Difference between revisions of "Coordinate system"
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The coordinate system is often used in geometry. The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers, (x,y). The cordinate system is organized to 4 quadrants. In the first quadrant, both (x,y) are positive. In the second quadrant x is negative, while y is positive.In the third quadrant, both (x,y) are negative. Finally, in the fourth quadrant x is positive while y is negative. To find the slope of a line on the coordinate system with points <math>(x_1, y_1); (x_2, y_2)</math> we need to compute <math>\frac{y_2-y_1}{x_2-y_2}</math>. The slope is usually expressed as <math>m</math>. Also, there is the point-slope form which states <math>y=m \cdot x + b</math> for some real numbers b,x,y. Also, if two lines are perpendicular, then product of the slopes is -1. (For example, the slope could be -3/4 and 4/3.) The coordinate system is also widely useful were right triangle, since there is 90 degrees. | The coordinate system is often used in geometry. The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers, (x,y). The cordinate system is organized to 4 quadrants. In the first quadrant, both (x,y) are positive. In the second quadrant x is negative, while y is positive.In the third quadrant, both (x,y) are negative. Finally, in the fourth quadrant x is positive while y is negative. To find the slope of a line on the coordinate system with points <math>(x_1, y_1); (x_2, y_2)</math> we need to compute <math>\frac{y_2-y_1}{x_2-y_2}</math>. The slope is usually expressed as <math>m</math>. Also, there is the point-slope form which states <math>y=m \cdot x + b</math> for some real numbers b,x,y. Also, if two lines are perpendicular, then product of the slopes is -1. (For example, the slope could be -3/4 and 4/3.) The coordinate system is also widely useful were right triangle, since there is 90 degrees. | ||
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| + | Related: | ||
| + | Cartesian coordinate system | ||
| + | A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. | ||
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| + | One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. | ||
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| + | Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius. | ||
| + | The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4. | ||
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| + | Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing | ||
Revision as of 00:23, 25 January 2016
The coordinate system is often used in geometry. The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers, (x,y). The cordinate system is organized to 4 quadrants. In the first quadrant, both (x,y) are positive. In the second quadrant x is negative, while y is positive.In the third quadrant, both (x,y) are negative. Finally, in the fourth quadrant x is positive while y is negative. To find the slope of a line on the coordinate system with points 
we need to compute 
. The slope is usually expressed as 
. Also, there is the point-slope form which states 
for some real numbers b,x,y. Also, if two lines are perpendicular, then product of the slopes is -1. (For example, the slope could be -3/4 and 4/3.) The coordinate system is also widely useful were right triangle, since there is 90 degrees.
Related: Cartesian coordinate system A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y − b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing
