Nnamdi Okpala

Posted on Nov 3

Optimizing Descent — An Exploration of the Brachistochrone Problem

optimizing descent
By Nnamdi Michael Okpala (okpalan)

Introduction to the Brachistochrone Problem
1.1 Historical Context
1.2 Computationally Efficient Approximation
Approach and Optimization
2.1 Core Steps of the Algorithm
2.2 Parameter Explanation
Implementation on the HTML5 Canvas
Conclusion 5.References

— -

1. Introduction to the Brachistochrone Problem

1.1 Historical Context

The Brachistochrone problem, initially posed by Johann Bernoulli in 1696, is a fascinating question in the calculus of variations. It seeks to identify the curve of fastest descent between two points under the influence of gravity. Notable mathematicians, including Isaac Newton and Gottfried Wilhelm Leibniz, tackled this problem, with Newton famously finding the solution overnight. The answer is the cycloid, the path traced by a point on the circumference of a rolling wheel.

Bernoulli’s challenge sparked significant interest in the mathematical community, leading to the development of variational calculus — a field that analyses optimal paths and surfaces. The implications of this problem extend beyond pure mathematics; its principles have influenced various domains, including physics, engineering, and even economics. For instance, understanding the optimal path for descent has applications in roller coaster design, where engineers use these principles to maximize speed and minimize construction costs while ensuring safety. Thus, the Brachistochrone problem serves as a cornerstone in the study of motion, demonstrating the interconnectedness of mathematics and the physical world.

1.2 Computationally Efficient Approximation

In my exploration of the Brachistochrone problem, I developed an approach that provides a computationally efficient approximation to the classic solution. This method leverages geometric properties and trigonometric relationships to optimize the descent time, making it faster and more memory-efficient compared to traditional methods.

To achieve this, I analysed the cycloid’s defining characteristics, such as its curvature and the angles at which it descends. By applying trigonometric functions, I could derive key equations that allow for a simplified representation of the cycloid path. This approximation involves using weighted averages of points along the trajectory, enabling quick calculations of the optimal descent path without the need for complex numerical integration techniques.

For example, instead of calculating the exact cycloidal path for every point, I focus on determining critical points that influence the curve’s shape. By dynamically adjusting the steepness of the curve through a parameter called angle Factor, users can visualize and experiment with different configurations. This not only makes the computation faster but also provides insights into how variations in the curve’s properties impact descent time. Ultimately, this approach retains the essence of the Brachistochrone problem while enhancing its computational efficiency, making it more accessible for educational and practical applications.

In the best-case scenario, the descent along the hypotenuse reaches the point where it meets the curve at a right angle, representing the optimal transition between the straight-line path and the cycloidal curve. This geometric relationship highlights the significance of right angles in minimizing descent time, as it facilitates a smooth and efficient transition along the curve. We are trying to maximize the curve as ths is key to the optimization.

— -

2. Approach and Optimization

2.1 Core Steps of the Algorithm

To achieve the optimal solution, I broke down my approach into the following core steps:

Set the First Point (Starting Point):
Let’s denote this point as ( P_1(x_1, y_1) ).
Set the Second Point (Ending Point):
This will be referred to as point ( P_2(x_2, y_2) ).
Connect Points to Form a Triangle:
I connect ( P_1 ) and ( P_2 ) with a line segment, forming the hypotenuse of a triangle. A third point is chosen along the horizontal or vertical line of either ( P_1 ) or ( P_2 ) to create a right triangle or another type.
Determine if the Triangle is a Right Angle:
The next step involves checking if the triangle formed is a right triangle at the intersection of the two lines. This can be done using the dot product for verifying orthogonality.
Weighted Average and Sine Calculation:
I calculate the weighted average (centroid) of the points as follows:[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) ]
Finally, I multiply by the sine of the target angle, incorporating the weighted influence of each point: [ G \cdot \sin(\theta) ]

— -
In other words:
Here are the steps for implementing this solution to approximate the Brachistochrone curve with a weighted average approach:

— -

2.2 Parameter Explanation

The algorithm could a parameter called angleFactor, which allows users to adjust the steepness and shape of the curve dynamically by moving points. By modifying this parameter, readers can experiment with the curve’s properties, gaining insights into how different configurations impact the descent time.

— -

3. Traditional Implementation on the HTML5 Canvas

To illustrate my approach, I created an HTML5 canvas implementation of the traditional solution to the Brachistochrone problem. This visualization allows users to observe the cycloid curve and understand its significance in relation to my computational method.

<!DOCTYPE html>

Brachistochrone - Cycloid Visualization

 
canvas { 
border: 1px solid black; 
display: block; 
margin: 0 auto; 
} 

 
const canvas = document.getElementById('myCanvas'); 
const ctx = canvas.getContext('2d');
<div class="highlight"><pre class="highlight plaintext"><code> // Brachistochrone parameters
const start = { x: 50, y: 50 }; // Start point
const end = { x: 750, y: 550 }; // End point
const numPoints = 100; // Number of points to calculate on the curve

// Function to draw the cycloid
function drawCycloid() {
    ctx.clearRect(0, 0, canvas.width, canvas.height); // Clear canvas

    // Draw start and end points
    ctx.fillStyle = 'green';
    ctx.beginPath();
    ctx.arc(start.x, start.y, 5, 0, Math.PI * 2);
    ctx.fill();
    ctx.fillStyle = 'red';
    ctx.beginPath();
    ctx.arc(end.x, end.y, 5, 0, Math.PI * 2);
    ctx.fill();

    // Draw the cycloid curve
    ctx.beginPath();
    for (let i = 0; i &amp;lt;= numPoints; i++) {
        const t = (i / numPoints) * Math.PI; // Parameter t from 0 to π
        const x = start.x + t * (end.x - start.x) / Math.PI; // Cycloid X-coordinate
        const y = start.y - (1 - Math.cos(t)) * (start.y - end.y) / 2; // Cycloid Y-coordinate

        // Move to the first point
        if (i === 0) {
            ctx.moveTo(x, y);
        } else {
            ctx.lineTo(x, y);
        }
    }
    ctx.strokeStyle = 'blue';
    ctx.stroke(); // Draw the curve
}

// Draw the initial cycloid
drawCycloid();

</script>
</code></pre></div>
</body> 
</html> 
— -
<h2>
<a name="time-complexity-of-traditional-brachistochrone-implementation" href="#time-complexity-of-traditional-brachistochrone-implementation" class="anchor">
</a>
Time Complexity of Traditional Brachistochrone Implementation.
</h2>

Calculating Points on the Cycloid: Given the physics involved, calculations are made based on the cycloid equations derived from parametric equations.

Numerical Integration: In practice, numerical methods (like Runge-Kutta) might be used to simulate the curve’s shape.

— -
<h2>
<a name="my-raw-optimized-endraw-impelementation" href="#my-raw-optimized-endraw-impelementation" class="anchor">
</a>
My <code>Optimized</code> Impelementation
</h2>

<!DOCTYPE html> 
<html lang="en"> 
<head> 
<meta charset="UTF-8"> 
<title>Optimal Brachistochrone Visualization</title> 
<style> 
canvas { 
border: 1px solid black; 
} 
</style> 
</head> 
<body> 
<canvas id="myCanvas" width="800" height="600"></canvas> 
<script> 
const canvas = document.getElementById('myCanvas'); 
const ctx = canvas.getContext('2d');
<div class="highlight"><pre class="highlight plaintext"><code> // Set the start and end points
const start = { x: 50, y: 50 }; // Start point
const end = { x: 750, y: 550 };

// Calculate the mid-point weighted average
const mid = {
    x: (start.x + end.x) / 2,
    y: (start.y + end.y) / 2
};

// Function to draw the right-angled triangle approximation
function drawTriangle() {
    ctx.beginPath();
    ctx.moveTo(start.x, start.y);      // Start point
    ctx.lineTo(mid.x, end.y);          // Right angle corner
    ctx.lineTo(end.x, end.y);          // End point
    ctx.closePath();
    ctx.strokeStyle = "blue";
    ctx.stroke();
}

// Function to draw the approximate 
// trajectory (curve)
function drawTrajectory() {
    ctx.beginPath();
    ctx.moveTo(start.x, start.y); // Start point
    ctx.quadraticCurveTo(mid.x, end.y, end.x, end.y); // Control point at mid
    ctx.strokeStyle = "red";
    ctx.stroke();
}

// Draw everything
drawTriangle();
drawTrajectory();

</script>
</code></pre></div>
</body> 
</html>
<h3>
<a name="stepbystep-implementation" href="#stepbystep-implementation" class="anchor">
</a>
Step-by-Step Implementation
</h3>

<ol>
<li>Set Points A and B: 
— Define the positions for points ( A ) and ( B ) on a 2D plane. ( A ) will be the higher (starting) point, and ( B ) the lower (ending) point of the descent.</li>
<li>Construct a Triangle: 
— Draw a triangle with ( A ) and ( B ) forming the endpoints of the hypotenuse. 
— At the midpoint of the hypotenuse, form an angle (the opposite angle) where the two lines meet. This forms the structure from which the curve will be calculated.</li>
<li>Construct a Spline Using Weighted Average: 
— To approximate the descent curve, compute a spline based on the weighted average ( W ) of points along the hypotenuse. 
— Use the formula ( W = (1 — t) \cdot C + t \cdot P ), where: 
— ( C ) is the midpoint (or weighted midpoint) of ( A ) and ( B ) along the hypotenuse. 
— ( P ) is the point opposite to the angle where the lines meet. 
— ( t ) is a parameter (0 to 1) that interpolates between ( C ) and ( P ), dynamically adjusting the position on the spline.</li>
<li>Minimize the Spline Using the Projection Point: 
— Optimize the descent time by adjusting ( t ), refining the spline to minimize travel distance based on the angle. 
— Multiply by the projection ( P ) to fine-tune the curve’s shape based on the opposite angle, ensuring a path that represents the quickest descent under gravity.</li>
</ol>

Time Complexity: The key operations in my approach include:
<h3>
<a name="calculating-midpoint" href="#calculating-midpoint" class="anchor">
</a>
Calculating Midpoint:
</h3>

𝑂(1)* O(1) since it uses direct calculations to dictate midpoint. 
Drawing Shapes: Each shape (triangle and curve) is drawn in a constant amount of time for a single render operation. 
— - 

<div class="highlight"><pre class="highlight python"><code>
import numpy as np
import matplotlib.pyplot as plt

# Define the start and end points for the Brachistochrone problem
start = np.array([50, 50])
end = np.array([750, 550])

# Midpoint weighted average — midpoint of the start and end points
mid = (start + end) / 2

# Function to calculate the right-angled triangle approximation
def draw_triangle_approximation(start, end, mid):
plt.plot([start[0], mid[0], end[0]], [start[1], end[1], end[1]], ‘b — ‘, label=”Triangle Approximation”)
plt.scatter(start, color=”green”, label=”Start Point”)
plt.scatter(end, color=”red”, label=”End Point”)

# Function to calculate the approximate curve trajectory using a quadratic spline
def draw_quadratic_spline(start, end, mid):
t_values = np.linspace(0, 1, 100)
curve_x = (1 — t_values)  ((1 — t_values)  start[0] + t_values  mid[0]) + t_values  ((1 — t_values)  mid[0] + t_values  end[0])
curve_y = (1 — t_values)  ((1 — t_values)  start[1] + t_values  mid[1]) + t_values  ((1 — t_values)  mid[1] + t_values  end[1])

plt.plot(curve_x, curve_y, ‘r-’, label=”Quadratic Spline Approximation”)
plt.legend()

# Plot the triangle approximation and spline approximation on the same graph
plt.figure(figsize=(10, 8))
draw_triangle_approximation(start, end, mid)
draw_quadratic_spline(start, end, mid)
plt.xlabel(“X”)
plt.ylabel(“Y”)
plt.title(“Brachistochrone Problem — Optimized Triangle and Spline Approximations”)
plt.grid(True)
plt.show()
</code></pre></div>

Quadratic Spline Approximation and Optimized Traingle
<h2>
<a name="4-conclusion" href="#4-conclusion" class="anchor">
</a>

Conclusion <a name=”conclusion”></a> </h2>

My exploration of the Brachistochrone problem has not only deepened my understanding of mathematical concepts but has also revealed the intricate connections between diverse fields, such as physics and geometry. The insight into how vectors can form curves and splines related to descent problems was pivotal in developing a solution that is both innovative and efficient. I am eager to share this knowledge with others and inspire further exploration of the exciting intersections between mathematics and science.

Happy Code Optimizing.

DEV Community

Optimizing Descent — An Exploration of the Brachistochrone Problem

Table of Contents

1. Introduction to the Brachistochrone Problem

2. Approach and Optimization

2.1 Core Steps of the Algorithm

2.2 Parameter Explanation

3. Traditional Implementation on the HTML5 Canvas

Top comments (0)

Read next

Standup Serverless Jenkins on Fargate with Terraform - Part 1: Networking

Understanding Fallback and Receive Functions in Solidity

Personal Injury Claim's History History Of Personal Injury Claim

Beginner's Guide to Managing Files in Amazon S3 with Go