Parametric Curves: The Basics

7/31/2019 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

Parametric Curves: The Basics

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