答案1
这是一个起点。
可以按如下方式编译。
首先，保存以下代码，灵感来自分割补丁程序，在一个名为的文件中crop3D.asy：
import three;
/**********************************************/
/* Code for splitting surfaces: */
struct possibleInt {
int value;
bool holds;
}
int operator cast(possibleInt i) { return i.value; }
restricted int maxdepth = 20;
restricted void maxdepth(int n) { maxdepth = n; }
surface[] divide(surface s, possibleInt region(patch), int numregions,
bool keepregion(int) = null) {
if (keepregion == null) keepregion = new bool(int region) {
return (0 <= region && region < numregions);
};
surface[] toreturn = new surface[numregions];
for (int i = 0; i < numregions; ++i)
toreturn[i] = new surface;
void addPatch(patch P, int region) {
if (keepregion(region)) toreturn[region].push(P);
}
void divide(patch P, int depth) {
if (depth == 0) {
addPatch(P, region(P));
return;
}
possibleInt region = region(P);
if (region.holds) {
addPatch(P, region);
return;
}
// Choose the splitting function based on the parity of the recursion depth.
triple[][][] Split(triple[][] P) = (depth % 2 == 0 ? hsplit : vsplit);
patch[] Split(patch P) {
triple[][][] patches = Split(P.P);
return sequence(new patch(int i) {return patch(patches[i]);}, patches.length);
}
patch[] patches = Split(P);
for (patch PP : patches)
divide(PP, depth-1);
}
for (patch P : s.s)
divide(P, maxdepth);
return toreturn;
}
surface[] divide(surface s, int region(triple), int numregions,
bool keepregion(int) = null) {
possibleInt patchregion(patch P) {
triple[][] controlpoints = P.P;
possibleInt theRegion;
theRegion.value = region(controlpoints[0][0]);
theRegion.holds = true;
for (triple[] ta : controlpoints) {
for (triple t : ta) {
if (region(t) != theRegion.value) {
theRegion.holds = false;
break;
}
}
if (!theRegion.holds) break;
}
return theRegion;
}
return divide(s, patchregion, numregions, keepregion);
}
/**************************************************/
/* Code for cropping surfaces */
// Return 0 iff the point lies in box(a,b).
int cropregion(triple pt, triple a=O, triple b=(1,1,1)) {
real x=pt.x, y=pt.y, z=pt.z;
int toreturn=0;
real xmin=a.x, xmax=b.x, ymin = a.y, ymax=b.y, zmin=a.z, zmax=b.z;
if (xmin > xmax) { xmin = b.x; xmax = a.x; }
if (ymin > ymax) { ymin = b.y; ymax = a.y; }
if (zmin > zmax) { zmin = b.z; zmax = a.z; }
if (x < xmin) --toreturn;
else if (x > xmax) ++toreturn;
toreturn *= 2;
if (y < ymin) --toreturn;
else if (y > ymax) ++toreturn;
toreturn *= 2;
if (z < zmin) --toreturn;
else if (z > zmax) ++toreturn;
return toreturn;
}
//bool keepregion(int region) { return (region == 0); }
// Crop the surface to box(a,b).
surface crop(surface s, triple a, triple b) {
int region(triple pt) {
return cropregion(pt, a, b);
}
return divide(s, region=region, numregions=1)[0];
}
// A rectangular solid with opposite vertices a, b:
surface surfacebox(triple a, triple b) {
return shift(a)*scale((b-a).x,(b-a).y,(b-a).z)*unitcube;
}
bool containedInBox(triple pt, triple a, triple b) {
return cropregion(pt, a, b) == 0;
}
// Crop a path3 to box(a,b).
path3[] crop(path3 g, triple a, triple b) {
surface thebox = surfacebox(a,b);
path3[] toreturn;
real[] times = new real[] {0};
real[][] alltimes = intersections(g, thebox);
for (real[] threetimes : alltimes)
times.push(threetimes[0]);
times.push(length(g));
for (int i = 1; i < times.length; ++i) {
real mintime = times[i-1];
real maxtime = times[i];
triple midpoint = point(g, (mintime+maxtime)/2);
if (containedInBox(midpoint, a, b))
toreturn.push(subpath(g, mintime, maxtime));
}
return toreturn;
}
path3[] crop(path3[] g, triple a, triple b) {
path3[] toreturn;
for (path3 gi : g)
toreturn.append(crop(gi, a, b));
return toreturn;
}
/***************************************/
/* Code to return only the portion of the surface facing the camera */
bool facingCamera(triple vec, triple pt=O, projection P = currentprojection, bool towardsCamera = true) {
triple normal = P.camera;
if (!P.infinity) {
normal = P.camera - pt;
}
if (towardsCamera) return (dot(vec, normal) >= 0);
else return (dot(vec, normal) <= 0);
}
surface facingCamera(surface s, bool towardsCamera = true) {
possibleInt facingregion(patch P) {
int n = 2;
possibleInt toreturn;
unravel toreturn;
bool facingcamera = facingCamera(P.normal(1/2, 1/2), pt=P.point(1/2,1/2), towardsCamera);
value = facingcamera ? 0 : 1;
holds = true;
for (int i = 0; i <= n; ++i) {
real u = i/n;
for (int j = 0; j <= n; ++j) {
real v = j/n;
if (facingCamera(P.normal(u,v), P.point(u,v), towardsCamera) != facingcamera) {
holds = false;
break;
}
}
if (!holds) break;
}
return toreturn;
}
return divide(s, facingregion, numregions=1)[0];
}
然后，将以下代码保存在名为的文件中（在同一目录中）surfacepaths.asy：
import graph3;
import contour;
// A bunch of auxiliary functions.
real fuzz = .001;
real umin(surface s) { return 0; }
real vmin(surface s) { return 0; }
pair uvmin(surface s) { return (umin(s), vmin(s)); }
real umax(surface s, real fuzz=fuzz) {
if (s.ucyclic()) return s.index.length;
else return s.index.length - fuzz;
}
real vmax(surface s, real fuzz=fuzz) {
if (s.vcyclic()) return s.index[0].length;
return s.index[0].length - fuzz;
}
pair uvmax(surface s, real fuzz=fuzz) { return (umax(s,fuzz), vmax(s,fuzz)); }
typedef real function(real, real);
function normalDot(surface s, triple eyedir(triple)) {
real toreturn(real u, real v) {
return dot(s.normal(u, v), eyedir(s.point(u,v)));
}
return toreturn;
}
struct patchWithCoords {
patch p;
real u;
real v;
void operator init(patch p, real u, real v) {
this.p = p;
this.u = u;
this.v = v;
}
void operator init(surface s, real u, real v) {
int U=floor(u);
int V=floor(v);
int index = (s.index.length == 0 ? U+V : s.index[U][V]);
this.p = s.s[index];
this.u = u-U;
this.v = v-V;
}
triple partialu() {
return p.partialu(u,v);
}
triple partialv() {
return p.partialv(u,v);
}
}
triple[] derivative(surface s, pair pt) {
patchWithCoords thepatch = patchWithCoords(s, pt.x, pt.y);
return new triple[] {thepatch.partialu(), thepatch.partialv()};
}
typedef triple paramsurface(pair);
paramsurface tangentplane(surface s, pair pt) {
patchWithCoords thepatch = patchWithCoords(s, pt.x, pt.y);
triple partialu = thepatch.partialu();
triple partialv = thepatch.partialv();
return new triple(pair tangentvector) {
return s.point(pt.x, pt.y) + (tangentvector.x * partialu) + (tangentvector.y * partialv);
};
}
guide[] normalpathuv(surface s, projection P = currentprojection, int n = ngraph) {
triple eyedir(triple a);
if (P.infinity) eyedir = new triple(triple) { return P.camera; };
else eyedir = new triple(triple pt) { return P.camera - pt; };
return contour(normalDot(s, eyedir), uvmin(s), uvmax(s), new real[] {0}, nx=n)[0];
}
path3 onSurface(surface s, path p) {
triple f(int t) {
pair point = point(p,t);
return s.point(point.x, point.y);
}
guide3 toreturn = f(0);
paramsurface thetangentplane = tangentplane(s, point(p,0));
triple oldcontrol, newcontrol;
int size = length(p);
for (int i = 1; i <= size; ++i) {
oldcontrol = thetangentplane(postcontrol(p,i-1) - point(p,i-1));
thetangentplane = tangentplane(s, point(p,i));
newcontrol = thetangentplane(precontrol(p, i) - point(p,i));
toreturn = toreturn .. controls oldcontrol and newcontrol .. f(i);
}
if (cyclic(p)) toreturn = toreturn & cycle;
return toreturn;
}
path3[] onSurface(surface s, path[] p) {
return sequence(new path3(int i) {return onSurface(s,p[i]);}, p.length);
}
/*
* This method returns an array of paths that trace out all the
* points on s at which s is parallel to eyedir.
*/
path[] paramSilhouetteNoEdges(surface s, projection P = currentprojection, int n = ngraph) {
guide[] uvpaths = normalpathuv(s, P, n);
//Reduce the number of segments to conserve memory
for (int i = 0; i < uvpaths.length; ++i) {
real len = length(uvpaths[i]);
uvpaths[i] = graph(new pair(real t) {return point(uvpaths[i],t);}, 0, len, n=n);
}
return uvpaths;
}
private typedef real function2(real, real);
private typedef real function3(triple);
triple[] normalVectors(triple dir, triple surfacen) {
dir = unit(dir);
surfacen = unit(surfacen);
triple v1, v2;
int i = 0;
do {
v1 = unit(cross(dir, (unitrand(), unitrand(), unitrand())));
v2 = unit(cross(dir, (unitrand(), unitrand(), unitrand())));
++i;
} while ((abs(dot(v1,v2)) > Cos(10) || abs(dot(v1,surfacen)) > Cos(5) || abs(dot(v2,surfacen)) > Cos(5)) && i < 1000);
if (i >= 1000) {
write("problem: Unable to comply.");
write(" dir = " + (string)dir);
write(" surface normal = " + (string)surfacen);
}
return new triple[] {v1, v2};
}
function3 planeEqn(triple pt, triple normal) {
return new real(triple r) {
return dot(normal, r - pt);
};
}
function2 pullback(function3 eqn, surface s) {
return new real(real u, real v) {
return eqn(s.point(u,v));
};
}
path3[] levelcurve(function3 f, surface s, real value=0) {
function2 fparam = pullback(f, s);
path[] paramcurve = contour(fparam, (0,0), uvmax(s), new real[] {value})[0];
return onSurface(s, paramcurve);
}
/*
* returns the distinct points in which the surface intersects
* the line through the point pt in the direction dir
*/
triple[] intersectionPoints(surface s, pair parampt, triple dir) {
triple pt = s.point(parampt.x, parampt.y);
triple[] lineNormals = normalVectors(dir, s.normal(parampt.x, parampt.y));
path[][] curves;
for (triple n : lineNormals) {
function3 planeEn = planeEqn(pt, n);
function2 pullback = pullback(planeEn, s);
guide[] contour = contour(pullback, uvmin(s), uvmax(s), new real[]{0})[0];
curves.push(contour);
}
pair[] intersectionPoints;
for (path c1 : curves[0])
for (path c2 : curves[1])
intersectionPoints.append(intersectionpoints(c1, c2));
triple[] toreturn;
for (pair P : intersectionPoints)
toreturn.push(s.point(P.x, P.y));
return toreturn;
}
/*
* Returns those intersection points for which the vector from pt forms an
* acute angle with dir.
*/
int numPointsInDirection(surface s, pair parampt, triple dir, real fuzz=.05) {
triple pt = s.point(parampt.x, parampt.y);
dir = unit(dir);
triple[] intersections = intersectionPoints(s, parampt, dir);
int num = 0;
for (triple isection: intersections)
if (dot(isection - pt, dir) > fuzz) ++num;
return num;
}
bool3 increasing(real t0, real t1) {
if (t0 < t1) return true;
if (t0 > t1) return false;
return default;
}
int[] extremes(real[] f, bool cyclic = f.cyclic) {
bool3 lastIncreasing;
bool3 nextIncreasing;
int max;
if (cyclic) {
lastIncreasing = increasing(f[-1], f[0]);
max = f.length - 1;
} else {
max = f.length - 2;
if (increasing(f[0], f[1])) lastIncreasing = false;
else lastIncreasing = true;
}
int[] toreturn;
for (int i = 0; i <= max; ++i) {
nextIncreasing = increasing(f[i], f[i+1]);
if (lastIncreasing != nextIncreasing) {
toreturn.push(i);
}
lastIncreasing = nextIncreasing;
}
if (!cyclic) toreturn.push(f.length - 1);
toreturn.cyclic = cyclic;
return toreturn;
}
int[] extremes(path path, real f(pair) = new real(pair P) {return P.x;})
{
real[] fvalues = new real[size(path)];
for (int i = 0; i < fvalues.length; ++i) {
fvalues[i] = f(point(path, i));
}
fvalues.cyclic = cyclic(path);
int[] toreturn = extremes(fvalues);
fvalues.delete();
return toreturn;
}
path[] splitAtExtremes(path path, real f(pair) = new real(pair P) {return P.x;})
{
int[] splittingTimes = extremes(path, f);
path[] toreturn;
if (cyclic(path)) toreturn.push(subpath(path, splittingTimes[-1], splittingTimes[0]));
for (int i = 0; i+1 < splittingTimes.length; ++i) {
toreturn.push(subpath(path, splittingTimes[i], splittingTimes[i+1]));
}
return toreturn;
}
path[] splitAtExtremes(path[] paths, real f(pair P) = new real(pair P) {return P.x;})
{
path[] toreturn;
for (path path : paths) {
toreturn.append(splitAtExtremes(path, f));
}
return toreturn;
}
path3 toCamera(triple p, projection P=currentprojection, real fuzz = .01, real upperLimit = 100) {
if (!P.infinity) {
triple directionToCamera = unit(P.camera - p);
triple startingPoint = p + fuzz*directionToCamera;
return startingPoint -- P.camera;
}
else {
triple directionToCamera = unit(P.camera);
triple startingPoint = p + fuzz*directionToCamera;
return startingPoint -- (p + upperLimit*directionToCamera);
}
}
int numSheetsHiding(surface s, pair parampt, projection P = currentprojection) {
triple p = s.point(parampt.x, parampt.y);
path3 tocamera = toCamera(p, P);
triple pt = beginpoint(tocamera);
triple dir = endpoint(tocamera) - pt;
return numPointsInDirection(s, parampt, dir);
}
struct coloredPath {
path path;
pen pen;
void operator init(path path, pen p=currentpen) {
this.path = path;
this.pen = p;
}
/* draws the path with the pen having the specified weight (using colors)*/
void draw(real weight) {
draw(path, p=weight*pen + (1-weight)*white);
}
}
coloredPath[][] layeredPaths;
// onTop indicates whether the path should be added at the top or bottom of the specified layer
void addPath(path path, pen p=currentpen, int layer, bool onTop=true) {
coloredPath toAdd = coloredPath(path, p);
if (layer >= layeredPaths.length) {
layeredPaths[layer] = new coloredPath[] {toAdd};
} else if (onTop) {
layeredPaths[layer].push(toAdd);
} else layeredPaths[layer].insert(0, toAdd);
}
void drawLayeredPaths() {
for (int layer = layeredPaths.length - 1; layer >= 0; --layer) {
real layerfactor = (1/3)^layer;
for (coloredPath toDraw : layeredPaths[layer]) {
toDraw.draw(layerfactor);
}
}
layeredPaths.delete();
}
real[] cutTimes(path tocut, path[] knives) {
real[] intersectionTimes = new real[] {0, length(tocut)};
for (path knife : knives) {
real[][] complexIntersections = intersections(tocut, knife);
for (real[] times : complexIntersections) {
intersectionTimes.push(times[0]);
}
}
return sort(intersectionTimes);
}
path[] cut(path tocut, path[] knives) {
real[] cutTimes = cutTimes(tocut, knives);
path[] toreturn;
for (int i = 0; i + 1 < cutTimes.length; ++i) {
toreturn.push(subpath(tocut,cutTimes[i], cutTimes[i+1]));
}
return toreturn;
}
real[] condense(real[] values, real fuzz=.001) {
values = sort(values);
values.push(infinity);
real previous = -infinity;
real lastMin;
real[] toReturn;
for (real t : values) {
if (t - fuzz > previous) {
if (previous > -infinity) toReturn.push((lastMin + previous) / 2);
lastMin = t;
}
previous = t;
}
return toReturn;
}
最后，这是实际创建图像的代码（使用前两个模块）。将其保存在morse_theory.asy与其他两个文件位于同一目录中的名为（例如）的文件中，然后asy morse_theory在命令行中运行。
settings.outformat="png";
settings.render=16;
settings.prc=false;
usepackage("lmodern");
usepackage("amssymb");
defaultpen(fontsize(10pt));
unitsize(1cm);
import graph3;
/**********************************************/
/* Code for splitting surfaces: */
import crop3D;
/************************************************/
/* Code for drawing a level curve on a surface: */
import surfacepaths;
/************************************************/
/* The actual drawing: */
currentprojection = orthographic(X+.2Z);
real r = 1.4;
real R = 2.6;
real rc = (r+R)/2;
real rtube = (R-r)/2;
transform3 yscale = scale(1,.8,1);
surface torus = yscale * surface(rotate(angle=90,rc*Y,rc*Y+Z)*Circle(c=rc*Y, r=rtube, normal=Z, n=8), c=O, axis=X, n=8);
maxdepth(10);
surface torusFront = facingCamera(torus, towardsCamera=false);
maxdepth(17);
real[] dividerHeights = sort(new real[] {-rc+.1, -r, -rc+2rtube-.1});
int region(triple t) {
real z = t.z;
for (int i = 0; i < dividerHeights.length; ++i) {
if (z < dividerHeights[i]) return i;
}
return dividerHeights.length;
}
int numregions = dividerHeights.length + 1;
surface[] torusCells = divide(torusFront, region, numregions);
pen transparency = opacity(0.8);
material[] materials = new material[] {material(black+transparency),
material(gray(1/3)+transparency, ambientpen=gray(1/3)),
material(gray(2/3)+transparency, ambientpen=gray(2/3)),
material(white+transparency, ambientpen=white)};
draw(torusCells[0], surfacepen=materials[0]);
draw(torusCells[1], surfacepen=materials[1]);
draw(torusCells[2], surfacepen=materials[2]);
draw(torusCells[3], surfacepen=materials[3]);
real height(triple pt) { return pt.z; }
for (real h : dividerHeights) {
draw(levelcurve(height, torus, h));
}
path3[] circularBands = levelcurve(planeEqn(pt=O,normal=Y+.3X), torus, 0);
real maxdivide = dividerHeights[dividerHeights.length - 1];
real mindivide = dividerHeights[0];
path3[] bandsToDraw = crop(circularBands, (-5,-5, mindivide), (5, 5, maxdivide));
// In our case, as it happens, there is only one band--but let's make sure of that.
assert(bandsToDraw.length == 1);
path3 bandToDraw = bandsToDraw[0];
draw(bandToDraw);
triple bandPoint = relpoint(bandToDraw, 0.25);
//dot(bandPoint);
draw(maxdivide*Z {(-1,-.4,-.2)} .. {(-.4,.6,-1)} bandPoint, arrow=ArcArrow3(DefaultHead2,emissive(black)), L=Label("$e^k$", position=BeginPoint, align=ENE));
real y = R;
draw((0,y,-R) -- (0,y,R), L=Label("$\mathbb{R}$",position=EndPoint,align=NE), arrow=Arrow3(TeXHead2,emissive(black)));
//draw bars with size specified in postscript coordinates
draw(shift(0,y,dividerHeights[0]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c-\varepsilon$",position=EndPoint,align=E));
draw(shift(0,y,dividerHeights[1]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c$",position=EndPoint,align=E));
draw(shift(0,y,dividerHeights[2]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c+\varepsilon$",position=EndPoint,align=E));
transform3 shiftright = shift(0,6,0);
real perturbdist = abs(min(torus).y)/2;
path perturbpath = (-2*perturbdist,0) --- (-perturbdist,0) .. (0,.08) .. (perturbdist,0) --- (2*perturbdist,0);
real perturbedHeight(triple pt) {
real z = pt.z, y=pt.y;
if (abs(y) > perturbdist) return z;
else return z + point(perturbpath,times(perturbpath,y)[0]).y;
}
real perturbedDivide = -r + .01;
region = new int(triple t) {
if (perturbedHeight(t) < perturbedDivide) return 0;
else if (height(t) < perturbedDivide) return 1;
else if (height(t) < dividerHeights[2]) return 2;
else return 3;
};
surface[] newTorusCells = divide(torusFront, region, numregions=3);
newTorusCells.push(torusCells[3]);
for (int i = 0; i < newTorusCells.length; ++i)
draw(shiftright*newTorusCells[i], surfacepen=materials[i]);
draw(shiftright * crop(levelcurve(perturbedHeight, torus,perturbedDivide), (0,-5,-5), (5,5,5)));
draw(shiftright * crop(levelcurve(height, torus, perturbedDivide), (0,-5,-5), (5,5,5)));
draw(shiftright * levelcurve(height, torus, maxdivide));
draw(shiftright * crop(Circle(c=-rc*Z, r=rtube, normal=Y), (5,5,perturbedDivide-perturbedHeight(O)), (0,-5, 5)));
draw(shiftright * ((0,R,-R) -- (0,R,maxdivide)), Bars3, arrow=Arrows3(DefaultHead2), L=Label("$M \leq c + \varepsilon$", position=MidPoint, align=E));
答案2
附件是 Inkscape 代码（下载代码并以 .svg 扩展名保存）。您可以根据需要自由修改它。生成的图像如下所示。
<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!-- Created with Inkscape (http://www.inkscape.org/) -->
<svg
xmlns:osb="http://www.openswatchbook.org/uri/2009/osb"
xmlns:dc="http://purl.org/dc/elements/1.1/"
xmlns:cc="http://creativecommons.org/ns#"
xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
xmlns:svg="http://www.w3.org/2000/svg"
xmlns="http://www.w3.org/2000/svg"
xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
width="800"
height="800"
id="svg2"
version="1.1"
inkscape:version="0.48.4 r9939"
sodipodi:docname="torus.svg"
inkscape:export-filename="torus.png"
inkscape:export-xdpi="600"
inkscape:export-ydpi="600">
<defs
id="defs4">
<marker
inkscape:stockid="Arrow2Mstart"
orient="auto"
refY="0.0"
refX="0.0"
id="Arrow2Mstart"
style="overflow:visible">
<path
id="path4128"
style="fill-rule:evenodd;stroke-width:0.62500000;stroke-linejoin:round"
d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
transform="scale(0.6) translate(0,0)" />
</marker>
<marker
inkscape:stockid="Arrow2Lend"
orient="auto"
refY="0.0"
refX="0.0"
id="Arrow2Lend"
style="overflow:visible;">
<path
id="path4125"
style="fill-rule:evenodd;stroke-width:0.62500000;stroke-linejoin:round;"
d="M 8.7185878,4.0337352 L -2.2072895,0.016013256 L 8.7185884,-4.0017078 C 6.9730900,-1.6296469 6.9831476,1.6157441 8.7185878,4.0337352 z "
transform="scale(1.1) rotate(180) translate(1,0)" />
</marker>
<inkscape:path-effect
effect="spiro"
id="path-effect4663"
is_visible="true" />
<inkscape:path-effect
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