Taxicab Geometry Math Formula Guide

Understanding Taxicab Geometry Basics

Taxicab Geometry, also called Manhattan Geometry or L1 Geometry, is a branch of mathematics that redefines the concept of distance. In classical Euclidean geometry, the distance between two points is based on the straight-line metric. However, Taxicab Geometry replaces this idea with a metric that only allows horizontal and vertical movement. This approach reflects how movement works in grid-based systems, such as city streets, delivery routing, or robot navigation.

The term “Taxicab” originates from the way taxis must move through city blocks—restricted to streets arranged in perpendicular directions. Because diagonal movement is not possible in such a system, this geometry offers a more realistic model for many modern applications.

In this expanded comprehensive article, similar in structure to a Chemical Ink Formula Guide, we will explore the formal definition, formulas, applications, shapes, comparisons with Euclidean geometry, real-world uses, advanced examples, and extended problem-solving techniques. All formulas are written using MathJax-compatible LaTeX.

Definition and Basic Formula of Taxicab Distance

For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the Taxicab distance is: \[ d_T = |x_2 - x_1| + |y_2 - y_1| \] This formula measures how far you must travel along a grid where movement is constrained to perpendicular directions.

Why Use Absolute Values?

The absolute value ensures that distance is always non-negative. Regardless of whether a coordinate increases or decreases, the movement distance is always a positive quantity. This fits the intuitive notion of how far a taxi or robot must travel in each direction.

Example 1: Basic Distance

Let \( A(2, 5) \) and \( B(9, 1) \): \[ d_T = |9 - 2| + |1 - 5| = 7 + 4 = 11 \] Whereas Euclidean distance: \[ d_E = \sqrt{7^2 + (-4)^2} = \sqrt{65} \] In many real-world situations, the Taxicab distance of 11 is a more functional measurement than \( \sqrt{65} \).

Understanding Shapes in Taxicab Geometry

Taxicab Circles

A Taxicab circle is defined as: \[ |x - a| + |y - b| = r \] This results in a diamond-shaped figure rotated 45 degrees relative to the axes.

Example 2: Circle of Radius 4

Centered at the origin: \[ |x| + |y| = 4 \] Boundary points include:

• \( (4,0) \)
• \( (0,4) \)
• \( (-4,0) \)
• \( (0,-4) \)

Connecting the points forms a square-like diamond.

Taxicab Balls

The filled region: \[ |x - a| + |y - b| \le r \] Forms a filled diamond. This differs significantly from Euclidean balls, which are circular.

Comparing Metrics: Euclidean vs Taxicab

The relationship between the two metrics is essential and often intersects with geometric concepts such as those discussed in Golden Angle Math Formula Explained. Taxicab Geometry simplifies certain calculations while complicating others.

One-Dimensional Comparison

In one dimension, the metrics match: \[ d_E = d_T = |x_2 - x_1| \] But differences arise only in two or more dimensions.

Two-Dimensional Comparison

Euclidean distance corresponds to straight-line travel, while Taxicab distance corresponds to axis-aligned movement:

- Euclidean circle → smooth curve - Taxicab circle → rotated square - Euclidean shortest path → unique straight segment - Taxicab shortest path → infinitely many paths

Table of Differences

Feature	Euclidean	Taxicab
Distance	Straight-line	Sum of orthogonal distances
Geometry Shape	Circles are round	Circles become diamonds
Shortest Path	Unique	Multiple shortest paths
Movement Restrictions	Free direction	Axis-aligned only

The Geometry of Lines in Taxicab Space

Although the concept of a line is unchanged, the geometry of shortest paths and distances behaves differently.

Multiple Shortest Paths

If the Taxicab distance between two points is \( p + q \), any path composed of \( p \) horizontal and \( q \) vertical moves in any order is a shortest path.

Example 3: Counting Shortest Paths

Between points: \[ A(0,0),\quad B(5,3) \] Taxicab distance: \[ d_T = 5 + 3 = 8 \] The number of shortest paths is: \[ \binom{p+q}{p} = \binom{8}{5} = 56 \] So 56 distinct shortest routes exist.

Triangles in Taxicab Geometry

Triangles can look stretched or skewed. Traditional formulas for perimeter and area still apply, but distances differ.

Example 4: Taxicab Triangle

For triangle with points:

• \( A(0,0) \)
• \( B(3,0) \)
• \( C(3,4) \)

Sides:

\[ d_T(A,B)=3,\quad d_T(B,C)=4,\quad d_T(A,C)=7 \] Perimeter: \[ P = 3 + 4 + 7 = 14 \] Even though the shape looks like a Euclidean right triangle, Taxicab Geometry changes the diagonal side length.

Taxicab Norm in Higher Dimensions

Taxicab Geometry generalizes to any number of dimensions. For points: \[ (x_1, x_2, ..., x_n) \] The L1 norm: \[ \|x\|_1 = \sum_{i=1}^n |x_i| \] This makes Taxicab Geometry essential in high-dimensional mathematics.

Example 5: 4D Distance

Between points \( P(1,2,3,4) \) and \( Q(4,1,7,0) \): \[ d_T = |1-4| + |2-1| + |3-7| + |4-0| \] \[ = 3 + 1 + 4 + 4 = 12 \]

Shapes Beyond Circles: Ellipses, Hyperbolas, and Parabolas

Taxicab Ellipse

Defined by: \[ |x - a_1| + |y - b_1| + |x - a_2| + |y - b_2| = k \] This shape becomes a polygon rather than a smooth ellipse.

Taxicab Parabola

Defined as points where distance to a focus equals distance to a line. Because both distances involve absolute values, the shape becomes piecewise linear.

Real-World Applications of Taxicab Geometry

1. Navigation and GPS Routing

In urban environments, navigation tools often compute L1-based routes due to grid-based street networks.

2. Logistic Optimization

Delivery and supply-chain routing frequently use Taxicab metrics for more accurate travel estimates.

3. Machine Learning and Data Science

L1 regularization and Lasso Regression rely on the Taxicab norm: \[ \|w\|_1 = \sum |w_i| \] It encourages sparsity in machine learning models.

4. Robotics and Grid-Based Pathfinding

Robots in grid environments use Taxicab distances for efficient motion planning.

5. Maze and Puzzle Algorithms

Games like grid mazes or tactical boards use Taxicab distance to compute movement cost.

6. Urban Planning

City designers use Taxicab metrics to estimate pedestrian or vehicle travel time across rectangular road networks.

Advanced Problem: Distance with Obstacles

Obstacles modify the shortest-path calculations. If a rectangular region blocks direct travel, the path must be computed around it.

Example 6: Obstacle Problem

Travel from \( A(0,0) \) to \( B(10,8) \). Obstacle: \[ 3 \le x \le 7,\quad 4 \le y \le 9 \] Straight Taxicab distance: \[ d_T = 10 + 8 = 18 \] But this path crosses the obstacle. Two main bypass options:

Path Below the Obstacle

Go to \( y = 4 \), then go horizontally, then go upward. This adds extra vertical movement.

Path to the Left

Go around the left boundary at \( x = 3 \). This changes total distance depending on which side minimizes the detour.

Real-world systems compute these detours using Taxicab-optimized routing algorithms like A*.

Grid Geometry and Combinatorics

Taxicab Geometry ties deeply into combinatorics, especially in counting unique shortest paths.

Example 7: Larger Path Counting

Compute shortest paths from \( (0,0) \) to \( (10,7) \): \[ d_T = 10 + 7 = 17 \] Number of shortest paths: \[ \binom{17}{10} \] This combinatorial insight is also used in probability and lattice path theory.

Taxicab Area and Perimeter

While area calculations remain identical to Euclidean formulas, perimeters differ due to distance changes.

Example 8: Perimeter of a Diamond

Taxicab circle radius = \( r \) Vertices: \( (r,0), (0,r), (-r,0), (0,-r) \)

Each side is length: \[ d_T((r,0), (0,r)) = |r-0| + |0-r| = r + r = 2r \] Perimeter: \[ P = 4 \cdot 2r = 8r \] Compared to Euclidean: \[ P = 2\pi r \]

Isometries in Taxicab Geometry

Translations

Always preserve distance: \[ |(x+k) - (a+k)| = |x-a| \]

Reflections

Reflections across axes preserve Taxicab distance.

Rotations

Only rotations of multiples of 90 degrees preserve the Taxicab metric. Other rotations distort distances.

Optimization Problems Using Taxicab Geometry

Median Point (1-Median Problem)

Given a set of points, the point minimizing the sum of Taxicab distances is the coordinate-wise median.

Example 9: Find Taxicab Median

Points: \( (1,4), (3,5), (7,2), (4,8) \)

Median of x-coordinates = 3.5 Median of y-coordinates = 4.5 So the optimal location is: \[ (3.5, 4.5) \]

Taxicab Geometry is a powerful and flexible alternative to traditional Euclidean geometry. It provides a better model for grid-based movement, optimization problems, logistics, robotics, machine learning, and many real-world systems. By redefining distance as the sum of horizontal and vertical motions, this geometry opens the door to new insights and applications across mathematics and applied sciences.

From shapes like diamonds and polygonal ellipses to complex optimization problems, Taxicab Geometry offers a rich, unique, and practical framework. This expanded exploration has covered foundational concepts, advanced mathematical structures, computational examples, and real-world relevance—demonstrating its significance and versatility in modern mathematical study.

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