Curves in 3-Space
As previously defined, every point in the
space R 3 is uniquely determined by a position
vector x = (x1 , x2 , x3).
Then in order to introduce the concept of a curve we use a real vector
function: x = x(t) and this will have components:
x1 = x1 (t)
x2 = x2 (t)
x3 = x3 (t)
Which are single-valued functions of real variable t
defined in the interval: I : a <
t < b. To every value of t a point
of R 3 is associated whose position vector is
x(t).
Then the vector function determines a point
set M in R 3 which we call a parametric
representation of the set M, and the variable t is called the parameter of the
representation. Given the above and that the trivial case of M = 1 point is
discarded while the vector function must have multiple derivatives, we may
assume:
1) The functions xi = xi (t) (i=1,2,3.....)
are r (> 1) times continuously differentiable in I where
the value of r will depend on the problem under consideration.
2) For every value of t in I, at least one of the
three functions:
xi (t) = d xi (t)/
dt
is different from zero.
Definition: Arc
of a curve:
A point set in space R 3
which can be represented by the allowable representations of an equivalence
class is called an arc of a curve. The functional correspondence of the
points of an arc to the value of t- given by an allowable representation x =
x(t) is continuous.
If an arc is simple the correspondence between the
points and the values of the parameter t is one to one. In this case not
only is the functional relation of the points to the values of t continuous but
also the inverse relation, i.e. the relation of the values of t to the points.
Definition: curve:
A point set is called a curve if it can be represented by
an equivalence class of the form x = x(t) whose interval I is not assumed to be
closed or bounded, but is such that one always obtains the arc of a curve if
the values of the parameter t are restricted to any closed and bounded
subinterval of I.
A curve is said to be closed if it possesses at least one
representation which is periodic, i.e. of the form:
x(t + w ) = x(t)
Thus the circle: x(t) = (r cos t, r
sin t, 0) with r 2
= 4 is an example of a
simple closed curve.
We have: x1 = r cos t , x 2 = r sin t, x 3 = 0
-> x 1 2 + x 2 2 = r 2 (cos 2 t +
sin 2 t) = 4
So,
the circle is closed and in the x 1 x 2 -plane, e.g.
Special curves: A few special curves and their representations are now
examined. The main proviso here is to point out that there are curves of
the form x = x(t) that
cannot always be represented as a whole in the forms: x 2 = x 2 (x 1), x 3 =
x 3 (x 1).
This difficulty will occur when one value of
the independent variable corresponds to several values of the dependent one,
since the concept of a function requires a 1:1 onto correspondence of dependent
to independent variable. Thus, in the case of the circle, a representation of
the form just highlighted would be:
x 2 = + Ö( r 2
- x 1 22) x 3 = 0
If we choose the + sign, we have a semicircle in the upper
half plane. If we choose the negative sign we have a semicircle in the other
half plane.
Ellipse with center
at origin of coordinate system in space. In the parametric representation
we can write (analogous to that for the circle): x(t) = (a cos t, b sin t, 0)
Or:
x 1 2 / a 2 + x 2 2 / b 2 =
1, x 3 =
0
If the principal axes have lengths 2a and 2b (say with a = 4, b = Ö8), respectively and coincide with the x 1 and x 2 axes, respectively, we obtain the graph shown below:
Note that if a= b = r we just recover a
circle:
x(t) = (r
cos t, r sin t, 0)
The folium of Descartes:
This can be expressed in parametric form as:
x(t) = (3t/
1 + t3 , 3t2/ 1 + t3 ,
0)
This curve will be found to lie in the 1st, 2nd and 4th quadrant of the x 1 - x 2 plane as shown below:
By inspection the reader will also see it has
a double point at (x 1 , x 2) = (0, 0) . Note also that part of the
curve in the 2nd quadrant corresponds to values of t from -1 to 0, while the
loop in the first quadrant corresponds to values between 0 and ¥. In the 4th quadrant the t-values range from - ¥ to -1.
The circular helix:
The parametric form here is:
x(t) = (r cos t, r sin t, ct) c ≠ 0.
The orthogonal projection of the helix into the
x 1 x 2 - plane, e.g.
is the circle:
x 1 2
+ x 2 2
- r 2 = 0, x 3 = 0
which is the intersection of this plane with
the cylinder of revolution on which the helix lies. Projecting the helix
orthogonally into the x2 x 3 - plane we obtain the sine curve:
x2 - r sin (x 3 /c) = 0, x 2 = 0
A cosine curve will be obtained by projecting the helix
orthogonally into the x 1 x 2 - plane.
Suggested Problems:
1) Sketch the graphs of;
x 2 = - Ö( 8 2 - x 1 2)
And:
x 2 = - Ö( 16 2 - x 1 2)
On the same Cartesian axes.
2)(a) Write the polar form of the equation of the line:
3 x 1 + 4 x 2 = 5
b)Determine the polar (r, q) equation for :
x 1 2
+ x 2 2 - 2ax 2 =
0, a ≠ 0
And sketch the resulting curve.
