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struct curve {
real a=0;
real b=8;
real y2(real x) {
return x^3+a*x+b;
}
real disc() {
return -16*(4*a*a*a+27*b*b);
}
real lowx () {
return sqrt(-a/3);
}
int comps() {
if (a < 0) {
real x=sqrt(-a/3);
return y2(x) < 0 ? 2 : 1;
}
return 1;
}
void locus(picture pic=currentpicture, real m, real M, int n=100,
pen p=currentpen) {
path flip(path p, bool close) {
path pp=reverse(yscale(-1)*p)..p;
return close ? pp..cycle : pp;
}
path section(real m, real M, int n) {
guide g;
real width=(M-m)/n;
for(int i=0; i <= n; ++i) {
real x=m+width*i;
real yy=y2(x);
if (yy > 0)
g=g..(x,sqrt(yy));
}
return g;
}
if (comps() == 1) {
draw(pic,flip(section(m,M,n),false),p);
}
else {
real x=lowx(); // The minimum on x^3+ax+b
if (m < x)
draw(pic,flip(section(m,min(x,M),n),true),p);
if (x < M)
draw(pic,flip(section(max(x,m),M,n),false),p);
}
}
pair neg(pair P) {
return finite(P.y) ? yscale(-1)*P : P;
}
pair add(pair P, pair Q) {
if (P.x == Q.x && P.x != Q.x)
return (0,infinity);
else {
real lambda=P == Q ? (3*P.x^2+a)/(2*P.y) : (Q.y-P.y)/(Q.x-P.x);
real Rx=lambda^2-P.x-Q.x;
return (Rx,(P.x-Rx)*lambda-P.y);
}
}
}
import graph;
import math;
size(0,200);
curve c; c.a=-1; c.b=4;
pair oncurve(real x)
{
return (x,sqrt(c.y2(x)));
}
picture output;
axes();
c.locus(-4,3,.3red+.7blue);
pair P=oncurve(-1),Q=oncurve(1.2);
pair PP=c.add(P,P),sum=c.add(P,Q);
save();
drawline(P,Q,dashed);
drawline(c.neg(sum),sum,dashed);
dot("$P$", P, NW);
dot("$Q$", Q, SSE);
dot(c.neg(sum));
dot("$P+Q$", sum, 2SW);
add(output,currentpicture.fit(),(-0.5cm,0),W);
restore();
save();
drawline(P,c.neg(PP),dashed);
drawline(c.neg(PP),PP,dashed);
dot("$P$", P, NW);
dot(c.neg(PP));
dot("$2P$", PP, SW);
add(output,currentpicture.fit(),(0.5cm,0),E);
shipout(output);
restore();
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