Question

Compute the length of a path. Some object is moving along a path in the plane. At \(n\) points of time we have recorded the corresponding \((x, y)\) positions of the object: \(\left(x_{0}, y_{0}\right),\left(x_{1}, y_{2}\right), \ldots,\left(x_{n-1}, y_{n-1}\right) .\) The total length \(L\) of the path from \(\left(x_{0}, y_{0}\right)\) to \(\left(x_{n-1}, y_{n-1}\right)\) is the sum of all the individual line segments \(\left(\left(x_{i-1}, y_{i-1}\right)\right.\) to \(\left.\left(x_{i}, y_{i}\right), i=1, \ldots, n-1\right)\) : $$ L=\sum_{i=1}^{n-1} \sqrt{\left(x_{i}-x_{i-1}\right)^{2}+\left(y_{i}-y_{i-1}\right)^{2}} $$ Make a function pathlength \((x, y)\) for computing \(L\) according to the formula. The arguments \(\mathrm{x}\) and \(\mathrm{y}\) hold all the \(x_{0}, \ldots, x_{n-1}\) and \(y_{0}, \ldots, y_{n-1}\) coordinates, respectively. Test the function on a triangular path with the four points \((1,1),(2,1),(1,2)\), and \((1,1) .\) Name of program file: pathlength.py.

Short answer

The path length is approximately 3.41.

Step-by-step solution

Understanding the Problem

The task is to calculate the total length of a path in the plane, given by a series of coordinate points. This path is composed of straight line segments connecting these points. For each segment, the length can be found using the distance formula between two points.

Examining the Distance Formula

The distance between two points \((x_{i-1}, y_{i-1})\) and \((x_i, y_i)\) in a 2D plane is calculated using the formula: \[ d = \sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2} \]This formula is derived from the Pythagorean theorem.

Designing the Function

The function `pathlength(x, y)` will accept two lists of coordinates, \(x\) and \(y\), corresponding to the x-coordinates and y-coordinates of the points, respectively. The function will iterate through all consecutive pairs of points, calculate the distance between them, and accumulate these distances to get the total length \(L\).

Implementing the Function

In the program file `pathlength.py`, implement the following code: ```python import math def pathlength(x, y): L = 0 for i in range(1, len(x)): dx = x[i] - x[i-1] dy = y[i] - y[i-1] L += math.sqrt(dx**2 + dy**2) return L ``` This uses Python's `math.sqrt` function for calculating the square root.

Testing the Function

Test the `pathlength` function with the provided triangular path points: ```python x = [1, 2, 1, 1] y = [1, 1, 2, 1] L = pathlength(x, y) print(f'The path length is {L:.2f}') ``` This test checks if the function correctly computes the length of the path made by these points.

Analyzing the Test Result

By substituting the coordinates into our function, we find the segment lengths: 1. From (1,1) to (2,1) has length 1.2. From (2,1) to (1,2) has length \(\sqrt{2}\), approximately 1.41.3. From (1,2) to (1,1) has length 1.Adding these gives a total path length of approximately 3.41.

Key concepts

Distance Formula

When measuring the length between two points in a 2D plane, the distance formula becomes exceptionally useful. The formula is given by \[ d = \sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2} \]. This formula stems from the Pythagorean theorem, which relates the sides of a right-angle triangle.

• Horizontal and Vertical Differences: The expressions \((x_i - x_{i-1})\) and \((y_i - y_{i-1})\) represent the differences in the x and y coordinates between the two points.
• Hypotenuse Calculation: These differences form two sides of a right triangle with the distance between the points being the hypotenuse, hence the use of the square root to find it.

Understanding this concept is key to applying this formula in geometry, physics, or even computer graphics. The distance formula helps us seamlessly compute how far apart points are, paving the way for further geometric calculations.

Python Function

Creating a Python function to compute path lengths is not only efficient but also straightforward. With Python, we can design reusable pieces of code that solve specific problems. The task is to create a function called `pathlength` that computes the cumulative path length given a series of

• Function Definition: The function `pathlength(x, y)` receives two lists, x and y, each containing respective x and y coordinates of points.
• Looping through Points: By iterating through the range of points, starting from 1 (since we're looking at pairs), we calculate the distance between each consecutive pair of points.
• Distance Calculation: Using the `math.sqrt` function, the distance is calculated for each segment of the path.
• Sum Accumulation: This distance is then added to the total, which is initialized to zero before the loop.

This approach neatly takes advantage of Python's powerful list operations and its math module, resulting in clear and concise code.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that utilizes algebraic formulas to represent and understand geometric figures and their properties in the coordinate plane. When dealing with problems involving paths, understanding coordinate geometry is beneficial.

• Coordinate System: Typically, we use the Cartesian coordinate system, which allows any point in the plane to be described by an ordered pair of numbers \((x, y)\).
• Points and Lines: In this system, geometric concepts like distance, midpoint, and slope can be easily calculated by formulas involving these coordinate points.
• Path Description: The sequence of points along a path can be visualized as a series of straight lines or segments in this space.

Coordinate geometry offers systematic techniques to analyze shapes and distances, providing a balance between algebra and geometry. It is the foundation for more complex studies involving motions and shapes in higher dimensions.

2D Plane

A 2D plane is a flat, two-dimensional surface extending infinitely in two directions, often defined in mathematics using a grid system where every location is given by two coordinates, typically x and y. Here are a few important points:

• Basic Concept: Think of the 2D plane as a piece of graph paper. Each point on this graph can be identified by two numbers corresponding to its horizontal (x-axis) and vertical (y-axis) positions.
• Graphing Figures: The 2D plane allows us to visually represent equations, shapes, and paths, making complex mathematical ideas easier to comprehend.
• Applications: In coordinate geometry, we often plot points, draw lines, and understand the relationship between different lines and shapes.

By studying the 2D plane, we gain insights not only into basic shapes and their dimensions but also into more complex scenarios involving motion, intersection, or alignment of paths and curves. Mastery of this concept empowers one to tackle a wide range of mathematical and real-world problems.