Euclidean distance

1. Course:
Data Visualization
Instructor: Mr. Abdul Baqi Malik
Lecture No.08 AI-SP-23
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2. ❑Euclidean distance is a measure of the straight-line distance between two points in Euclidean
space. It is the most common and familiar distance metric, often referred to as the "ordinary"
distance.
❑Euclidean Distance gives the distance between any two points in an n-dimensional plane. Euclidean distance
between two points in the Euclidean space is defined as the length of the line segment joining the two
points.
❑Euclidean distance is like measuring the straightest and shortest path between two points.
❑It tells how far apart the two points are without any turns or bends, just like a bird would fly directly from
one spot to another. This metric is based on the Pythagorean theorem and is widely utilized in various fields
such as machine learning, data analysis, computer vision
Euclidean distance
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3. ❑Euclidean Distance Formula
❑Consider two points (x1, y1) and (x2, y2) in a 2-dimensional space; the Euclidean Distance between them is
given by using the formula:
Euclidean distance
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4. ❑Euclidean Distance in 3D
❑If the two points (x1, y1, z1) and (x2, y2, z2) are in a 3-dimensional space, the Euclidean Distance between
them is given by using the formula:
Euclidean distance
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5. ❑Euclidean Distance in nD
❑In general, the Euclidean Distance formula between two points (x11, x12, x13, ...., x1n) and (x21, x22, x23, ...., x2n)
in an n-dimensional space is given by the formula:
• i Ranges from 1 to n
• d is Euclidean distance
• (x11, x12, x13, ...., x1n) is Coordinate of First Point
• (x21, x22, x23, ...., x2n) is Coordinate of Second Point
Euclidean distance
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6. ❑ Euclidean Distance Formula is derived by following the steps added below:
❑ Step 1: Let us consider two points, A (x1, y1) and B (x2, y2), and d is the distance between the two points.
❑ Step 2: Join the points using a straight line (AB).
❑ Step 3: Now, let us construct a right-angled triangle whose hypotenuse is AB, as shown in the figure below.
Euclidean Distance Formula Derivation
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7. Example
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8. Example
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9. Example
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10. Example
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11. Practice Questions
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12. ❑Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is
ignored, and the distance between two points is instead defined to be the sum of the absolute
differences of their respective Cartesian coordinates, a distance function (or metric) called
the taxicab distance, Manhattan distance, or city block distance.
Manhattan geometry
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13. ❑Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is
ignored, and the distance between two points is instead defined to be the sum of the absolute
differences of their respective Cartesian coordinates, a distance function (or metric) called
the taxicab distance, Manhattan distance, or city block distance.
Manhattan geometry
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14. Manhattan geometry
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15. Manhattan geometry
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16. Manhattan geometry
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17. Euclidean Distance and Manhattan Distance
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18. ❑The Jaccard index is a statistic used for gauging the similarity and diversity of sample sets. It is defined in
general taking the ratio of two sizes (areas or volumes), the intersection size divided by the union size, also
called intersection over union (IoU).
Jaccard similarity
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19. Jaccard similarity
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20. Jaccard similarity
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21. Jaccard similarity
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22. Jaccard similarity
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23. Jaccard similarity
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