX using metapost, trust me!

numeric u;
color yellow;
u := 1pc;
yellow = red+green;

verbatimtex \input amssym.tex etex;

% sphere
beginfig(0)
draw fullcircle scaled 4u;
draw (-2u,0)..(0,-.7u)..(2u,0);
draw (-2u,0)..(0,.7u)..(2u,0) dashed evenly;
endfig;

% plane + cylinder
beginfig(1)
draw (0,-u)--(0,3u)--(u,4u)--(u,0)--cycle;

draw (3u,0)..(4u,-.5u)..(5u,0);
draw (3u,0)..(4u,.5u)..(5u,0) dashed evenly;

draw (3u,4u)..(4u,4.5u)..(5u,4u);
draw (5u,4u)..(4u,3.5u)..(3u,4u);

draw (3u,0)--(3u,4u);
draw (5u,0)--(5u,4u);
endfig;

beginfig(2)
path p[];
z[0] = (2u,4u);
z[1] = (2.5u,3u);
z[2] = (1.5u,2.5u);
z[3] = (4u,0u);
z[4] = (6.5u,u);
z[5] = (7.u,1.75u);
z[6] = (6.75u,0.5u);

p[0] = z[0]..z[1]..z[2];
p[1] = z[1]..z[3]..z[4];
p[2] = z[5]..z[4]..z[6];

p[3] = z[0]--(point 0.75*length(p[1]) of p[0]);
p[4] = z[2]--(1.75u,-2u);
p[5] = z[6]--(6.5u,-2u);
p[6] = z[5]--(point 0.9*length(p[1]) of p[2]);
for i=0 upto 6:
draw p[i];
endfor;

draw (4u,-u){up}..(point 0.6*length(p1) of p1);
draw (point 0.6*length(p1) of p1)..{down}(5.125u,-.75u) dashed evenly;
draw (1.75u,-2u)..(4u,-u)..(6.5u,-2u);

p7 = ((0,0)--(2u*unitvector(direction 0 of p[6]))) shifted point (length p6) of p[6];
draw p[7] dashed evenly;

p8 = ((0,0)--(3.5u*unitvector(direction 0 of p3))) shifted point (length p3) of p3;
draw p8 dashed evenly;

p9 = (point (length p8) of p8)..(5.125u,-.75u)..(point (length p7) of p7);
draw p9 dashed evenly;

endfig;

beginfig(3)
picture Rn;
picture sphere;
picture torus;
% R^n
Rn = image(
for i=1 upto 3:
draw (-.5u,i*u)--(3.5u,i*u) withcolor 0.75white;
draw (i*u,-.5u)--(i*u,3.5u) withcolor 0.75white;
endfor;
drawdblarrow (-.5u,0)--(3.5u,0);
drawdblarrow (0,-.5u)--(0,3.5u);
);
% sphere
sphere = image(
draw fullcircle scaled 4u;
draw ((-2u,0)..(0,-.7u)..(2u,0));
draw ((-2u,0)..(0,.7u)..(2u,0)) dashed evenly;
);
% torus
torus = image(
path hole;
path uhole;
draw
(0,2u)..(-2u,0)..(0,-2u)..(2u,-u)..(4u,-2u)..(6u,0)..(4u,2u)..(2u,u)..cycle;
hole = halfcircle rotated 180 scaled 2u shifted (4.5u,0);
draw hole;
uhole = (point 0.1*length(hole) of hole)
..(0.5[(point 0.1*length(hole) of hole),(point 0.9*length(hole) of hole)]+(0,.5u))
..(point 0.9*length(hole) of hole);
draw uhole;
);

draw Rn;
draw Rn shifted (20u,-5u);

draw torus shifted (8u,4u);

path localPatch;
path imageOfF;
path preimageOfG;
path imageOfG;
path imageOfPsi;
path intersection;
localPatch = (fullcircle scaled 2u shifted (1.5u,1.5u));
fill localPatch withcolor 0.75[blue,white];
draw localPatch dashed evenly;

z[0] = (((u,0)--(u,2u)) intersectionpoint localPatch);
draw (((0,u)--(u,u)) intersectionpoint localPatch)
--(((2u,u)--(4u,u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((0,2u)--(2u,2u)) intersectionpoint localPatch)
--(((2u,2u)--(4u,2u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((u,0)--(u,2u)) intersectionpoint localPatch)
--(((u,2u)--(u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((2u,0)--(2u,2u)) intersectionpoint localPatch)
--(((2u,2u)--(2u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];

imageOfF = fullcircle xscaled 3u yscaled 1.5u shifted (8u,4u);
fill imageOfF withcolor 0.75[blue,white];
draw imageOfF dashed evenly;

picture preG;

preG = image(

preimageOfG = unitsquare scaled 2u rotated 45 shifted (22u,-5u);

fill preimageOfG withcolor 0.5[yellow,white];
fill fullcircle xscaled u yscaled 5u rotated -23 shifted (22u,-3u)
withcolor 0.5[green,white];
draw (((21u,-4u)--(21u,-5u)) intersectionpoint preimageOfG)
--(((21u,-5u)--(21u,-3u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((22u,-2u)--(22u,-3u)) intersectionpoint preimageOfG)
--(((22u,-4u)--(22u,-8u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((23u,-3u)--(23u,-3.5u)) intersectionpoint preimageOfG)
--(((23u,-3.5u)--(23u,-8u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((21u,-4u)--(22u,-4u)) intersectionpoint preimageOfG)
--(((22u,-4u)--(25u,-4u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((21u,-3u)--(22u,-3u)) intersectionpoint preimageOfG)
--(((22u,-3u)--(25u,-3u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw preimageOfG dashed evenly;

clip currentpicture to preimageOfG;
);
draw preG;

imageOfG = (19u,2.5u)--(20u,3u)--(21.5u,3u)--(21u,2.5u)--cycle;

fill imageOfG withcolor 0.5[white,yellow];

imageOfPsi = fullcircle xscaled 2u yscaled u rotated -45
shifted (20u,3u);
fill imageOfPsi withcolor 0.75[blue,white];

numeric t[];
(t[0],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,3u));
(t[1],whatever) = imageOfPsi intersectiontimes ((21u,2.5u)--(19u,2.5u));
(t[2],whatever) = imageOfPsi intersectiontimes ((20u,3u)--(23u,3u));
(t[3],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,2.5u));

intersection = (subpath(t0,t3) of imageOfPsi)--%(20u,3u)--
(subpath(t1,t2) of imageOfPsi)--(20u,3u)--cycle;

fill intersection withcolor 0.5[green,white];
draw imageOfPsi dashed evenly;
draw imageOfG dashed evenly;
draw sphere shifted (20u,4u);

p[0] := (point 0.25*length(localPatch) of localPatch)..(4u,4u)
..(point 0.5*length(imageOfF) of imageOfF);
p[1] := (point 0 of imageOfF)..(12u,5u)..(19.5u,3.5u);
p[2] := (21.5u,3u)..(23u,0)..(23u,-3u);
p[3] := (point .75*length(localPatch) of localPatch)..(1.5u,-u)
..(10u,-4u)..(21.5u,-4.5u);
drawarrow p[0];

drawarrow p1;

drawarrow p2;

drawarrow p3 dashed evenly;

label.top(btex $x$ etex, point 0.5*length(p0) of p0);
label.top(btex $\varphi$ etex, point 0.75*length(p1) of p1);
label.rt(btex $y^{-1}$ etex, point 0.5*length(p2) of p2);
label.bot(btex $y^{-1}\circ\varphi\circ x$ etex, point 0.6*length(p3) of p3);

label(btex $U$ etex, (1.5u,1.5u)) withcolor 0.5[black,blue];

draw (8u,4u) withpen pencircle scaled 3;
label.lft(btex $p$ etex, (8u,4u)) withcolor 0.75[blue,black];
label.bot(btex $M$ etex, (12u,2u));

label.rt(btex $N$ etex, (22u,4u));

label.lft(btex $\Bbb{R}^{n}$ etex, (20u,-2u));

label.lft(btex $\Bbb{R}^{m}$ etex, (0,3u));
endfig;

end;