




In two dimensional space, such as the x  y plane, a linear equation determines
a line. That is, the set of (locus of) points obeying it is a straight line.
For example, x  y = 0 describes a forty five degree line passing through the
origin. A curve or path is often described by similarly by one equation, which
related x and y, but is not necessarily linear. A path is a directed curve,
so that at any point it is aimed in some direction. Thus x^{2 }+ y^{2
}= 4 characterizes a circle of radius 2 around the origin. We typically
make this a path by going around it in the counterclockwise direction in the
xy plane.
For the most part we consider simple smooth paths: a smooth path
is one that can be described by a well behaved equation; a simple one
is one that does not intersect itself.
When we use the word path we will usually mean a simple smooth path which starts
at some point a and ends at some point b. When a and b
are the same we call the path closed. Our paths will generally be connected
but not always.
Thus the equation (x^{2} + y^{2 } 1)(x^{2 }+ y^{2
} 4) = 0 defines two concentric circles of radius 1 and 2. In orienting
such a pair of curves into a path it is necessary to be consistent in how we
go around them. We do this by introducing a zipper between them. When
this zipper is closed it is invisible and the path behaves as two separate paths.
When it is open one has to orient the two circles in a manner consistent with
both forming one big cycle; thus if one goes around the outer one counterclockwise,
when one reaches the zipper one goes down it and then must go around the inner
one clockwise and then back up the zipper to form one consistently oriented
path.
We often encounter such curves or paths through equations. We also meet them
described in terms of a parameter. Thus our radius 2 circle path around the
origin can be described by a parameter t as follows:
r = x i+y j; x = 2cos t, y = 2sin t, 0 t 2.