**Motion in Space**

**Copyright © 2001 by James F. Hurley, University of Connecticut Department of Mathematics, Unit 3009, Storrs CT 06269-3009. All rights reserved.**

This worksheet is a continuation of
*ParaCurves*
, and focuses on the motion of a particle along a parametric curve

**x**
(
*t*
) =
*x*
(
*t*
)
**i**
+
*y*
(
*t*
)
**j**
+
*z*
(
*t*
)
**k**
= (
*x*
(
*t*
),
*y*
(
*t*
),
*z*
(
*t*
)).

in space. Consider, for instance, the first curve in
*ParaCurves*
, the circular helix

**x**
(
*t*
) = (
).

Its veclocity and acceleration vectors are

**v**
(
*t*
) = (
) and
**a**
(
*t*
) = (
).

Since for any value of
*t*
the speed of a particle moving along the curve is

||
**v**
(
*t*
) || =
=
,

the formula for the unit tangent vector at any time
*t*
is

**T**
(
*t*
) =
(
).

To find the unit normal vector
**N**
(
*t*
), first differentiate
**T**
(
*t*
), and then divide by its length, which here is simply the constant
. The calculation therefore yields

[
**T**
(
*t*
)] =
(
), so
** N**
(
*t*
) =
**i**
**j**
.

The following Maple routine plots the helix, along with its unit tangent and normal vectors at the point corresponding to
*t*
=
/2. It draws the unit tangent vector by plotting the segment of the tangent line

** x**
=
**x**
(
*t*
) =
**x**
(
/2) +
*t*
**T**
(
/2) = (0, 1,
/2) +
*t*
(
)

at the point
*P*
(0, 1,
/2) in the direction of the unit tangent vector
**T**
= (
) for
*t*
ranging from 0 to 1. (
*P*
corresponds to
*t*
= 0, and
*t*
= 1 gives the end of
**T**
(
/2) drawn from initial point
*P*
.) It uses the same approach for
**N**
(
/2): starting from
*P*
it plots the normal line

**x**
=
**x**
(
*t*
) =
**x**
(
/2) +
*t *
**N**
(
/2) = (0, 1,
/2) +
*t*
(
).

for
*t*
between 0 and 1. (Again, the unit normal vector ends at the point corresponding to t = 1.)

`> `
**with (plots):
curve := spacecurve( [cos(t), sin(t), t], t = 0..2*Pi, color = black, axes=boxed, numpoints = 150, labels = ["x", "y", "z"] ):
utan := spacecurve( [-t/sqrt(2), 1, Pi/2 + t/sqrt(2)], t = 0..1, color = blue, axes = boxed, numpoints = 150 ):
unorm := spacecurve( [0, 1 - t, Pi/2], t = 0..1, color = green, axes = boxed, numpoints = 150 ):
xaxis := spacecurve( [t, 0, 0, t = -1..1, color = magenta] ):
yaxis := spacecurve( [0, t, 0, t = -1..1, color = magenta] ):
zaxis := spacecurve( [0, 0, t, t = -1..6, color = magenta] ):
labx := textplot3d([1.2, 0, -.2, `x`], color = magenta):
laby := textplot3d([0,1.2, -.2, `y`], color = magenta):
labz := textplot3d([0, 0, 6.5, `z`], color = magenta):
display(curve, utan, unorm, xaxis, yaxis, zaxis, labx, laby, labz);**

**The Moving Trihedral.**
The
* binormal vector*
has formula

**B**
(
*t*
) =
(
)

=
(1, 0, 1) for
*t*
=
/2.

The next routine uses the technique of the preceding one to generate a plot that includes
**B**
(
/2). It thus gives shows the moving trihedral at the point
**x**
(
/2) = (0, 1,
/2).

`> `
**with (plots):
curve := spacecurve( [cos(t), sin(t), t], t = 0..Pi, color = black, axes=boxed, numpoints = 150, labels = ["x", "y", "z"] ):
utan := spacecurve( [-t/sqrt(2), 1, Pi/2 + t/sqrt(2)], t = 0..1, color = blue, axes = boxed, numpoints = 150 ):
unorm := spacecurve( [0, 1 - t, Pi/2], t = 0..1, color = green, axe
**