Distance Formula - Revision history

Khanh at 18:38, 14 November 2021

2021-11-14T18:38:59Z

Lila at 20:57, 18 October 2021

2021-10-18T20:57:55Z

Lila at 20:56, 18 October 2021

2021-10-18T20:56:56Z

Khanh: Created page with "* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length], OpenStax"

2021-09-26T02:20:16Z

Created page with "* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length], OpenStax"

New page

* [https://openstax.org/books/calculus-volume-3/pages/3-3-arc-length-and-curvature Arc Length], OpenStax

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 <math display=block>d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2}.</math>
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Euclidean_distance Euclidean distance], Wikipedia
+Content obtained and/or adapted from:

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 The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates. Thus if <math>p</math> and <math>q</math> are two points on the real line, then the distance between them is given by:
 <math display=block>d(p,q) = |p-q|.</math>
-A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:<ref name=smith />
+A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:
 <math display=block>d(p,q) = \sqrt{(p-q)^2}.</math>
-In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.<ref name=smith />
+In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.
 === Two dimensions ===
@@ Line 18: / Line 18: @@
 This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from <math>p</math> to <math>q</math> as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.
-It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of <math>p</math> are <math>(r,\theta)</math> and the polar coordinates of <math>q</math> are <math>(s,\psi)</math>, then their distance is<ref name=cohen /> given by the law of cosines:
+It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of <math>p</math> are <math>(r,\theta)</math> and the polar coordinates of <math>q</math> are <math>(s,\psi)</math>, then their distance is given by the law of cosines:
 <math display=block>d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}.</math>
@@ Line 31: / Line 31: @@
 <math display=block>d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2}.</math>

← Older revision		Revision as of 20:56, 18 October 2021
Line 1:		Line 1:
−	* [~~https~~://~~openstax~~.~~org~~/~~books~~/~~calculus~~-~~volume~~-3/~~pages~~/3-3-~~arc~~-~~length~~-~~and~~-~~curvature Arc Length~~], ~~OpenStax~~	+	[[File:Euclidean distance 2d.svg\|thumb\|upright=1.35\|Using the Pythagorean theorem to compute two-dimensional Euclidean distance]]
		+	The '''Euclidean distance''' between two points in Euclidean space is the length of a line segment between the two points.
		+	It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the '''Pythagorean distance'''. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
		+
		+	The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.
		+
		+	== Distance formulas ==
		+	=== One dimension ===
		+	The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates. Thus if <math>p</math> and <math>q</math> are two points on the real line, then the distance between them is given by:
		+	<math display=block>d(p,q) = \|p-q\|.</math>
		+	A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:<ref name=smith />
		+	<math display=block>d(p,q) = \sqrt{(p-q)^2}.</math>
		+	In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.<ref name=smith />
		+
		+	=== Two dimensions ===
		+	In the Euclidean plane, let point <math>p</math> have Cartesian coordinates <math>(p_1,p_2)</math> and let point <math>q</math> have coordinates <math>(q_1,q_2)</math>. Then the distance between <math>p</math> and <math>q</math> is given by:
		+	<math display=block>d(p,q) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2}.</math>
		+	This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from <math>p</math> to <math>q</math> as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.
		+
		+	It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of <math>p</math> are <math>(r,\theta)</math> and the polar coordinates of <math>q</math> are <math>(s,\psi)</math>, then their distance is<ref name=cohen /> given by the law of cosines:
		+	<math display=block>d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}.</math>
		+
		+	When <math>p</math> and <math>q</math> are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used:
		+	<math display=block>d(p,q)=\|p-q\|.</math>
		+
		+	=== Higher dimensions ===
		+	[[File:Euclidean distance 3d 2 cropped.png\|thumb\|upright=1.2\|Deriving the <math>n</math>-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem]]
		+	In three dimensions, for points given by their Cartesian coordinates, the distance is
		+	<math display=block>d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}.</math>
		+	In general, for points given by Cartesian coordinates in <math>n</math>-dimensional Euclidean space, the distance is
		+
		+	<math display=block>d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_i - q_i)^2+\cdots+(p_n - q_n)^2}.</math>