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 ) = ( cos(t), sin(t), t ).


Its veclocity and acceleration vectors are

v ( t ) = ( -sin(t), cos(t), 1 ) and a ( t ) = ( -cos(t), -sin(t), 0 ).


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


||
v ( t ) || = sqrt(sin^2*t+cos^2*t+1) = sqrt(2) ,

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

T ( t ) = 1/sqrt(2) ( -sin(t), cos(t), 1 ).


To find the unit normal vector N ( t ), first differentiate T ( t ), and then divide by its length, which here is simply the constant 1/sqrt(2) . The calculation therefore yields

d/dt [ T ( t )] = 1/sqrt(2) ( -cos(t), -sin(t), 0 ), so N ( t ) = -cos(t) i -sin(t) j .


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

x = x ( t ) = x ( Pi /2) + t T ( Pi /2) = (0, 1, Pi /2) + t ( -1/sqrt(2), 0, 1/sqrt(2) )

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


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


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);

[Maple Plot]


The Moving Trihedral. The binormal vector has formula B ( t ) = T ( t ) N ( t ). For each value of t , the three vectors T ( t ), N ( t ) and B ( t ) define a local right-handed coordinate system with base point ( x ( t ), y ( t ), z ( t )). This leads to the term moving trihedral for the array of these three vectors emanating from the point x (t). For the circular helix x ( t ) = ( cos(t), sin(t), t ) , the binormal vector is

B ( t ) = 1/sqrt(2) ( sin(t), -cos(t), 1 )

=
1/sqrt(2) (1, 0, 1) for t = Pi /2.


The next routine uses the technique of the preceding one to generate a plot that includes
B ( Pi /2). It thus gives shows the moving trihedral at the point x ( Pi /2) = (0, 1, Pi /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 BookFinder.com: Search for New, Out of Print and Used Books

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[Maple Plot]

Tangental and Normal Components of Acceleration. To decompose a (1) into its tangential and normal components at the point P (1, 1, 1) corresponding to t = 1, just use the foregoing calculaations:

a[T] (1) = [ (a v)/(v v)] v = [22/14] (1, 2, 3) = (11/7, 22/7, 33/7).

Then (with pointed brackets enclosing the coordinates of vectors to accommodate Maple's syntactical requirements in mathematical expressions),


a[N] (1) = a(1)-a[T] (1) = <0,2, 6>-<11/7,22/7, 33/7> = <-11/7,-8/7, 9/7> .

From this, it follows that

N (1) = [ 1/|| a[N] (1) || ] a[N] (1) = 1/sqrt(266) <-11,-8, 9> .

The next routine plots the accleration and its tangential and normal components at t = 1.

> with (plots):
curve := spacecurve( [t, t^2, t^3], t = 0..2, color = black, axes=boxed, numpoints = 150, labels = ["x", "y", "z"] ):
accel := spacecurve( [1, 1 + 2*t, 1 + 6*t], t = 0..1, color = green, axes = boxed, numpoints = 150 ):
asubt := spacecurve( [1 + 11*t/7, 1 + 22*t/7, 1 + 33*t/7], t = 0..1, color = blue, axes = boxed):
asubn := spacecurve( [1 - 11*t/7, 1 - 8*t/7, 1 + 9*t/7], t = 0..1, color = blue, axes = boxed ):
parall1 := spacecurve( [18/7 - 11*t/7, 29/7- 8*t/7, 40/7 + 9*t/7], t = 0..1, color = cyan, axes = boxed ):
parall2 := spacecurve( [-4/7 + 11*t/7, -1/7 + 22*t/7, 16/7 + 33*t/7], t = 0..1, color = cyan, axes = boxed ):
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, accel, asubt, asubn, parall1, parall2, xaxis, yaxis, zaxis, labx, laby, labz);

[Maple Plot]

>


Calculation of Curvature from the Normal Component of Acceleration. The decomposition of a into its tangential and normal components provides a quick and often convenient way to calculate the curvature K . This avoids ugly differentiating or computation of a cross product. Above the normal component of acceleration is

(ds/dt)^2 K (1) N (1) = 14 K (1) 1/sqrt(266) <-11,-8, 9> = 1/7 <-11,-8, 9> ,


It follows immediately then that

14 K (1) 1/sqrt(266) = 1/7 , and so K (1) = sqrt(266)/98 = sqrt(38)/(14*sqrt(7)) .


As you can check, that agrees with the result of using the formula from Theorem 3.10 in Section 2.3.