A Cartesian coordinate is an ordered tuple of numbers (represented by (x,y,z)
in 3 dimensions) which describes the distance from the origin to the point
measured along each axis. The values can be positive or negative. The axes
of the graph are perpendicular, and intersect at one point (the origin).
This system can be generalized to represent
points in n dimensions. A point in n dimensions is represented by an n-tuple
(or vector) (x1,x2,...,xn). A graph in n dimensions consists of n axes, all
of which are perpendicular to each other and intersect at the origin.
These are also sometimes called rectangular coordinates.
What are polar coordinates?
A polar coordinate (in 2 dimensions) is an ordered pair (r, theta) which are defined as follows.
For a point, P, r is the distance from the origin to P. Theta
is the angle between the x-axis and the line segment from P to
the origin. Notice that there is more than one valid value for theta. For
example, if the angle between the x-axis and the line segment is 30 degrees,
valid values of theta are (in degrees): 30, (30+360) = 390, (390+360) = 750,
etc.
The use of polar (instead of Cartesian) coordinates can simplify some calculations.
What is the origin?
The origin is the point (0,0,...,0) at which all axes of the coordinate system
intersect.
What are the principle axes?
Any line can be used as an axis for performing transformations.
Howver,
the principle axes are those which define the coordinate system. For
example, in 2 dimensions, the lines x = 0 and y = 0 are the principle axes.
How do I convert between Cartesian and polar coordinates?
To convert from polar (r, theta) to Cartesian (x,y):
To convert from Cartesian to polar:
These relationships are derived from results in trigonometry.
What is a quadrant?
What is an octant?
A two dimentional grid can be divided into four areas called quadrants.
They are defined by the signs of the points found in each area.
For a point (a,b):
