Euclidean Distance

Definition and Calculation

• Euclidean Distance: The length of the line segment between two points in Euclidean space.
• Calculated using Cartesian coordinates and the Pythagorean theorem.
• Also known as the Pythagorean distance.
• Formula: For points (A(x_1, y_1)) and (B(x_2, y_2)), distance (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

Historical Background

• Named after Greek mathematicians Euclid and Pythagoras.
• In Greek geometry, distances were represented as line segments, not numbers.
• Connection to Pythagorean theorem recognized in the 18th century.

Distance Between Objects

• For non-point objects, distance is the smallest distance between any pair of points from the objects.
• Formulas exist for specific cases:
  • Distance from a point to a line/plane.
  • Distance between two lines.
• Generalized to abstract metric spaces in advanced mathematics.

Properties of Euclidean Distance

• Symmetric: Distance is the same regardless of the order of points.
• Positive: Distance between distinct points is positive; zero if same point.
• Triangle Inequality: Direct distance is the shortest.

Squared Euclidean Distance

• Often used in applications to simplify calculations by omitting the square root.
• Important in statistics for methods like least squares.
• Not a metric space as it doesn't satisfy the triangle inequality.

Applications and Extensions

• Used in diverse fields like statistics, cluster analysis, and optimization.
• Part of Euclidean geometry and distance geometry.
• Related to Euclidean norm in vector spaces, and remains unchanged under space rotations.
• Euclidean distance matrix stores squared distances for finite point sets.
• Compared with other distances like Chebyshev, Taxicab, and Minkowski distances.

Euclidean Distance in History and Theory

• Euclidean distance is a basic concept in metric spaces and geometry.
• Despite its ancient roots, its mathematical definition evolved with Cartesian coordinates and non-Euclidean geometry.
• Integral in understanding distances in higher-dimensional spaces.