PreCalc*G Spirals
By Damian Ferraro and Dan Curran
Table of Contents:
Understanding the Equations:
- r = n(Θ-c)
- r = n/Θ-c
- r = n^(Θ-c)
Imagining Hypothetical Situations
- As c gets infinitely larger, what happens to the graph?
Conversion to the Cartesian Form
Conclusion
Understanding the Equations:
r = n(Θ-c)
Where n=1 and c=0
r = n/(Θ-c)
Where n=1 and c=0
r=n^(Θ-c)
Where n=1 and c=0
r = n(Θ-c)
The equation r = n(Θ-c) where n is 1 and c=0 results in a spiral. The variable n is a scalar in this case, since it is multiplying the Θ in the parentheses, making the spiral bigger or smaller. The x-intercepts also move as a result of the n value moving. As n gets farther from zero, the spiral gets bigger (meaning that -n also increases the size of the spiral). When n hits 0, the graph disappears because then r=0.
The c value is an angle that being subtracted from Θ and as it gets more negative, the graph also gets bigger. The c value also acts as a rotator to the spiral, changing the points on the graph as if it were a circle.
Finally, the spiral itself is limited to 0 to 2π, making Θ's values act as if they are rotating in a circular motion while r gets infintely bigger.
c=5, n=1
c=1, n=3
n=2, c=0
This graph has an asymptote at y=2
c = 1.6, n = 1
The line in QIII intersects with the original spiral.
r = n/(Θ-c)
Similar to the polar equation above, (Θ-c) have the same purpose but now n is dividing this value instead of multiplying it. Yet again, as n gets farther away from zero, the graph increases in size and when the n value equals 0, the graph disappears. When c = 0, the n value acts as an asymptote as the graph goes across the x-axis. Also, negative n values create the same graph but it was reflected about the y-axis.
As for the c value, it acts as an angle again, and it rotates the spiral around in a circular motion. When c is 0, the graph acts as a regular spiral seen in the top image to the left, but when c increases from 0, another line that is staggered from the spiral joins in on the graph. If n=1, this new line intercepts at the original spiral at c = 1.6. (See image 2).
Yet again, Θ is restricted to 0 to 2π, creating the circular motion similar to the first equation, while r gets infintely bigger.
c=10, n=1: The line from QIII and the original spiral join in a loop.
r=n^(Θ-c)
Unlike the other two polar equations, this one only shows up when n > 0. This is due to the fact that a negative n squared does not work for this equation. Also, when n = 1 and c = All real numbers, the graph represents a circle with a radius of 1. This is because 1 with an expontent always equals 1. Once n is greater than 1, the graph increases is size and n becomes a scalar again.
C also acts as a rotater of the graph but when c gets farther from 0 in the negative direction, it horizontally moves the graph across the x-axis. When c gets infintiely more positive, the graph continues to rotate in the same way the other equations rotated.
When n is between 0 and 1, the c value acts of a scalar while also rotating the graph at the same time. The c value in this case acts as an unwinder of the spiral itself. This is also true once n > 1 and c acs as a scalar while n rotates the graph. In this case, the c value and n value rotate the graph and increase the size of the graph at the same time. The n value creates a way bigger increase from one value to the next, while c creates a smaller dilation from one value to the next.
n = 1.5, c=10
n = 1 c = All real numbers
n = 0.5, c = 5
As c gets farther away from 0 in the positive direction, the graph continues to rotate and gets larger in size. The n value is smaller and it gets bigger as this value gets closer to 1, and eventually infintely bigger.
Imagining Hypothetical Situations:
As c gets larger, what happens to the graph?
r = n(Θ-c)
In r = n(Θ-c), the c value gets infintely larger and once it gets to a point where it is very big, eventually the spiral turns into a thick ring. When c = 1000, the spiral itself comes closer together while the n value still acts as a scalar. In the image show, the c value of 1000 acts as a circle, but the outline itself is very thick and the x-intercepts also hit at -1000 and 1000. C, in this case, almost acts as a radius for the circle. In this image, n = 1, but if n were to get farther away from zero, it would still act as a scalar. Although this shape does look like a ring, it is actually a spiral that is just very close together as c increases.
r = n/(Θ-c)
In r = n / (Θ-c), the same effect to the graph happens when c = 1000, but in this case, the graph gets smaller and smaller as c increases. N = 1 and it still acts as a scalar in the same way, but c acts as a scalar in the other direction. As c gets closer to 0, the spiral gets bigger but as c hits 1000 and farther on, the graph gets smaller. In the image to the right, c = 1000 and n =1 and the x = intercepts are not 1000, but -0.001 and 0.001 representing the thousandths digit. This is interesting because it is the opposite in the r = n(Θ-c) equation. The reason for the change in x-intercepts is through the function of the variable n. N acts as a coefficient in the equation above, but in this equation, it acts as a divisor and because of that, the intercept is 1/1000 instead of 1000.
r = n^(Θ-c)
Yet again, the same effect happens to the n^(Θ-c), but there is no ring this time. When n = 1 in this equation, it creates a circle with a center of (0,0) and a radius of one. Any value for c does not change the graph in this case, so in this image, n = 1.3 instead of 1. Another thing to point out in this equation is that the c value makes the graph very smaller as it goes from one value to the next increasing value. In this case, c = 150 and n =1.3. The same spiral in the representation of this function above is shown, but as c increases, the graph gets expontentially smaller. The c value acts as a scalar yet again in this example, and so does the n value. The only difference is that the c value makes the graph smaller as if increases while the n value still makes the graph bigger as it increases.
Conversion to the Cartesian Form:
Cartesian Form of r = n(Θ-c)
The r = n/(Θ-c) as a Cartesian form is represnted by the graph to the right. It is seen almost as two teardrops which are reflected across the y-axis. The c value still acts as a rotator but when c gets farther from 0, the graph rotates clockwise and then the two tear drops turn into two arcs that at first are apart but slowly start to get closer together to eventually form a circle. The math for finding the Cartesian Form of the equation is shown below:
n=1, c=0
n = 1 , c = 0
n = 1, c = 1000
Cartesian Form of r = n/(Θ-c)
The r = n/(Θ-c) as a Cartesian form is represnted by the graph to the right. It is seen almost as two rays of sunlight which are reflected across the x-axis. The c value still acts as a rotator but when c is negative, the graph rotates clockwise until about -1.5. Once the c value gets more negative, the graph slowly comes together and the two arcs eventually form a circle. The n value still acts as a scalar and n cannot equal 0. The math for finding the Cartesian Form of the equation is shown below:
Cartesian Form of r = n^(Θ-c)
The r = n^(Θ-c) as a Cartesian form is represnted by the graph to the right. It is seen as a circle when n = 1, and c = 0 but when n increases, it creates a shape almost similar to a double edged sword. The c value still acts as a rotator and once a full rotation has occured, the arcs on the graph slowly seperate from each other. In this case, the n value acts similar to a scalar, but does not actually dilation the image in the same way as it was first graphed. The math for finding the Cartesian Form of the equation is shown below:
n = 1, c = 0
n = 10, c = 1000
Conclusion
The three polar equations of the spiral are very interesting because they all seem quite similar, but one thing is altered to make the equations different. One equation is a simple distributive representation while there is another that is almost the same, but the n acts as a divisor instead of a coefficient. Then, there is also a third equation that is exponential. Each of these unique equations all have different things that make them distinctive, especially in the way that their values change. The c values in each equation have some similarities but also some very distinguishing differences. The n value, on the other hand, seems to always be a scalar for each equation and this is because it acts as the variable that makes the graph either bigger or smaller. All of these equations have some forms of spirals, but some of these equations actually have different shapes when the n and c value are at certain points. These special characteristics in some of the functions show how Pre-Calculus is not just finding the one answer and moving on. These equations show more exploration and discoveries behind the variables written on the screen. This project helped us understand that to really understand math, you must think outside the box, and think of the questions that have not yet been answered in the world.