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what is euclidean distance with relation to pixels in image processing (tutorial attached)

Please find the attached pdf on image processing and neighbours of a pixel.

"Each of them are at Euclidean distance of
1.414 from P"

What does this mean? What is Eucildean distance?

thanks in advance
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jtiernan2008
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jtiernan2008
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jtiernan2008Author Commented:
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NovaDenizenCommented:
If one point is at (x1,y1) and another point is at (x2,y2), then the euclidean distance between the points is sqrt((x1 - x2)^2 + (y1 - y2)^2).  Euclidean distance is the same as the normal everyday distance that we are used to.

1.414 is close to sqrt(2).  When one point is at (x,y) and another point is at (x+1, y+1) the distance is ((x+1 - x)^2 + (y+1 - y)^2) = sqrt(1 + 1) = sqrt(2) ~= 1.414.
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jtiernan2008Author Commented:
=> "Euclidean distance is the same as the normal everyday distance that we are used to. "

Yes I see the formulae for the line but why do they bother to call it Euclidean distance. What does this mean?
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NovaDenizenCommented:
Euclid was, of course, a famous old mathematician.  He published the famous work Elements, which included this definition of distance.

It used to be that the geometry taught in Elements was just referred to as Geometry.  But as mathematicians discovered that there can be other types of geometries besides the one that we see in our everyday life, they realized that "geometry" should be a more abstract term, and the geometry described in Euclid's Elements should specifically be called Euclidean Geometry.  Thus, the distance formula from Euclidean Geometry is called euclidean distance.
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