The curve does change in a small but important way which we will be discussing shortly. From a quick glance at the values in this table it would look like the curve, in this case, is moving in a clockwise direction. But is that correct? We can check our first impression by doing the derivative work to get the correct direction. The only way for this to happen is if the curve is in fact tracing out in a counter-clockwise direction initially.
Note that this is only true for parametric equations in the form that we have here. This is directly counter to our guess from the tables of values above and so we can see that, in this case, the table would probably have led us to the wrong direction.
So, once again, tables are generally not very reliable for getting pretty much any real information about a parametric curve other than a few points that must be on the curve. Outside of that the tables are rarely useful and will generally not be dealt with in further examples. So, why did our table give an incorrect impression about the direction? Doing this gives the following equation and solution,.
So, what is this telling us? In fact, this curve is tracing out three separate times. This is why the table gives the wrong impression. The speed of the tracing has increased leading to an incorrect impression from the points in the table. So, we saw in the last two examples two sets of parametric equations that in some way gave the same graph.
Yet, because they traced out the graph a different number of times we really do need to think of them as different parametric curves at least in some manner. In some of the later sections we are going to need a curve that is traced out exactly once. We can eliminate the parameter much as we did in the previous two examples. We will need to be very, very careful however in sketching this parametric curve.
We will NOT get the whole parabola. So, it is clear from this that we will only get a portion of the parabola that is defined by the algebraic equation. Below is a quick sketch of the portion of the parabola that the parametric curve will cover. To finish the sketch of the parametric curve we also need the direction of motion for the curve.
This is not the only range that will trace out the curve however. Any of them would be acceptable answers for this problem. All travel must be done on the path sketched out. This means that we had to go back up the path. The only way for that to happen on this particular this curve will be for the curve to be traced out in both directions.
Contrast this with the ellipse in Example 4. However, the curve only traced out in one direction, not in both directions. So, to finish this problem out, below is a sketch of the parametric curve. Note that we put direction arrows in both directions to clearly indicate that it would be traced out in both directions.
It is more than possible to have a set of parametric equations which will continuously trace out just a portion of the curve. We will often use parametric equations to describe the path of an object or particle. Completely describe the path of this particle. Eliminating the parameter this time will be a little different. This gives,. This time the algebraic equation is a parabola that opens upward. So, again we only trace out a portion of the curve. Here is a quick sketch of the portion of the parabola that the parametric curve will cover.
Here is that work. Here are a few of them. We should give a small warning at this point. Because of the ideas involved in them we concentrated on parametric curves that retraced portions of the curve more than once. Do not, however, get too locked into the idea that this will always happen. Many, if not most parametric curves will only trace out once. The first one we looked at is a good example of this. That parametric curve will never repeat any portion of itself. There is one final topic to be discussed in this section before moving on.
However, there are times in which we want to go the other way. Parametric Equations A rectangular equation, or an equation in rectangular form is an equation composed of variables like x and y which can be graphed on a regular Cartesian plane.
Solution: Assign any one of the variable equal to t. Example 2: Eliminate the parameter and find the corresponding rectangular equation. Subjects Near Me. The negative sign is needed to reverse the orientation of the curve. If the negative sign were not there, we would have to imagine the wheel rotating counterclockwise. Adding these equations together gives the equations for the cycloid.
In this graph, the green circle is traveling around the blue circle in a counterclockwise direction. A point on the edge of the green circle traces out the red graph, which is called a hypocycloid.
These equations are a bit more complicated, but the derivation is somewhat similar to the equations for the cycloid. This ratio can lead to some very interesting graphs, depending on whether or not the ratio is rational.
The result is a hypocycloid with four cusps. In these cases the hypocycloids have an infinite number of cusps, so they never return to their starting point.
These are examples of what are known as space-filling curves. Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However, perhaps the strangest name for a curve is the witch of Agnesi.
Why a witch? Maria Gaetana Agnesi — was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in , which included an interesting curve that had been studied by Fermat in Let O denote the origin.
Choose any other point A on the circle, and draw the secant line OA. The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines.
In probability theory, the curve describes the probability density function of the Cauchy distribution. In this project you will parameterize these curves. Earlier in this section, we looked at the parametric equations for a cycloid, which is the path a point on the edge of a wheel traces as the wheel rolls along a straight path.
In this project we look at two different variations of the cycloid, called the curtate and prolate cycloids. Recall that we considered a tenacious ant trying to get home by hanging onto the edge of a bicycle tire. We have assumed the ant climbed onto the tire at the very edge, where the tire touches the ground. As we have discussed, we have a lot of flexibility when parameterizing a curve. In this case we let our parameter t represent the angle the tire has rotated through.
After a while the ant is getting dizzy from going round and round on the edge of the tire. So he climbs up one of the spokes toward the center of the wheel. By climbing toward the center of the wheel, the ant has changed his path of motion. As shown in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant. As before, we let t represent the angle the tire has rotated through.
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