In this lesson we will deal with different type of coordinate system
their properties and how to transform from one coordinate system to other.
Coordinate system is a method of representing points in a space of given dimensions by coordinates. In general it is represented by "the point with coordinates (x,y)" or "the curve with equation f(x,y)".
The different types of coordinate systems are:
1.>Cartesian coordinates in the Plane
A Cartesian coordinate system is one of the simplest and most useful systems of coordinates.Cartesian coordinates are also known as rectangular coordinates. It consists of two fixed, mutually perpendicular, graduated lines called axis intersecting at a point call Origin.
In it the "address"
or location of any point P in space is represented by two real numbers which are the positions of the perpendicular projections from the point on to axes.
If one coordinate is denoted x and the other y, the axes are called the x-axis and the y-axis. Any point P is represented as P=(x,y), where x and y are the x axis and y axis coordinates. Usually the x-axis is drawn horizontal, with x increasing towards the right, and the y-axis is drawn vertical, with y increasing upwards. The point x=0, y=0 is the origin O, where the axes intersect.
Example in Cartesian coordinates, P=(4,3), Q=(-1.3,2.5), R=(-3.5,-3.5) and S=(3.5,-1). The axes divide the plane into four quadrants: P is in the first quadrant, Q in the second, R in the third, and S in the fourth.
2.>Polar Coordinates in the Plane
A polar coordinate system is constructed in the plane by choosing a point O called the pole and through it a directed straight line, the initial line. A point P is located by specifying the directed distance OP and the angle through which the initial line must be rotated to match in the position and direction of line OP.
Below is shown a diagram representing all angles in a full circle. The concentric circles are measure to the distance r of the coordinate.
Now, P is defined by two numbers: the distance
measured from the center and the angle A the ray OP makes with a fixed ray
originating at O called axis. The angle is generally measured in anticlockwise direction and is only defined up to a multiple of 360° or
Also sometimes for convenience, the condition
allows r to be a signed distance, so (r,A) and (-r, A+180°) represent the same point. The angles in anticlockwise direction is taken as positive and in clockwise direction is taken as negative.
Example among the possible sets of polar coordinates for P are: (10, 30°), (10, 390°) and (10, -330°). Among the sets of polar coordinates for Q are: (2.5, 210°) and (-2.5, 30°).
Relations between Cartesian and Polar Coordinates
Consider a system of polar coordinates and a system of Cartesian coordinates with the same origin. Assume the initial ray of the polar coordinate system coincides with the positive x-axis, and that the ray A=90 degrees coincides with the positive y-axis. Then the polar coordinates (r,A) and the Cartesian coordinates (x,y) of the same point are related as follows:
3.> Oblique Coordinates in the Plane
The following generalization of Cartesian coordinates is sometimes useful. Consider two axes (graduated lines), intersecting at the origin but not necessarily perpendicularly. Let the angle between them be w.
Same as cartesian system in this system of oblique coordinates, a point P is given by two real numbers indicating the positions of the projections from the point to each axis, in the direction of the other axis, just the axis makes an angle w between them. The first axis (x-axis) is generally drawn horizontally. The case when w=90 degrees is a special case which will yield a Cartesian coordinate system.
Example in oblique coordinates, P=(4,3), Q=(-1.3,2.5), R=(-3.5,-3.5), S=(3.5,-1), which we can compare to points in the cartesian coordinate system.
Relations between two oblique coordinate systems
Let the two oblique coordinate systems (
) and (
), with angles
, share the same origin, and suppose the
-axis makes an angle theta with the x-axis. The coordinates (
) and (
of a point in the two systems are related by
This formula also covers passing from a Cartesian system to an oblique system and vice versa, by taking
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