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Titlu referat: Coordonate carteziene

Nivel referat: liceu

Descriere referat:
Coordonate carteziene
A coordinate system is a method by which a set
of numbers is used to locate the position of a point. The numbers are called
the point's coordinates. In a coordinate system, a single point corresponds to
each set of coordinates. Coordinate systems are used in analytic geometry to
study properties of geometric objects with algebraic techniques.
When an object having a finite number of
degrees of freedom is considered among all the objects of that kind, the object
in question can be conveniently characterized and distinguished from the other
objects by a set of coordinatesÑthat is, a set of numbers, one for each degree
of freedom. For example, a point in a plane has two degrees of freedom, so that
the point has two coordinates with respect to any coordinate system of the
plane.
There are many different coordinate systems.
Usually the geometry and symmetry of a problem will suggest an appropriate
coordinate system. Common coordinate systems are Cartesian (after RenŽ
Descartes) coordinates and polar coordinates in two -dimensional space and
Cartesian, spherical, and cylindrical coordinates in three-dimensional
space.
Coordinate Systems in Two
Dimensions
Through an arbitrary point O in the plane, two
mutually perpendicular lines, usually horizontal and vertical, are drawn. The
x-axis is taken to be horizontal, the y-axis is vertical, and point O is called
the origin. The portion of the x-axis to the right of the origin is the
positive x-axis, and the part of the y-axis above the origin is called the
positive y-axis. The two axes (coordinate axes) divide the plane into four
quadrants: the upper right (first), the upper left (second), the lower left
(third), and the lower right (fourth). The x-coordinate, or abscissa, of a
point P in the plane is the
perpendicular distance of P
from the y-axis. It is positive if P is to the right of the y-axis, negative if P is to the left, and zero if P is on the y-axis. The y-coordinate, or
ordinate, of P is analogously
the perpendicular distance of P from the x- axis. It is, respectively, positive, negative, or zero
if P is above, below, or on
the x-axis. The ordered pair (x, y) represents the coordinates of P in the coordinate system thus defined.
The point P with coordinates
(x, y) is symbolically
represented as P (x, y). This
system is called a two- dimensional, or plane, Cartesian coordinate
system.
A polar coordinate system in two dimensions is
a system determined by a fixed point O, called the pole, and an axis through
it, called the initial line. A point P in the plane can then be fixed by specifying two quantities: (1)
the angle < through which
the axis must be rotated in the counterclockwise direction so as to pass
through P, and (2) the
positive distance r of the
point P from the pole. The
notation P (r, <) is used to
represent P in polar
coordinates r and
<.
If P (x,
y) is the Cartesian representation of P, and P (r,
<) is the polar
coordinate representation of the same point, and if the origin and x-axis of
the Cartesian system coincide, respectively, with the pole and the initial line
of the polar coordinate system, then the two systems are related by x = r cos
<, y = r sin <, and r = the square root of
x6 + y6, tan < = y/x. These are the equations of
transformation from one system to another.
Coordinate Systems in Three
Dimensions
Three mutually perpendicular lines (the
coordinate axes) are drawn through an arbitrary point O, the origin, in space.
The axes are called the x-axis, y-axis, and z-axis. The plane containing the
x-axis and the y-axis is the xy-plane (a coordinate plane) and the z-axis is a
normal (a line that is perpendicular) to this plane. The other two coordinate
planes are defined likewise. The x-coordinate of a point P is the perpendicular distance of
P from the yz-plane. The
y-coordinate and the z-coordinate are defined similarly. The three coordinate
planes divide all space into octants. If P is a point in the first octant, all the coordinates of
P are positive.
The system thus described is a
three-dimensional Cartesian system, and P (x, y,
z) is the Cartesian representation of P with coordinates x, y, z with respect
to a fixed frame of reference. For every point there corresponds uniquely a set
of three real numbers, and vice versa.
The spherical coordinate system in space is a
system that locates a point P
by its distance from a fixed point O (the pole), and by two angles that
describe the orientation of the segment OP. The coordinate system is fixed by
two perpendicular half-lines through O. One of these is the polar axis. The
plane that contains the two half-lines is called the initial meridian plane.
The spherical coordinates of P are (r, <,
ñ), where r is the length
of OP, < is the angle from the initial
meridian plane to the plane through the polar axis and OP, and ñ is the angle from the polar axis to
OP. The spherical system is
usually aligned with a Cartesian system in which the pole is the origin. The
polar axis coincides with the z-axis, and the initial meridian plane with the
xz-plane. The equations x = r sin ñ cos <, y = r
sin <, and z = r cos
ñ express the relation
between the two systems.
A coordinate system consisting of a plane with
polar coordinates and a z-axis through the pole, or origin, perpendicular to
the plane is called a cylindrical polar coordinate system. A point P is located in space by its distance
z from the plane and the
polar coordinates (r, <) of the foot of the perpendicular
from P to the plane. The two
systems are related by the equations x = r cos <, y = sin <, and z = z.
V. K. Balakrishnan
Bibliography: Fuller, Gordon,
Analytic Geometry (1993);
Leithold, Louis, Before Calculus, 2d ed. (1990).
i
iCopyright (c) 1997 Grolier Interactive Inc.



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