Do Parametric Circles Draw Clockwise

We talk over the nuts of parametric curves.

The idea of parametric equations

Consider an ant itch along a flat surface like a floor of a edifice. Suppose we want to draw the ant'southward position and the path it takes as it moves. We could introduce an origin as well as a fix of and axes on the floor. Nosotros could approximate the ant equally a point and so for each moment in time we could describe the ant's position by some pair of and coordinates. Every bit time passes, the particular path followed by the pismire would describe a curve in the plane.

See the interactive below: Apply the slider for to trace out the path of the pismire.

Suppose we want to depict this curve. Since the pismire'southward path could be complicated with lots of meandering and backtracking, it is unlikely nosotros would be able to depict equally a function of or every bit a office of . We still might be able to detect an equation in and that describes the curve as just a fix of points in the airplane. Nosotros phone call such an equation a Cartesian description of the curve since it gives the curve in terms of Cartesian coordinates and . The human relationship between the two coordinates could exist very complicated.

Having an equation involving and would give us a static view of the curve. We see the path of the ant, all at once, like the chemical trail left behind by the pismire. Although useful, in that location are many natural questions that we would not exist able to answer using this characterization of the curve. When was the pismire at a given betoken ? How fast was the ant traveling at that point? Did the emmet terminate at any indicate in time? But having the trail left by ant leaves out a great deal of useful information.

For each moment in fourth dimension, we take some unique associated indicate in the aeroplane respective to the emmet's position at that given fourth dimension. We could draw the position of the ant as a pair of functions that depend on :

These are chosen parametric equations for the curve . In these equations we call the parameter. In many examples, the parameter will be time only the parameter can have other interpretations also. At present given specific functions for and we could answer questions about the location of the ant at a given fourth dimension or nosotros could decide the velocity of the ant at a given moment. This gives the states a dynamic view of the curve in that it emphasizes how the curve is traced out every bit the parameter changes. In fact, we can retrieve of a parametric curve as a new kind of role where is the input and the output is a point in the airplane. We won't stress this interpretation at the moment just later in your studies you will render to this notion every bit your primary manner of thinking of curves given parametrically.

Visualizing parametric equations

Suppose nosotros are are given a parametric equation algebraically, such equally How can we visualize the curve traced out in -airplane?

Think back to when yous first learned how to graph a bend like . I'm pretty certain you used a so-called "T-chart," and if , I bet information technology looked something similar this:

Eventually yous did this enough times that y'all simply learned the basic geometric shape of such a parabola. Similarly you lot have memorized the bones shapes of the graph for many other functions.

Suppose now we want to graph a curve given parametrically.

With a parametric plot, both and are now functions of a third parameter, nosotros'll call it , often thought of every bit fourth dimension. In the same way, we tin can make a chart. Here is the input and and are the outputs of the 2 different functions and . Graphing these points, we will go:

It'due south non obvious that should take been a line. If instead we had used the parameter , we could trace out the same bend as

Generally, when both and are lines, the curve will ever exist a line. Tin can you lot find the Cartesian equation of this line?

If y'all solve for in the equation, yous tin can replace with an equation in terms of in the equation.

Generally graphing parametric curves are more difficult since the and values are both changing as varies and it can be hard to meet how and are related to ane some other. At that place are some curves other than lines that you should exist able to visualize.

Consider the following parametric equations: every bit varies over the interval .

Although the parameter is , it tin can be helpful here to call back of it as an angle. The bending is initially and you increment the angle from to . You should then recognize that these equations but give the and coordinate of a point on the unit circle. Nosotros can imagine these parametric equations equally "drawing" the unit of measurement circle as changes

Brand a table showing how the circle is being plotted as runs from to :

Is the circle "drawn" in a clockwise or counterclockwise mode?

clockwise counterclockwise

Once the parameter has reached , the circle has been traced out exactly one time.

The management in which the curve is traced out every bit the parameter increases is called the positive orientation.

In the case of the circle above, what is the positive orientation?

clockwise counterclockwise

Using parametric equations allows united states to describe many curves which would be likewise difficult to describe otherwise.

Consider the following parametric equations where .

Every bit the parameter increases, the curve rotates like it will trace out the circle. Still instead of having a stock-still amplitude that describes how far the circle is from the origin, the factor in front end of the cosine and sine term can be idea of as a varying radius that increases equally increases. Thus as the parameter increases, the curve rotates in a counterclockwise mode like the circle but with a radius that keeps getting larger and larger. This leads to the following spiral shape.

Do the parametric equations

as runs from to ascertain a office of ?

No, because the graph does non pass the vertical line test. Yeah, it is a function of , because for each input , there is exactly one output value, an ordered pair.

For the graph of a circle, is not a office of . Withal, it can be a part of that maps

where the domain is and the elements of the range consist of ordered pairs.

Projectile Motion

Parametric descriptions of curves occur in many applications. For example, in physics, a common scenario is to accept a object acted upon past forces. Nosotros use information near the initial state of the system likewise as the laws of motion to solve for how the object will behave. This description volition naturally be equations which describe how the position of the object, in a coordinate system, volition change with time. Therefore, parametric descriptions of curves are very natural way to describe the move of objects. Although nosotros will focus on describing curves parametrically in the aeroplane, the ideas we comprehend tin can be easily generalized to three dimensions and higher.

In differential calculus, you lot likely discussed projectile motion in ane dimension. For case, suppose you launch a ball straight upwards into the air. Yous then desire to find the position of the object with respect to time. Using integration and the fact that the ball has a constant acceleration with respect to gravity, nosotros can find the trajectory of the brawl.

We cull the -centrality so that the origin is at footing level and the positive direction on the -axis is upwards.

At present the acceleration due to gravity is constant and we take . Call the initial velocity (this is the intial velocity in direction). So we can integrate and apply the intial velocity to find the velocity

Compute with initial condition .

Suppose it is launched from a height . Integrating once more and using the intial position, we obtain

Now suppose nosotros want to launch the brawl at an angle instead of straight up. The brawl'south path volition accept both a vertical and horizontal component, and then this is at present a 2 dimensional trouble. Nosotros launch our ball with a sure initial speed . Nosotros once more choose a coordinate arrangement. Choose the origin at a fixed point on the ground level, choose the -axis oriented upwards, and choose our -axis to be the direction in which we launch our projectile. Ignoring air resistance, the trajectory of our object will be confined to this -plane. Suppose our object is launched from an initial position in our coordinate organization. Can nosotros describe the move of the ball?

It turns out the motion of the object can be analyzed in the and directions separately. We can draw the position of the ball every bit

The bending that we launch our ball determines an the initial velocity in each components. Take to be the initial velocity along the management and to be the initial velocity forth the direction. In the direction, our ball will behave just similar a brawl thrown directly upward. Using our previous work on ane dimensional projectile motility, we obtain

Now nosotros want to understand the behavior in the direction. There are no forces in the management so the acceleration is . Using the initial velocity in the direction and integrating the dispatch , we find . Integrating over again and using the initial position , we obtain :

This gives us the parametric description of the trajectory of the ball in the airplane with respect to fourth dimension.

The curve that the projectile follows is parabolic. However the parametric description gives united states more information. Information technology gives us a dynamic description that tells us the position of the particle at each moment of time.

Explore changing the initial velocities and in the and directions and see how that changes the trajectory of the ball by use the slider to trace the curve.

Multiple parametrizations of a given curve

Given a curve , there are many different sets of parametric equations that trace out .

Previously we saw that we can parametrize the circle as

as varies over the interval , traces out the circle in a counterclockwise fashion.

Suppose we supersede by in these equations.

where runs over the interval . Let's brand a table.

Now the positive orientation of the curve is clockwise. Notice that if nosotros replace by any expression in , we will trace out (a portion of) the aforementioned curve, but mayhap at a unlike speed and/or in a different direction.

Which of the post-obit parametric equations describe the line ?

and for and for and for and for and for and for

Which of the following parametric equations draw the circumvolve ?

and for and for and for and for and for and for

When is information technology piece of cake to detect a parametrization of a curve?

Suppose you lot are given a curve and yous want to detect a parametrization for . There are certain dainty curves where we can hands come up up with an explicit parametrization.

Graphs of functions

If yous are given the graph of a function then it is like shooting fish in a barrel to notice a parametric description of the bend. The key idea is that the independent variable is already acting like a parameter so nosotros can utilise

In this way, nosotros can easily parametrize any curve given as the graph of a function.

Can you use the technique described immediately to a higher place to limited as a parametric office?

Lines

Suppose nosotros are given 2 points and . Suppose we are interested in parametrizing the line segment from to . There is a approved style to parameterize the line segment then that the parameter is at and is equal to at . In a previous example, we have seen that if both and are lines then the curve traced out is a line. Thus it suffices to construct a line so that and . Similarly we construct a line and so that and . This leads to the equations

where we let vary over the interval .

If nosotros allow vary over the entire real line in this case, then we become the unique line that passes through the points and .

Utilize this method to discover a parametric equation for a line segment that starts at the indicate and ends at the signal .

Let for in the interval .

Circles

The standard form for a circle centered at a point with radius is given by One problem with the standard form for a circumvolve is that it is somewhat difficult to notice points on the circle. A parametric equation representing a circumvolve solves this problem.

Give a parametric equation representing the circle and explain why your answer is correct.

This is the circle of radius centered at the point . Here we set

as runs from to . To see that our answer is right, we can "plug" information technology back into the implicit equation for the circumvolve. Write with me:

by the Pythagorean identity. Since our functions satisfy the class of the circle, our solution is correct.

One day while trying to graph a unit circle, you accidentally write down

What happens now? Do you lot still go a circle? How is this different from what we did in the previous question?

you withal plot a unit of measurement circle in a counterclockwise style, with the aforementioned starting and ending points you plot a unit circle but in a clockwise fashion, with the same starting and ending points you still plot a unit circumvolve in a counterclockwise manner, but the starting and catastrophe points are different you plot a unit circle but in a clockwise fashion, but the starting and ending points are unlike this no longer plots a circle

In mathematics, when parameterizing closed curves (like circles), the convention is to draw them in a "counterclockwise" direction. This is called the positive orientation.

If you parameterize your closed curves in a clockwise direction, you may find your "answers" are off by a gene of .

What should and exist to parameterize the circle in a counterclockwise style, with respective to ?

Converting from parametric to cartesian

We mentioned earlier that graphing a curve from a given set of parametric equations direct from can be quite difficult. Unremarkably computational software is needed. However in that location are some occasions where we can eliminate the parameter and obtain an equation involving but and .

Here are some bones strategies to try:

• Solve for .
• Solve for a part of .
• Utilize a trigonometric identity.

In each case the process that we are using is chosen elimination of a parameter.

We'll give several examples of how one actually eliminates a parameter.

Solving for the variable

In the first example, we'll solve for .

Let

Eliminate a parameter to express this curve purely in terms of and .

Here nosotros will solve for . Since it is easier, nosotros will solve for in this equation:

Now plug this into , and write

We run into in this situation that the set up of points traced out the curve (equally we vary over all possible values of ) is a parabola.

Solving for a mutual function

Sometimes instead of solving for the parameter directly, it is easier to solve for a function of the parameter that is mutual to both and .

Permit

Eliminate a parameter to express this curve purely in terms of and .

Here we will solve for a function of . Write

Now nosotros can rewrite every bit

Now nosotros encounter that

Solving for related functions

Let

Eliminate a parameter to express this curve purely in terms of and .

There is no common function of in the 2 equations. Furthermore solving for directly is possible merely there is a easier path frontwards. Notice that if we rearrange the equations slightly, so we tin make use of a trig identity to eliminate the expressions involving . We use the Pythagorean identity: So commencement isolate cosine and sine, and foursquare the equations. With we take

and with we have

Plugging this back into the Pythagorean identity, we see:

These techniques are useful to know since they tin can aid you more easily visualize certain elementary parametric curves. All the same, when yous eliminate the parameter it is of import to realize that you are losing information virtually how the curve is traced out.

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Source: https://ximera.osu.edu/csccmathematics/calculus2/parametricEquations/digInParametricEquations

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