billiard table 从角点开始45°发球，轨迹自交点数

hbghlyj

在 $p:q$ ($p,q\inZ_+$) 的矩形台球桌上，方向 45 度，从角点开始发球，轨迹有多少个自交点？

7:5有12个交点：
import graph;
picture pic;
size(100);

path bill = box((0,0),(7,5));

draw(bill);

pair s = (1e-3,1e-3), db, dt = (1,1), e = s+1e2*dt;
path traj = s--e;
path path = s;
real [] c;

for(int i:sequence(11)) {

  c = intersect(bill, traj);
  e = point(traj, c[1]);
  db = dir(bill, c[0]);
  path = path--e;  

  dt = -dt + 2*dot(dt,db)*db;

  s = e;
  e = s + 1e2*dt;

  for(pair a:intersectionpoints(s+1e-2*dt--e, path)){dot(pic,a,blue);}
  traj = (s+dt)--e;
}
draw(path,red);
add(pic);
7:3有6个交点：
import graph;
picture pic;
size(100);

path bill = box((0,0),(7,3));

draw(bill);

pair s = (1e-3,1e-3), db, dt = (1,1), e = s+1e2*dt;
path traj = s--e;
path path = s;
real [] c;

for(int i:sequence(9)) {

  c = intersect(bill, traj);
  e = point(traj, c[1]);
  db = dir(bill, c[0]);
  path = path--e;  

  dt = -dt + 2*dot(dt,db)*db;

  s = e;
  e = s + 1e2*dt;

  for(pair a:intersectionpoints(s+1e-2*dt--e, path)){dot(pic,a,blue);}
  traj = (s+dt)--e;
}
draw(path,red);
add(pic);

矩形內部格点数$(p-1)(q-1)$除以2

来自第8页

hbghlyj

在 $p:q$ ($p,q\inZ_+$) 的矩形台球桌上，方向 45 度，从一般的点开始发球，轨迹有多少个自交点？
mathcurve.com說，有$p(q-1)+q(p-1)$个自交点。如何證明

例如7:5有7×4+5×6=58个交点：
import graph;
picture pic;
size(100);

path bill = box((0,0),(7,5));

draw(bill);

pair s = (.7,1e-3), db, dt = (1,1), e = s+1e1*dt;
path traj = s--e;
path path = s;
real [] c;

for(int i:sequence(23)) {

  c = intersect(bill, traj);
  e = point(traj, c[1]);
  db = dir(bill, c[0]);
  path = path--e;  

  dt = -dt + 2*dot(dt,db)*db;

  s = e;
  e = s + 1e1*dt;

  for(pair a:intersectionpoints(s+1e-1*dt--e, path)){dot(pic,a,blue);}
  traj = (s+1e-3*dt)--e;
}
draw(path--cycle,red);
add(pic);

hbghlyj

Multiple Points on Lissajous's Curves in Two and Three Dimensions, Edward A. Hook, Annals of Mathematics

這裡有一個證明，但是没看懂……