Coordinates: Difference between revisions
(Created page with "A '''coordinate system''' is a system assigning a number (or tuple of numbers) to every point on an entity. The number (or tuple of numbers) assigned to a point is called the '''coordinate''' (or '''coordinates''') of the point. == Examples == Longitude and latitude are a coordinate system for the surface of the earth. If one picks two lines on a plane which meet at right angles, then one has implicitly chosen a Cartesian coordinate...") |
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Longitude and latitude are a coordinate system for the surface of the earth. | Longitude and latitude are a coordinate system for the surface of the earth. | ||
If one picks two lines on a [[Surface#Plane|plane]] which meet at right angles, then one | If one picks two [[oriented]] lines on a [[Surface#Plane|plane]] which meet at right angles, then one can use them to define a coordinate system for the plane, called "cartesian" coordinates. The cartesian coordinates of a point <math>p</math> on the plane are two numbers <math>(x_p, y_p)</math>, determined as follows. First, [[order]] the two lines using the right-hand rule: the [[cross product]] of the first line with second line should point away from you. Project <math>p</math> onto the first line, and let <math>x_p</math> denote the (signed) distance between its projection and the origin. Similarly, project <math>p</math> onto the second line, and let <math>y_p</math> denote the (signed) distance between its projection and the origin. | ||
Using a similar procedure, if one picks a [[ray]] on a plane, one can use it to define a "polar" coordinate system on the plane. | |||
Revision as of 04:25, 14 February 2024
A coordinate system is a system assigning a number (or tuple of numbers) to every point on an entity. The number (or tuple of numbers) assigned to a point is called the coordinate (or coordinates) of the point.
Examples
Longitude and latitude are a coordinate system for the surface of the earth.
If one picks two oriented lines on a plane which meet at right angles, then one can use them to define a coordinate system for the plane, called "cartesian" coordinates. The cartesian coordinates of a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} on the plane are two numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_p, y_p)} , determined as follows. First, order the two lines using the right-hand rule: the cross product of the first line with second line should point away from you. Project Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} onto the first line, and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_p} denote the (signed) distance between its projection and the origin. Similarly, project Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} onto the second line, and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_p} denote the (signed) distance between its projection and the origin.
Using a similar procedure, if one picks a ray on a plane, one can use it to define a "polar" coordinate system on the plane.