cartesian coordinate system
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Figure 1- Red circle with radius of 2 and center at the origin of rectangular coordinate system. The equation of this circle is. Mathematically, Cartesian coordinate system, also known as rectangular coordinate system, is an orthogonal coordinate system. With reference to figure 1, the two-dimensional rectangular coordinate system consists of two mutually perpendicular number axes, with 0 points coincident. In the plane, the coordinates of any point are set according to the coordinates of the corresponding point on the number axis. In a plane, the correspondence between any point and coordinates is similar to the correspondence between points and coordinates on the number axis.
Using rectangular coordinates, geometric shapes can be clearly expressed by algebraic formulas. The rectangular coordinates of each point of a geometric shape must obey this algebraic formula. For example, a circle with a radius of 2 and a center at the origin of a rectangular coordinate system. A circle can be expressed as.
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1 history
Two-dimensional coordinate system
Three-dimensional coordinate system
Four directions
4. 1 two-dimensional space
4.2 Three-dimensional space
5 vector
6 See also
7 references
8 reference catalogue
9 External connection
[Edit] History
The French mathematician Descartes founded the Cartesian coordinate system. 1637, Descartes published his masterpiece Methodology. This book is devoted to the study and discussion of western academic methods, providing many correct opinions and good suggestions, and making great contributions to the future development of western academics. In order to show the advantages and effects of the new method and help him carry out scientific research, he added another book, Geometry, to the appendix of Methodology. The study of Cartesian coordinate system appears in the book Geometry. Descartes' research on coordinate system combines algebra and Euclidean geometry, which has a key guiding force for later achievements in analytic geometry, calculus and cartography.
Two-dimensional coordinate system
Figure 2- Cartesian coordinate system. The coordinates of the four points in the picture are: green point:, red point:, blue point: and purple point:.
Figure 3- Four quadrants of rectangular coordinate system, counterclockwise from one quadrant to the other. The head of the coordinate axis symbolizes the infinite extension of the pointing. Referring to fig. 2, a two-dimensional rectangular coordinate system is usually set by two mutually perpendicular coordinate axes, which are usually called X axis and Y axis respectively; The intersection of two coordinate axes, called the Origin, is usually marked with O, which means "zero" and is also the first letter of the English word "origin". Each axis points in a specific direction. The coordinate axes of these two different lines define a plane, which is called xy plane, also called Cartesian plane. Usually, as long as the two axes are perpendicular to each other, it doesn't matter where to analyze the problem, but habitually (see the right picture), the X axis is placed horizontally, called the horizontal axis, which usually points to the right; The y axis is placed vertically, called the longitudinal axis, and usually points upward. This positional relationship between the two coordinate axes is called two-dimensional right-handed coordinate system, or right-handed coordinate system. If you draw this right hand tie on a piece of transparent paper, no matter how you rotate in the plane, what you get is called right hand tie; But if you turn the paper over, the coordinate system you see on the back is called "left-handed system". This is related to the nature of left-right alignment when looking in the mirror.
In order to know the distance between any point on the coordinate axis and the origin. Suppose that we can plot numerical values on the coordinate axis. Then, from the origin to the direction indicated by the coordinate axis, the numerical value is plotted on the coordinate axis every other unit length. The value is the number of descriptions and the positive integer distance from the origin; Similarly, we can also describe the negative integer distance from the origin, facing away from the direction pointed by the coordinate axis. The value depicted on the X axis is called X coordinate, also called abscissa, and the value depicted on the Y axis is called Y coordinate, also called ordinate. Although, here, these two coordinates are integers, corresponding to specific points of the coordinate axis. In proportion, we can generalize to every point of real coordinates and its corresponding coordinate axis. These two coordinates are rectangular coordinates of rectangular coordinate system, marked as.
The position of any point p on the plane can be uniquely represented by rectangular coordinates. As long as a straight line perpendicular to the X axis is drawn from point P, the X coordinate of point P can be obtained from the intersection of this straight line and the X axis. Similarly, you can find the Y coordinate of point P, so we can get the rectangular coordinate of point P, for example, refer to Figure 3, the rectangular coordinate of point P is.
Cartesian coordinate system can also be extended to three-dimensional space and higher dimensions.
Referring to Figure 3, the two coordinate axes of rectangular coordinate system divide the plane into four parts, called quadrants, which are numbered by Roman numerals respectively. By convention, the two coordinates of the quadrant are positive; The x coordinate of the quadrant is negative and the y coordinate is positive; Both coordinates of the quadrant are negative; The x coordinate of the quadrant is positive and the y coordinate is negative. Therefore, the number of quadrants is compiled counterclockwise from one quadrant to another.
[Edit] 3D coordinate system
Figure 4- Several coordinate planes of rectangular coordinate system. Red plane. Yellow plane. Blue plane. The z axis is vertical and is represented by white. The x-axis is displayed in green. The three coordinate planes intersect at point P (represented by a black ball), and the rectangular coordinates are about.
Figure 5-3D Cartesian coordinate system. The direction of the y axis is far away from the reader.
Figure 6-3D Cartesian coordinate system. The direction of the x axis is close to the reader. In the original two-dimensional rectangular coordinate system, a coordinate axis perpendicular to the X axis and the Y axis is added, which is called the Z axis. If these three coordinate axes satisfy the right-hand rule, you can get a three-dimensional rectangular coordinate system. The z axis, x axis and y axis are orthogonal to the origin. The position of any point p in three-dimensional space can be expressed by rectangular coordinates. For example, referring to fig. 5, the rectangular coordinates of two points P and Q are sum, respectively.
Xy- plane, yz- plane and xz- plane divide the three-dimensional space into eight parts, which are called octants. Unlike the four quadrants in a two-dimensional space, only one hexagon is limited to a number. The three coordinates of each point in the first divination limit are positive.
direction
Main items: direction
[Edit] Two-dimensional space
The x-axis and y-axis of rectangular coordinate system must be perpendicular to each other. The straight line containing the y axis is called the y line. In two-dimensional space, when we set the position and direction of the X axis, we also set the direction of the Y line. However, we still have to choose which of the two semi-straight lines of Y line whose origin is the same point is positive and which is negative. Both options determine the direction of the xy plane.
See figure 1. Usually, the direction we choose is that the X axis of positive value points horizontally to the right and the Y axis of positive value points vertically upward. This orientation is called forward orientation, standard orientation or right-handed orientation.
The right hand rule is a common memory method, which is specially used to identify the positive direction: put a half-clenched right hand on a plane, with the thumb up, and then all the other fingers point from the X axis to the Y axis.
The other orientation, which adopts the left-handed rule, is specially used to identify negative orientation or left-handed orientation: put a half-clenched left hand on the xy plane with the thumb up, and then all other fingers point from the Y axis to the X axis.
No matter what the direction of the coordinate axis is, when the coordinate system rotates at any angle, the direction remains the same.
[Edit] Three-dimensional space
Figure 7- Right hand rule.
Figure 8- Left hand direction, right hand direction. The x, y and z axes of a rectangular coordinate system must be perpendicular to each other. A straight line containing the z axis is called a z line. In three-dimensional space, when we set the position and direction of X axis and Y axis, we also set the direction of Z line. However, we still have to choose, on the Z line with the same point as the origin of * * *, which half line's point coordinates are positive and which one is negative? These two different coordinate systems are called right-handed coordinate system and left-handed coordinate system. Right-handed coordinate system is also called standard coordinate system or positive coordinate system.
The term right-handed coordinate system comes from the right-handed rule. First straighten the palm of your right hand with your fingers. Then, the middle finger yearns for the palm half space of the palm, at right angles to the index finger. Then thumb up, at right angles to middle finger and forefinger. Then the thumb, index finger and middle finger represent the X-axis, Y-axis and Z-axis of the right-hand coordinate system respectively. Similarly, the left-handed coordinate system can also be expressed by the left hand.
Fig. 8 attempts to show a left-handed coordinate system and a right-handed coordinate system. Because we use two-dimensional pictures to represent three-dimensional objects, the graphics will be distorted or blurred. Pointing to the lower right axis also means pointing to the reader; And the middle axis also means pointing in the direction the reader sees. The red circular curved arrow parallel to the xy plane (whose red arrow passes in front of the Z axis) indicates the direction of rotation from the X axis to the Y axis.
[edit] vector
Using rectangular coordinate system, any point P can be represented by a vector in three-dimensional space. We can imagine a vector as an arrow, the tail of the arrow is at the origin, and the front of the arrow is at point P. If the vector of point P is 0, then the rectangular coordinate is 0. So,
;
Among them, the unit vector and the positive infinite direction point to the X axis, the Y axis and the Z axis, respectively.
[Edit] See
oblong
hyperbola
parabola
generalized coordinates
canonical coordinate
Parallel coordinates
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orthogonal coordinate system
Two-dimensional coordinate system Cartesian coordinate system Parabolic coordinate system Bipolar coordinate system Biangular coordinate system Bicentric coordinate system Hyperbolic coordinate system Elliptic coordinate system
Three-dimensional coordinate system Cartesian coordinate system, cylindrical coordinate system, spherical coordinate system, three-dimensional parabolic coordinate system, parabolic coordinate system, oblate spherical coordinate system, long spherical coordinate system, ellipsoid coordinate system, elliptic cylindrical coordinate system, circular coordinate system, double spherical coordinate system, bipolar cylindrical coordinate system, conical coordinate system and flat ring cylindrical coordinate system. Nates flat disk ring coordinates double ring coordinates Cap ring coordinates
reference system
[edit] reference
Descartes, Rene. Paul Oskamp (translated). Lectures on methods, optics, geometry and meteorology. 200 1.
[Edit] Reference Table of Contents
Prime Minister Morse, Feshbach H( 1953). Theoretical physical methods, part I. New york: McGraw-Hill, p. 656. ISBN 0-07-0433 16-X。
Murphy general motors (1956). Mathematics of physics and chemistry. New york: D. van Nostrand, p. 177.
Korn GA，Korn TM( 196 1)。 Handbook of mathematics for scientists and engineers. New york: mcgraw-hill companies, pp. 55-79. ASIN B0000CKZX7。
Shore River, Shoboi (1967). Engineers' mathematical association. New york: springer Publishing House, p. 94.
Moon p, Spencer (1988). "Cartesian coordinates (x, y, z)", handbook of field theory, including coordinate system, differential equations and their solutions, revised 2nd edition. , 3rd edition. , new york: springer Publishing House, page 9–11(table 1.0 1). ISBN 978-0387 184302。
[Edit] External connection