Euclidean Distance Metric

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A Euclidean Distance Metric is a Minkowski distance metric with [math]d=2[/math] (that represents our intuitive notion of the intuitive notion of distance on a plane).



References

2012

  • http://en.wikipedia.org/wiki/Euclidean_distance#Definition
    • QUOTE: The Euclidean distance between points p and q is the length of the line segment connecting them ([math]\overline{\mathbf{p}\mathbf{q}}[/math]).

      In Cartesian coordinates, if p = (p1, p2,..., pn) and q = (q1, q2,..., qn) are two points in Euclidean n-space, then the distance from p to q, or from q to p is given by: :[math]\mathrm{d}(\mathbf{p},\mathbf{q}) = \mathrm{d}(\mathbf{q},\mathbf{p}) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2 + \cdots + (q_n-p_n)^2} = \sqrt{\sum_{i=1}^n (q_i-p_i)^2}.[/math]

      The position of a point in a Euclidean n-space is an Euclidean vector. So, p and q are Euclidean vectors, starting from the origin of the space, and their tips indicate two points. The Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector: :[math]\|\mathbf{p}\| = \sqrt{p_1^2+p_2^2+\cdots +p_n^2} = \sqrt{\mathbf{p}\cdot\mathbf{p}}[/math] where the last equation involves the dot product.

2011

  • (Wikipedia, 2011) http://en.wikipedia.org/wiki/L2_norm#Euclidean_norm
    • On Rn, the intuitive notion of length of the vector x = [x1, x2, ..., xn] is captured by the formula [math]\|\boldsymbol{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}.[/math] This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. The Euclidean norm is by far the most commonly used norm on Rn, but there are other norms on this vector space as will be shown below. However all these norms are equivalent in the sense that they all define the same topology. ...

2009

  • (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Euclidean_distance
    • In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space (even a Hilbert space). The associated norm is called the Euclidean norm. Older literature refers to this metric as Pythagorean metric. The technique has been rediscovered numerous times throughout history, as it is a logical extension of the Pythagorean theorem.

2008