Conics
Overview
The following equations
r = k/(a + bsinθ)
r = k/(a + bcosθ)
and their graphs have the special feature of illustrating all four conic sections through their changing variables. As the values of a, b, and k change, the graphs of the two equations demonstrate hyperbolas, parabolas, circles, and ellipses.
Conceptual Analysis
A simple evaluation of the graph includes finding its symmetries, asymptotes, and intercepts.
There is a reason why these two polar equations have such unique features. In order to construct a clear analysis of the relationship between the graphs and the equations, one should perceive the changes in graphs as a result of the changes in the variables: a, b, and k.
Interesting properties of the graph emerge when setting these variables equal to zero. It is known in the basics of polar expressions that the equation r = cos theta is the graph of a circle with a diameter of 1 and center located at polar coordinate (½,0).
Taking the example of only the cosine equation, if only variable a were to exist, the equation would become
r=a+cos θ
In this case, as the absolute value of a becomes greater, more curvature is added to the circle, making it a distorted shape.
If only variable b were to exist, the equation is set to be
r=bcos θ
The larger the absolute value of b, the larger the original circle becomes dilated. When b is zero, the graph disappears
If only variable k were to exist, the equation is changed to
r=k/cos θ
This time, adding the variable k shifts the graph into a vertical line. The change of k translates the line to the left/right. The graph disappears when k is at zero.
When a and k are added to the graph at the same time, the equation becomes
r=k/a+cos θ
At this point, the combination of the variables has already created a new feature, in which the graph can change to an eclipse or a hyperbola, depending on the values of a. On the other hand, k affects the graph by translating its vertex left/right. Moreover, an irregular dilation occurs near the pole.
When k and b are added to the graph without a at the same time, the equation becomes
r=k/bcos θ
In fact, the shape of this graph will be so easy to predict that it won’t require a graph to explain. As the graph of r=k/cos is a line, adding b to it only divides k by b, which still makes the equation a line. As values of k and b change, the line translates left/right.
In the last scenario, when a and b are on the graph without k, the equation becomes
r=a+bcos θ
This time, a very interesting but complex relationship between a and b is discovered. The curvature of the original circle equation seems to be affected by both a and b at the same time.
Effects of Parameters on Graph
There is clearly a connection between the parameters of variables in the equations and its transformational effects on the shapes of the curves.
k is the variable that dilates and translates the location of the curves. It affects the foci of each graph.
The relationship between a and b, known as eccentricity, is the decisive feature that defines the shape of each of the four conic sections.
- If a > b, then the shape of the graph will be an ellipse
- If a < b, then the shape of the graph will be a hyperbola
- If |a| = |b|, then the shape of the graph will be a parabola
- If b = 0, then the shape of the graph will be a circle
- If a = 0, then the shape of the graph will be a line
Contrast Between the Parent Equations
Sin Graph
Cosine Graph
The sin graph is identical to the cos graph, except the the sin graph is rotated pi/2 radians from the cos graph.
A potentially challenging but meaningful step of the investigation would be to find the intersections of the sine and cosine equations under different conditions in order to summarize a pattern.
In the end, there is still one question that remains unsolved: In Cartesian coordinates, inverse relations are graphs that reflect over the y = x line. The sine and cosine equations have the same visual on the graph. However, would the graphical features of inverse relations be the same with polar expressions?
A more interesting relationship between the sine and cosine equations is how one can make them overlap through rotating one of them by pi/2 so that they overlap each other. This connects to and proved what we learned from the unit circle.
Converting from Polar to Cartesian
In order to fully comprehend the equations, it is necessary to prove our observations with the Cartesian version of the equation:
We can tell the shape of the conic section based on its Cartesian equation and how they relate to the polar version of the equation.
r=k/(a+bsinθ)
r=k/(a+bcosθ)
For example, the Cartesian form of the equation looks like
b^2x^2 + b^2y^2 = k^2 - 2aky + a^2y^2
(b^2 - a^2) y2+2aky + (b^2x^2 - k^2) = 0
In the Cartesian form, the values regarding a, b, and k are constants, so the equation can change into a quadratic, ellipse, circle, or hyperbola depending on special values of a, b, and k.
What happens when real numbers are plugged into the equation?
CIRCLE
Knowing that the sine and the cosine equation more or less changes with the same pattern, we’ll use the sine equation. We know that the graph is a circle when b is 0. We will set up a hypothetical where
r=k/(a+bsinθ)=(1)/(2+0sinθ)
We will then try to convert it into a Cartesian
r=1/2+0sinθ
r=1/2
r^2=(1/2)2=1/4
x^2+y^2=1/4
We proved that the equation is indeed a circle in Cartesian form.
PARABOLA
We also know that the graph is a parabola when a = b, we can also set up an equation where
r=k/(a+bsinθ)=1/(1+1sinθ)
Convert to Cartesian
r=1/(1+sinθ)
r+rsinθ=1
r+y=1
x^2+y^2+y=1
x^2+y2=1-y
x^2+y^2=1-2y+y^2
x^2=1-2y
x^2-1=-2y
(-x^2+1)/2=y
y=-1/2x^2+1/2
We proved that the equation is indeed a parabola in Cartesian form.
ELLIPSE
We know that the graph is an ellipse when a > b, we can set up an equation where
r=k/(a+bsinθ)=1/(2+1sinθ)
Convert to Cartesian
r=1/(2+1sinθ)
2r+rsinθ=1
2x^2+y^2+y=1
- x^2+y^2=(1-y)/2
x^2+y^2=(1-2y+y^2)/4
4x^2+4y^2=1-2y+y^2
4x2+3y2=1-2y
4x^2+3y^2+2y=1
We proved that the equation is indeed an ellipse in Cartesian form.
HYPERBOLA
We know that the graph is a hyperbola when a < b, so we can set up an equation where
r=k/(a+bsinθ)=1/(1+2sinθ)
Convert to Cartesian
r=1/(1+2sinθ)
r+2rsinθ=1
x^2+y^2+2y=1
x^2+y^2=1-2y
x^2+y^2=1-4y+4y^2
x^2-3y^2=1-4y
x^2-3y^2+4y=1
We proved that the equation is indeed a hyperbola in Cartesian form. And we proved through Cartesian form that all the conjectures we had at the beginning through analyzing the graphs were indeed justified!
Extensive Exploration
Other than just analyzing the parent equations, we also explored other possibilities that change the equations into something completely strange but mesmerizing.
For example, If you add whole numbers to the equation, the curvature near the pole gets amplified and more circular.
The transformation of the equation can also be done in a unique way. For example, you can make k negative to reflect the cosine equation horizontally and the sine equation vertically.
One can also change theta to make the graph rotate. This is something that one wasn't able to accomplish in Cartesian expressions.
Now, through dissecting the seemingly complex equations into separate pieces, we have discovered the respective features of the variables.
Conclusion
In conclusion, these two polar curves are interesting and convenient ways to express all conic sections in one equation, as its Cartesian version would be much less comprehensible.