Question

(Advanced) Here is a puzzle problem that can be solved with either some fancy analytic geometry (calculus) or a (relatively) simple simulation. Suppose you are located at the exact center of a cube. If you could look all around you in every direction, each wall of the cube would occupy \(\frac{1}{6}\) of your field of vision. Suppose you move toward one of the walls so that you are now halfway between it and the center of the cube. What fraction of your field of vision is now taken up by the closest wall? Hint: Use a Monte Carlo simulation that repeatedly "looks" in a random direction and counts how many times it sees the wall.

Short answer

Use Monte Carlo simulation to estimate the fraction of vision occupied by the nearest wall.

Step-by-step solution

Understanding the Problem

You are at the center of a cube and move halfway towards one of the walls. Originally, each wall occupies \( \frac{1}{6} \) of your field of vision. We want to know the fraction of the field of vision taken up by the closest wall at the new position. We will use a Monte Carlo method to estimate this fraction.

Setting Up the Simulation

Set up a coordinate system within the cube. Assume the cube is centered at the origin and each edge has length 2, extending from -1 to 1 on each axis. You move from the center (0, 0, 0) to (0.5, 0, 0), halfway towards the +x wall at x = 1.

Generating Random Directions

For the Monte Carlo simulation, randomly generate a large number of unit direction vectors from the point (0.5, 0, 0). This can be done by sampling points on a unit sphere.

Counting Intersections

For each randomly generated direction, determine if it hits the wall at x = 1 before any other wall. This can be checked by extending the line from (0.5, 0, 0) in the direction vector until it intersects a wall.

Calculating the Fraction

The fraction of the field of vision taken up by the closest wall is the number of times the direction vector first hits the x = 1 wall divided by the total number of directions sampled.

Running the Simulation

Implement the above logic in a programming environment or using statistical software to simulate a large number of rays, such as 10,000 or more, for an accurate estimate of the fraction of the field of vision occupied by the closest wall.

Key concepts

Analytic Geometry

Analytic geometry is a field of mathematics where algebra is used to study geometric properties. It provides a way to represent geometric shapes and analyze their properties using algebraic equations. In this context, it helps us represent the position within a cube using coordinates. We assume the cube is centered at the origin of our coordinate system with edges spanning from -1 to 1 on each axis.

By doing so, we can easily calculate and understand movements within the cube, such as moving from the center (0, 0, 0) to (0.5, 0, 0). This process is crucial for solving visualization problems like knowing the portion of field of vision occupied by a wall after you've moved towards it. Working with coordinates helps break down the problem into manageable parts by calculating the directions and distances involved.

• Positions are represented as coordinates.
• Geometry is linked with algebraic laws.
• Clear calculation of movements and directions.

Random Direction Vector

In the Monte Carlo simulation, we use random direction vectors to simulate different angles of view, as if we were looking in every possible direction. A direction vector essentially tells us where we are looking from a particular point in space.

For instance, from halfway between the center and a cube wall at (0.5, 0, 0), these random directions are essential to understand which part of the wall is visible. By using random vectors, we ensure that our field of vision is evenly studied and that we accurately estimate the proportion of it that the wall occupies.

• Simulates various viewing angles.
• Ensures randomness in vector selection.
• Helps analyze field of vision extensively.

Unit Sphere Sampling

Unit sphere sampling is a technique used to generate random direction vectors, as needed for the simulation. Imagine a sphere with a radius of 1 centered at the origin of a coordinate system. This is the unit sphere.

In our exercise, we sample points on the surface of this unit sphere to find all possible direction vectors. This method ensures that each direction vector is of unit length, which guarantees consistency in our random sampling process. By sampling a large number of vectors, we mimic an unbiased view of all potential directions in space.

• Generates consistent direction vectors.
• Ensures even coverage of all directions.
• Aids in comprehensive field of vision analysis.

Field of Vision Calculation

Calculating the field of vision involves understanding how much of your total view is occupied by the closest wall. After producing numerous unit direction vectors and their interactions with the cube's walls, we determine the fraction of your view taken by the nearest wall.

This fraction is found by extending each direction vector until it hits a wall and counting how many times the first wall hit is the wall closest to the observer, i.e., x = 1 in our simulation setup. By doing this in a Monte Carlo simulation many times, say over 10,000 trials, we obtain a good approximation of this fraction.

• Determines visible part of the nearest wall.
• Relies on statistical estimation.
• Uses random directional interactions for accuracy.