### Investigating multiple reflections of a laser beam题号：144 难度： 50 中英对照

In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out.

The specific white cell we will be considering is an ellipse with the equation 4x2 + y2 = 100

The section corresponding to −0.01 ≤ x ≤ +0.01 at the top is missing, allowing the light to enter and exit through the hole.

The light beam in this problem starts at the point (0.0,10.1) just outside the white cell, and the beam first impacts the mirror at (1.4,-9.6).

Each time the laser beam hits the surface of the ellipse, it follows the usual law of reflection "angle of incidence equals angle of reflection." That is, both the incident and reflected beams make the same angle with the normal line at the point of incidence.

In the figure on the left, the red line shows the first two points of contact between the laser beam and the wall of the white cell; the blue line shows the line tangent to the ellipse at the point of incidence of the first bounce.

The slope m of the tangent line at any point (x,y) of the given ellipse is: m = −4x/y

The normal line is perpendicular to this tangent line at the point of incidence.

The animation on the right shows the first 10 reflections of the beam.

How many times does the beam hit the internal surface of the white cell before exiting?

### Code

public final class p144 {
public static void main(String[] args) {
long start=System.nanoTime();
String result = run();
long end=System.nanoTime();
System.out.println(result);
System.out.println( (end-start)/1000000 + "ms" );
}

static public String run(){
int res=0;
double xA=0.0;
double yA=10.1;
double xO=1.4;
double yO=-9.6;
while(Math.abs(xO)>0.01 || yO<0){
//Calculate the slope of A
double slopeA = (yO - yA) / (xO - xA);
//Calculate the slope of the ellipse tangent
double slopeO = -4*xO/yO;
//Calculate the slope of B
double tanA = (slopeA - slopeO)/(1+slopeA*slopeO);
double slopeB = (slopeO- tanA)/ (1+ tanA*slopeO);
//calculate intercept of line B
double interceptB = yO - slopeB * xO;
//solve the quadratic equation for finding
// the intersection of B and the ellipse
// a*x^2 + b*x + c = 0
double a = 4 + slopeB*slopeB;
double b = 2 * slopeB * interceptB;
double c = interceptB * interceptB - 100;
double ans1 = (-b + Math.sqrt(b * b - 4 * a * c)) / (2 * a);
double ans2 = (-b - Math.sqrt(b * b - 4 * a * c)) / (2 * a);
xA = xO;
yA = yO;
//Take the solution which is furtherst from x0
xO = (Math.abs(ans1 - xO) > Math.abs(ans2 - xO)) ? ans1 : ans2;
yO = slopeB * xO + interceptB;
res++;
}
return res+"";
}

}
354
0ms