Metric Geometry

Question 1

Metric Space

Answer 1

Question 2

Open Ball

Answer 2

Question 3

Convergence in Metric Space

Answer 3

Question 4

Compactness

Answer 4

Question 5

Lipschitz Function

Answer 5

Question 6

Hausdorff Distance

Answer 6

Question 7

Closed Ball

Answer 7

Question 8

Isometry

Answer 8

Question 9

Hausdorff Distance

Answer 9

Question 10

Bounded Set

Answer 10

Question 11

Application of Euclidean Distance

Answer 11

Examples:

Computer Graphics: Calculating the distance between pixels in image processing.
Machine Learning: Measuring similarity between data points.
Robotics: Navigating the shortest path between points.
Geography: Calculating the straight-line distance between two locations on a map.

Question 12

Properties of Euclidean Distance

Answer 12

Question 13

Taxicab (Manhattan) Distance

Answer 13

Question 14

Taxicab Distance in 2D

Answer 14

Question 15

Taxicab Distance in 3D

Answer 15

Question 16

Taxicab Distance in n-Dimensional Space

Answer 16

Question 17

Properties of Taxicab Distance

Answer 17

Question 18

Applications of Taxicab Distance

Answer 18

Examples:

Urban Planning: Used to model travel distances in cities with grid-like street layouts.
Computer Vision: Used in certain algorithms to measure the difference between images.
Machine Learning: Applied in clustering algorithms and as a distance metric in various models.

Question 19

Open and Closed Balls in Taxicab Geometry

Answer 19

Question 20

Comparison with Euclidean Distance

Answer 20