(To
control animations: place the cursor over the graphic, then use 'right
click' on a PC or 'control+click' on a MAC to select/deselect 'play'
and
'loop'.)
A Space Curve is precisely
that - a curve in 3-space. Following the same idea as in the case of
curves given parametrically in the plane, such a curve is usually
thought of as a function $t
\to (x(t),\, y(t), \, z(t))$ where $x(t),\, y(t),$
and $z(t)$
are functions of the parameter $t$ or
whatever parameter is used. Now that we've got used to
vectors, we shall write this as a vector function
When $t$
is time, the graph of r
could be the path of something travelling in 3-space like a
car on the ShockWave roller-coaster at Six Flags or even
someone climbing a spiral
staircase as in one of the banners for Quest; on the other hand, if r$(t)$ simply
represents static position, then the graph could arise as an
intersection
of surfaces, for
instance. Typical examples of these are
Recognize the building? They show the ways a space curve can viewed: as
a free-standing curve in 3-space, as a curve on a surface, and as the
intersection of two surfaces.
From a multi-variable calculus point of
view, a very important example is the vector function determined by a
cross-section of the graph of $z
= f(x,\,y)$ with a plane, say $y = Y$. Since
the graph of $f$
consists of all points $(x,\,
y,\, f(x,\,y))$ in 3-space as $x,\, y$ vary,
the intersection of this graph with $y = Y$
consists of all points $(x,\,Y,\,
f(x,\,Y)$, meaning that we've
fixed $y$
at $y = Y$
and only let $x$
vary. So the cross-section can now be thought
of
as the graph of the vector function
As
an example, consider first a cubic function of two variables:
$z
= f(x,\,y) = 3x - x^3 - y^2.$
Because the function contains an $x^3$
term, it's graph certainly won't be a quadric surface; in fact, we
haven't met it before. So how did we decide that the surface to the
right is it's graph? Well, we could have tried taking slices as was
done with quadric surfaces! Instead, a computer drawing program was
used
to create it. It's interesting to note that the program puts on a
surface mesh - basically, curves on the surface - to
provide greater spatial understanding. Soon we'll use contour
maps to help with the calculus for the same reason. But let's draw
our own curves on the
surface as graphs of particular vector functions. For instance, is it
clear where there is a cubic graph on the surface?
Here's one cubic graph obtained from a vector function. The
surface mesh and axes have been removed for
clarity.
If a cross-section is taken with the plane $y = 0$, then
the intersection of the graph of $f$
and this plane is the set of all points
$(x,\, 0,\,
f(x,\,0)) = (x,\, 0,\, 3x - x^3)$
as $x$ varies. Thus
the corresponding vector function is
It's graph is a cubic passing through the origin,
decreasing when $
x
\to -1$ from the left and decreasing again when
$ x
\to \infty$ as shown in the animation to the
right. (Clicking doesn't control this
animation.)
Suppose now that we take the
cross-section with the plane $x
= 0$. What
would be the corresponding vector function? Is it clear that it's graph
would be a parabola opening downwards on the surface?
An example
of a space curve that will occur several times is
On a stand-alone basis
showing only axes its graph is
which
gives some idea of its properties. What would probably help much more
help would be to see how
this space curve sits on a particular surface, say the graph
of $g(x, \, y,\,
z) = 0$. Can you imagine what such a surface might be - a
saddle, perhaps? Recall the trig indentities
Then
the space curve will lie on the graph
of $g(x, \, y,\,
z) = 0$ when each point $(x(t), \, y(t),\, z(t)$
lies on the graph of $g
= 0$. This occurs when $g (x(t),\,y(t),\,z(t) ) =
0$, which in familiar notation can be
written as the composition $g($r$(t)) = 0$.
Ah, now I see, the trig identities tell us that
