Distance Calculator - Distance Measurement

This physics calculator helps you calculate distances between points in 2D and 3D space using various distance metrics. Distance is a fundamental concept in physics and geometry.

Distance Definition

• Euclidean Distance: Straight-line distance between points
• Manhattan Distance: Sum of absolute differences (L1 norm)
• Chebyshev Distance: Maximum absolute difference (L∞ norm)
• Displacement: Vector from initial to final position

Distance Formulas

• Euclidean (3D): d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)
• Euclidean (2D): d = √((x₂-x₁)² + (y₂-y₁)²)
• Manhattan: d = |x₂-x₁| + |y₂-y₁| + |z₂-z₁|
• Chebyshev: d = max(|x₂-x₁|, |y₂-y₁|, |z₂-z₁|)
• Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)

Applications

• Physics: Position, displacement, and motion
• Geometry: Point-to-point measurements
• Navigation: GPS and routing
• Computer Science: Clustering and classification
• Engineering: Structural design and analysis
• Robotics: Path planning and obstacle avoidance

Important Notes

• Euclidean distance is always shortest
• Manhattan distance ≥ Euclidean distance
• Chebyshev distance ≥ Manhattan distance
• Distance is always non-negative
• Distance is symmetric: d(A,B) = d(B,A)

Distance Types

• Metric Distances: Satisfy triangle inequality
• Non-Metric Distances: May violate triangle inequality
• Weighted Distances: Different weights for dimensions
• Curved Space Distances: On spheres or manifolds
• Time-Dependent Distances: Varying with time

Distance Examples

• Pythagorean Theorem: 3-4-5 triangle
• Unit Circle: Distance from origin = 1
• City Blocks: Manhattan distance
• Chess Moves: Chebyshev distance
• Great Circle: Distance on Earth's surface

Coordinate Systems

• Cartesian: (x, y, z) coordinates
• Cylindrical: (r, θ, z) coordinates
• Spherical: (r, θ, φ) coordinates
• Polar: (r, θ) coordinates

Distance Properties

• Non-Negativity: d(A,B) ≥ 0
• Identity: d(A,A) = 0
• Symmetry: d(A,B) = d(B,A)
• Triangle Inequality: d(A,C) ≤ d(A,B) + d(B,C)
• Translation Invariance: d(A,B) = d(A+c, B+c)

Special Distances

• Hamming Distance: Number of differing bits
• Edit Distance: String similarity
• Mahalanobis Distance: Statistical distance
• Cosine Distance: Angular similarity
• Jaccard Distance: Set similarity