Download - Parametric Curves: The Basics
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Nick Herbert
Parametric Equations
Overview of Parametric Equations
Parametric equations are used to graph the position of a point in space on a
2-dimensional plane. The reason this is different than a function is because
functions can only have one y value for everyxvalue on the graph, but if one is
trying to graph the position of a thing like a fly, they must be able to show the fly
even if it backtracks. The way that parametric equations are able to graph multiple
points for eachxvalue is by actually having two functions, one that graphs the
vertical position of the point in space, and one that graphs the horizontal position of
the point in space, each in terms of time.
The graph of parametric equations is actually not a continuous line, but a
series of points, possibly an infinite series of coordinates. These coordinates are
decided by the output of thexfunction (horizontal position) and the output of they
function (vertical position) for a given t(t
is often time): [x(t),y(t)]. Figure 1 is a
simple example of a parametric equation.
Notice on the parametric circle in
Figure 1 that when anxvalue is selected,
there are multiple potentialyvalues.
Parametric equations cant simply be talked about in terms ofxlike a function can,
Figure 1
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for one value ofxcould have multiple values ofy. Instead they are both discussed in
terms of a constant like time (t), something that cannot have two values. Although a
fly may be in the same spot of the room at two different times, it cannot be in two
different spots at the same time.
Below are the most general parametric equations for a point moving around
a circle.
( )
( )
tis the variable, time ris the radius of the circle (only when rxand ryare the same: it is a
perfect circle)
kis the time it takes the point to make one full rotation is the radians counterclockwise from the right of the circle that the
point is starting at
a and b are the coordinates for the center of the circle: (a,b)o a and b can be equations if the center of the circle is moving
Figure 2 on the next page explains this in the context of a simple set of
parametric equations.
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( )
( )
Figure 2
Applying Parametric Equations to More Complex Movement The movement graphed in Figure 2 could display a piece of gum stuck to the
tire of a bike as the tire spins freely, however it does not show the movement of the
gum if a biker jumped on the bike and started riding down the road. This would add
another aspect of motion to the gum; it is going around the bike tire as well as down
the street. If the bike is moving down the street at 3/5 feet per second, the center
of the circle that the gum is rotating around isnt a fixed constant. Thexvalue
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(horizontal location of the center) is moving down the street at 3/5 feet per
second, while theyvalue (vertical location of the center) stays constant since the
road is flat. Figure 3 displays what the gums movement would look like in this
scenario by leaving a +1 in the x parameter as well as adding the speed, the start
position is kept the same as in Figure 2.
( )
Figure 3
By looking at the parametric equations that made this curve one can see that
the speed of the biker riding down the road was incorporated in a. If the biker also
had a vertical motion component because the road was uphill, his vertical speed
would be incorporated in b.
This handles objects that travel along flat surfaces at constant speeds, but
parametric equations only start getting interesting when the movement is a little
more complex. Parametric curves can show the movement of a point on a small
circle that is rolling around the outside of a bigger circle.
To do this, first some values must be supplied to describe the objects and
their exact movements. The bigger circle will have a radius of 5 and will be centered
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at the origin,
(0,0). The smaller
circle will have a
radius of 2,
complete one full
rotation every 5
seconds, and will
initially be
centered at (7,0).
These two circles
are shown in Figure 4 along with the path (red) created by the center of the smaller
circle (blue) as it rolls around the larger circle (black).
To determine the parametric equations for a point starting on the right side
of the smaller circle the first step is to look at the movement of the point in relation
to the small circle, and the movement of the center of the small circle as the circle
rolls. Using the general parametric equations for a point on a circle some initial
values can easily be plugged in for both of these.
( )
( )
Point in Relation To The Smaller Circle
Figure 4
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r=2 because the center is at (7,0) and the left side touches the larger circleat (5,0): 7-5=2
k=5 because the circle rotates once every 5 seconds a=7 and b=0 because (a,b) is the center and the center is at (7,0)
This allows one to create set of
parametric equations for this
circle, which is shown in Figure 5:
(
)
( )
Center of Smaller Circle in Relation to The Bigger Circle
r=7 because the center of the smaller circle is at (7,0) and it rotatesaround the center of the bigger circle at (0,0): 7-0=7
k=12.5 because the smaller circle takes 5 seconds to rotate once. Itscircumference is 4 (2r, r=2), meaning it travels 4 around the bigger
circle in 5 seconds. The bigger circle has a circumference of 10 (2r,
r=5), which is 2.5x greater than 4, meaning the circle takes 2.5x longer
to go around the big circle than to rotate once: 2.5*5=12.5
a=0 and b=0 because the center is the origin, (0,0) =0 because the point is starting on the far right of the circle
Figure 5
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With this information another set of parametric equations can be created to
create Figure 6.
+0
+0
Combining the Movements
The first correlation that must be understood is that the second set of
equations (that create Figure 6) forms the axis of oscillation for the point on the
smaller circle. Referring back to Figure 4 may aid in understanding this the point
on the smaller circle rotates around the center of the smaller circle, and the second
set of equations explain the movement of that center.
To incorporate this into the equations for the smaller circle, a and b will be
effected. Looking back to Figure 3 when the biker began moving forward, a included
an equation with t, specifically an equation that modeled the movement of the
center of the bike tire. In this situation the movement is modeled in the same way,
Figure 6
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only its not linear. a and b in the equations for the point on the smaller circle
become the parametric equations for the center of the smaller circle:
( ) ( )
( ) ( )
These equations
create the graph shown in
Figure 7. On first glance
this seems like a possible
plot of the movement of
the point, however with a
little critical analysis it
becomes obvious this
cannot be correct.
The point is on the far side of the small circle when it starts, which means it
will do a half rotation before the point touches the bigger circle. The smaller circle
has a circumference of 4, which means it will roll 2 distance around the bigger
circle when the point first touches the bigger circle. The bigger circle has a
circumference of 10, which means the point should touch the bigger circle before
its rolled way around. This graph implies that the point will touch the larger circle
after its passed the mark.
Figure 7
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The reason for this error is because the speed of the point rotating around
the smaller circle changes when the terrain the circle is rolling on changes when
the smaller circle is set rolling around the bigger circle, the point actually rotates
more than radians in a half turn (remembering that a full circle has 2 radians).
This confusing fact is displayed in Figure 8.
To find how much faster the point is moving one can calculate the extra
radians it has rotated in the same 2.5 seconds. When the point touches the bigger
circle, it has gone 2 distance around the 10 circumference of the bigger circle. 2
is 1/5 of 10, meaning that at 2.5 seconds the point is 1/5 around the full 2 radians
or 2/5 radians. Referring to Figure 8, this angle is shown by the blue curves. On the
Figure 8
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smaller circle, the point has gone radians, or half the way around, and another
2/5 radians. By adding these angles together one can find the total radians the
point has rotated around the smaller circle in 2.5 seconds: +2/5=7/5. The set of
equations we currently have say the point rotated 2 radians in 2.5 seconds, when
in reality it rotated 7/5 in 2.5 seconds. Below are the new equations and their
respective graph, Figure 9, with this alteration to account for the increased speed
caused by the terrain of the bigger circle.
( ) ( )
( ) ( )
The red line is
the curve of the point.
Figure 7 failed the
verification because
the point wasnt
shown touching the
circle at the correct
point, however in
Figure 9 the first
touch comes just
before the mark, suggesting that this is in deed the correct set of parametric
equations for this situation.
But what if the smaller circle where inside the bigger circle?
Figure 9
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Putting The Circle on The Inside
If we wanted to do the same
thing we just did, but instead have the
smaller circle rolling around the
inside of the bigger circle, we would
just have to change a couple things
(new situation shown in Figure 10).
The set of equations that graphed the
position of the center of the smaller
circle would no longer have a radius
of 7 (5+2), but would have a radius of 3 (5-2). The graph of the movement of the
center of the smaller circle is shown in Figure 10 in red. The adjusted set of
equations for the center of the smaller circle is below.
Now we have the a and b of our new set of parametric equations, we just
need to adjust the speed of the point because the circle is on the inside. This
adjustment is shown and explained in Figure 11 on the next page.
Figure 10
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To find the number of radians the point has moved we must subtract 2/5
from , which equals 3/5. However the direction of the rotation has also changed
and the smaller circle is now rotating clockwise. This can be integrated into the
speed component of the equations by simply placing a (-) before the 3. Now we can
create our new set of parametric equations and graph as shown in Figure 12.
( ) ( )
( ) ( )
Figure 11
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This graph displays two full rounds of the smaller circle, which completes the
shape. Figure 9 of the circle rolling on the outside does not appear complete because
the circle is shown only doing one round. The reason it takes two round for the
shape to complete is because it takes the small circle 5 seconds for every full
rotation, and 12.5 seconds for every round of the bigger circle: 12.5/5=2.5, which
means the smaller circle will have rotated 2.5x in one round of the bigger circle.
That half rotation puts the point on the opposite side of the smaller circle, but after
two rounds around the bigger circle, the smaller circle will have made an even 5
rotations, putting the point back at the starting position.
Figure 12
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On Figure 12, the point starts out touching the bigger circle. This means it
must do one full rotation around the small circle before it touches the bigger circle
again. The circumference of the smaller circle is 4, which means that the point will
touch the big circle 4 distance along its perimeter. The circumference of the big
circle is 10, and 4 is 2/5 of 10, meaning that the point should touch the bigger
circle after its made it the way around and before its gone around. Figure 12
(Figure 14 below demonstrates this with more clarity) shows that the first touch
after the starting point - comes in that top left quadrant of the graph, or between
and the way around the bigger circle. This verifies the set of parametric equations
used to display this situation.
Additional Graphs
Figure 13 below is a graph of two rounds of the smaller circle from Figure 9.
Figure 14 below is a graph of a one round of the smaller circle from Figure 12.
Figure 13Figure 14