# Problem 144: Investigating multiple reflections of a laser beam

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 4x^2 + y^2 = 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?

# My Algorithm

I solved this problem using a combination of vector computations and playing with the formula of an ellipse.

The structs Vector and Point are very simple. I could get rid of them and just use simple variables
but I like to have my code clean and organized instead of short and unreadable.

Each iteration of the main loop performs these tasks:
1. check the intersection to whether it belongs to the hole on top of the ellipse (and isn't actually a hole).
2. find the normal at point to
3. reflect the ray's direction in relation to the normal
4. compute next intersection of the new ray

I consider every intersection between -0.01 <= x <= +0.01 and y > 9.9 to be a valid exit point.
(it's even sufficient to ask for y > 0, but don't change the 9.9 to 10 because of the curving of the ellipse).

The reflection can be found very easily with the dot product (nice and short explanation: paulbourke.net/geometry/reflected/)

It took me quite some time to come up with a formula that computes the next intersection.

The ray has a slope s = y / x and any point on the ray is described as
(1) y_2 = s(x_2 - x_1) + y_1

The ellipse was defined as 4x^2 + y^2 = 100 which is
(2) dfrac{x^2}{5} + dfrac{y^2}{10} = 1 (that means a = 5 and b = 10)

The first (already known) intersection occurs at
(3) 4x_1^2 + y_1^2 = 1

The second (unknown) intersection will be at
(4) 4x_2^2 + y_2^2 = 1

I replace in the latter formula (4) the variable y_2 by the ray equation (1):
(5) 4x_2^2 + (s(x_2 - x_1) + y_1)^2 = 1

Since the the ellipse equation (2) is always 1 for each point, I can write:
(6) 4x_2^2 + (s(x_2 - x_1) + y_1)^2 = 4x_1^2 + y_1^2

(7) 4x_2^2 + s^2(x_2 - x_1)^2 + 2s(x_2 - x_1)y_1 + y_1^2 = 4x_1^2 + y_1^2

(8) 4x_2^2 + s^2(x_2 - x_1)^2 + 2s(x_2 - x_1)y_1 = 4x_1^2

(9) 4(x_2^2 - x_1^2) + s^2(x_2 - x_1)^2 + 2s(x_2 - x_1)y_1 = 0

(10) 4(x_2 - x_1)(x_2 + x_1) + s^2(x_2 - x_1)^2 + 2s(x_2 - x_1)y_1 = 0

Dividing by the term (x_2^2 - x_1^2):
(11) 4(x_2 + x_1) + s^2(x_2 - x_1) + 2s y_1 = 0

(12) 4x_2 + 4x_1 + s^2 x_2 - s^2 x_1 + 2s y_1 = 0

(13) 4x_1 - s^2 x_1 + 2s y_1 = -4x_2 - s^2 x_2

(14) 4x_1 - s^2 x_1 + 2s y_1 = x_2 (-4 - s^2)

(15) dfrac{4x_1 - s^2 x_1 + 2s y_1}{-4 - s^2} = x_2

Once I know x_2 it's a walk in the park to get y_2 with the help of (1).

## Alternative Approaches

There are several approaches to this problem:

• you can use trigonometric equations instead of vector computations.
• the intersection line/ellipse often leads to a quadratic equation which I believe is more prone to rounding errors

## Note

I even get the correct result when replacing double by float. That's surprising.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>
#include <cmath>

// a simple 2D vector
struct Vector
{
// create new vector
Vector(double x_, double y_) : x(x_), y(y_) {}
double x, y;

// compute length
double getLength() const
{
return sqrt(x*x + y*y);
}
// dot product
double dot(const Vector& other) const
{
return x * other.x + y * other.y;
}
};
// use a Vector for 2D points, too
typedef Vector Point;

int main()
{
// count reflections
unsigned int steps = 0;

// initial ray
Point from(0.0, 10.1);
Point to  (1.4, -9.6);

while (true)
{
// exit ellipse when x is between -0.01 and +0.01 on the upper side of the ellipse
if (to.x >= -0.01 && to.x <= 0.01 && to.y > 10 - 0.1) // vertical radius is 10, subtract a bit because of curving
break;

// find inward pointing vector of the ellipse at intersection point
Vector normal(-4 * to.x, -to.y);
// normalize vector length
double length = normal.getLength();
normal.x /= length;
normal.y /= length;

// current direction of the ray
Vector direction(to.x - from.x, to.y - from.y);
// degenerated case ? ray will exit after only one bounce
if (direction.x == 0)
{
steps++; // same as steps = 1 in this case
break;
}

// reflect ray at intersection considering the normal
// a short explanation can be found here: http://paulbourke.net/geometry/reflected/
Vector reflect(0, 0);
reflect.x = direction.x - 2 * direction.dot(normal) * normal.x;
reflect.y = direction.y - 2 * direction.dot(normal) * normal.y;

// slope of the reflected ray
double slope = reflect.y / reflect.x;

// previous endpoint becomes the start point of the next iteration
from.x = to.x;
from.y = to.y;

// find next intersection
// I derived that strange formula in the initial comment section
to.x = (4*from.x - slope*slope*from.x + 2*slope*from.y) / (-4 - slope*slope);
to.y = slope * (to.x - from.x) + from.y;

// count iterations
steps++;
}

// display result
std::cout << steps << std::endl;
return 0;
}


This solution contains 13 empty lines, 20 comments and 2 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

June 30, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 50% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

more about me can be found on my homepage, especially in my coding blog.
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