Blowup of a plane at a point

by Posted on

The following image illustrates the blowup of a plane at a point–an important construction in algebraic geometry.

The blowup of a plane at a point

\documentclass[border=10pt]{standalone}
\usepackage[inline]{asymptote}
\begin{document}
\begin{asy}[width=\the\linewidth,inline=true]
settings.outformat="pdf";
settings.render=0;
settings.prc=false;

usepackage("lmodern");
usepackage("fontenc","T1");
usepackage("amssymb");  // for the \mathbb command
defaultpen(fontsize(10pt));

import graph3;
size(400,400);
currentprojection=orthographic(5,-10,4);

real R=8;

struct scaler {
    private real factor;

    void operator init(real factor) {
        this.factor = factor;
    }

    real scale(real t) {return factor*atan(tan(t)/factor);}
    real invert(real t) {return tan(atan(t)*factor)/factor;}
}

scaler theScaler = scaler(6);

triple f(pair t) {
    real r = t.x;
    real theta = 2 * atan(t.y*2/pi);
//  real theta = -t.y;
    return (r*cos(theta),r*sin(theta),theScaler.scale(theta));
}

int resolution = 10;
real epsilon = .01;
real vmin = -pi/2;
real vmax = pi/2;
real umin = -R;
real umax = R;
splinetype[] Linear = new splinetype[] {linear, linear, linear};
splinetype[] ZMonotonic = new splinetype[] {notaknot, notaknot, monotonic};
surface sBack=surface(f,(umin,vmin),(0,vmax),nu=resolution, nv=2*resolution,  usplinetype=Linear, vsplinetype = ZMonotonic);
surface sFront = surface(f, (0,vmin), (umax,vmax), nu=resolution, nv=2*resolution, usplinetype=Linear, vsplinetype=ZMonotonic);

pen meshpen = heavygray + linewidth(0.2);

material surfacepen = 
    material(diffusepen=lightgray+opacity(0.5), 
        emissivepen=gray(0.3),
        specularpen=gray(0.2));

draw(sBack, surfacepen=surfacepen, meshpen=meshpen);
draw(f((0,vmin)) -- f((0,vmax)), darkgray+linewidth(1.0));   // the exceptional divisor
draw(sFront, surfacepen=surfacepen, meshpen=meshpen);


pen planePen = black+linewidth(0.3);

triple bottomPoint = f((0,vmin));
triple planeCenter = 2.0*bottomPoint;
draw((bottomPoint-.6Z)--(planeCenter+.6Z), arrow=Arrow3(TeXHead2), p=linewidth(0.9),
     L="$\pi_1$");

real planeZ = planeCenter.z;

triple h(pair t) {
    return (t.x, t.y, planeZ);
}

triple g(pair t) {
    triple projectFrom = f(t);
    return h((projectFrom.x, projectFrom.y));
}
triple g(real tx, real ty) { return g((tx, ty)); }

real planeRadius = R+1;
surface thePlane = surface(h, (-planeRadius,-planeRadius),(planeRadius,planeRadius),
    nu=1);

path3 planeOutline = h((-planeRadius,-planeRadius)) -- h((-planeRadius,planeRadius)) -- h((planeRadius,planeRadius)) -- h((planeRadius,-planeRadius)) -- cycle;

for (real u = 0; u <= R; u += R/resolution)
  draw(circle(planeCenter, u), planePen);
for (real v = vmin; v < vmax; v += (vmax-vmin)/(2*resolution)) {
  draw(g(umin,v) -- g(umax,v), planePen);
}
draw(planeOutline, p=planePen);

//Embed the label "\mathbb P^2" on the plane:
real labelScale = 1.5;  
Label planeLabel = Label(scale(labelScale, labelScale*1.3, 1)*"$\mathbb P^2$", fontsize(10pt));
Label placedPlaneLabel = shift((planeRadius-1.2),(planeRadius-1.5),planeCenter.z)*planeLabel;

label(planeLabel, position = (planeRadius-1.2, planeRadius-1.5, planeCenter.z));

\end{asy}
\end{document}

Source: TeX.SE
Author: Charles Staats (License)