Conic Sections

About 2 years ago, I wrote a post about plotting parabolas and quadratic equations, because I was learning quadratics that year in my math work. Now that I’m almost done with my sophomore year of high school, I’ve moved on to more complicated things.

Conic Sections

There are 3 types of conic sections. An ellipse, a parabola, and a hyperbola .

Three or Four?

Eh. It depends. There are technically 3, but circles and ellipses are sometimes counted as separate conic sections. In here, I count them as one (ish)

You get a conic section by intersecting 2 cones and a plane in various ways.

Figure 1

Parabolas

I covered parabolas in a previous post, but I’ll do a quick review in this post as well. Parabolas are the curve formed when the plane is at the same angle as the side of the cone. It is #3 in the Figure 1

The simplest equation for a parabola is a quadratic equation:

$$y=a{x}^2+bx+c$$

This will be very easy to plot. When making this a function, we can’t include the $y$, and will just set the function equal to $a{x}^2+bx+c$ for some $a$, $b$, and $c$.

A Function

We will be using the following equation for this example: $$x^2-4x+2$$ This won’t require any rearranging of the equation, or any extra steps, so we will be able to turn it into a function.

q <- function(x) {
  x^2 - 4*x + 2
}

It only has one variable, so it is a function with $x$.

Plotting

Now, we need a familiar library, tidyverse.

library(tidyverse)

We can start plotting with ggplot(), and add geom_function() as layer. Within geom_function(), we need to set the function equal to $q$ so that ggplot can take the right numbers generated.

ggplot() +
  geom_function(fun = q)

This makes a really zoomed in plot. I want to see the whole parabola though. We can set different limits using xlim() and ylim(). We will also add some vertical and horizontal lines to make the different quadrants of the Cartesian plane more apparent, and change the color of the parabola to make it more visible. I’m also going to switch the order so the parabola shows up on top.

ggplot() +
  geom_hline(yintercept = 0) +
  geom_vline(xintercept = 0) +
  geom_function(fun = q, color = "red") +
  xlim(-10, 10) +
  ylim(-10, 10) 

Hmm… I don’t love where the breaks are, and I don’t like how the Cartesian plane is made of rectangles.

ggplot() +
  geom_hline(yintercept = 0) +
  geom_vline(xintercept = 0) +
  geom_function(fun = q, color = "red") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))

We can fix that with coord_fixed and scale_x/y_continuous. We also no longer need the x/ylim layers after the geom

Let’s add labels and a theme to finish up.

ggplot() +
  geom_hline(yintercept = 0) +
  geom_vline(xintercept = 0) +
  geom_function(fun = q, color = "red") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  labs(x = "X",
       y = "Y")+
  theme_linedraw()+
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

That was pretty easy, because we only had one variable to contend with. From now on, things will get a little bit trickier.

Circles and Ellipses

In Figure 1, the circle is #1 and the ellipse is #2

Circles centered at the origin

Circle equations centered at the origin ($0$, $0$) are pretty simple:

$$x^2+y^2=r^2$$ Rearranging (just a little)

In functions, we can’t have an equals sign, so we’ll need to move the $r^2$ to the other side. The equation then becomes:

$$x^2+y^2-r^2=0$$ Y in the function

If we assign this equation with any $r$ to a function like last time, it will not work. We need to add a y to the argument, like this:

h <- function(x, y) {
  x^2 + y^2 - 4     
}

Note

The equation above is $x^2+y^2=4$

But we still can’t plug this function in to ggplot. Something that’s more complicated with multiple variables (that I’ve found, if there are easier ways please let me know) is that you have to make your own data for any extra variables.

Making a Y

Thankfully, making a $y$ is not a lot of work. We don’t have to type in every single number from $-\infty$ to $+\infty$. We just have to give a range of numbers. We could use a variety of commands, like tibble, but I found that expand_grid works best. We don’t need very many numbers, so $-7$ to $7$ will work just fine for our purposes. We will also have to do this for the rest of the conic sections.

The way we make our $y$ should look something like this:

circle_data <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100))

Make sure you save it to an object, otherwise all the numbers won’t be easily viewable, and will get lost. We are technically making a dataset (though we are really just reminding R to list a bunch of numbers), so we need an x and a y. We also need a z, which will be the product of the x and y in the function $h$ we defined earlier. We can do this with mutate

circle_data <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = h(x, y))

Plot at the origin

We can finally plot this circle! We can start like we normally do, with ggplot and a geom.

ggplot(circle_data, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "purple")

Wait a minute. That’s… not a circle? Don’t worry, it is actually a circle, we just need to add a few more things to the graph to make it be extra circle-y. We will also fix what the limits are and make it look just a little bit nicer.

Inside geom_contour there is something called breaks. What exactly is that for? There are 10,000 observations in circle_data and we don’t use all of them for this graph. We actually only use the observations where z is closest to 0. If the breaks was set to 1, for example, it would take all observations where z is close to/equal to 1. That’s kind of like adding a number to the radius. It’s a similar equation, but won’t produce the circle we are looking for.

ggplot(circle_data, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "purple") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  labs(x = "X",
         y = "Y") +
  theme_linedraw() +
  theme(panel.grid.minor = element_blank(),
        axis.title.y = element_text(angle = 0, vjust = 0.5))

Awesome!!! We made a circle with radius 4, centered around the origin, which is what our original equation was!

Circles not centered at the origin

So, the standard equation for a circle is:

$$x^2+y^2=r^2$$ But that is only centered around $0,0$. To make a circle centered…not…there, we need to use a slightly different equation.

When a circle, ellipse, or hyperbola is not centered around the origin, we simply shift the $x$ and $y$ by adding, subtracting, and multiplying.

The new circle equation is

$$(x-h{)}^2+(y-k{)}^2=r^2$$

For some $h$, $k$, and $r$. The center of the circle is $h,k$ and the radius is $r$.

Let’s take the same equation from the last graph, and change some things about it.

$$x^2+y^2=4$$

I want this new circle to be 4 over and 3 down from the original circle.

$$(x-4{)}^2+(y+3{)}^2-4$$

Rearranging for Functions

Remember, we need to subtract the radius and move it to the other side of the equation in order to turn it into a function

We can turn this into a function $t$

t <- function(x, y) {
  (x - 4)^2 + (y + 3)^2 - 4
}

We need to make our $y$, so we’ll do that the same way as before

circle_data2 <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = t(x, y))

Be careful!

We need to use our function $t$ when making the z column in circle_data_2

Now we can plot it the same exact way as before. It should be the same size, but in a different place.

ggplot(circle_data2, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "blue") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  labs(x = "X",
       y = "Y") +
  theme_linedraw() +
  theme(panel.grid.minor = element_blank(),
        axis.title.y = element_text(angle = 0, vjust = 0.5))

Now that we’ve done 2 circles, we should be getting into a bit of a rhythm. All we really need to change is what the function is!

Ellipses

The standard form equation for an Ellipse is:

$$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1$$

This should look really similar to the circle equation:

$$(x-h{)}^2+(y-k{)}^2$$ As I mentioned at the very very beginning, circles are a special form of ellipse. An ellipse becomes a circle when $a=b$. If $a>b$, the ellipse is horizontal. If $b>a$, the ellipse is vertical.

Vertical Ellipse

We will be using the equation

$$\frac{(2x-2{)}^2}{4^2}+\frac{(3y+3{)}^2}{8^2}-1$$

p <- function(x, y) {
  (2*x - 2)^2/4^2 + (3*y + 3)^2/8^2 - 1     
}

Now, we go through the same process of making our $y$ and $z$.

ellipse_data <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = p(x, y))

And we graph the same way:

ggplot(ellipse_data, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "violetred2") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  labs(x = "X",
         y = "Y") +
  theme_linedraw() +
  theme(panel.grid.minor = element_blank(),
        axis.title.y = element_text(angle = 0, vjust = 0.5))

Horizontal Ellipse

See if you can graph this equation:

$$\frac{(3x+5{)}^2}{8^2}+\frac{(3y-2{)}^2}{4^2}=1$$

Solution

Here’s the solution!

v <- function(x, y) {
  (3*x + 5)^2/8^2 + (3*y - 2)^2/4^2 - 1     
}

ellipse_data2 <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = v(x, y))

ggplot(ellipse_data, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "springgreen3") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Hyperbolas

This is the last kind of conic section. It is #4 in Figure 1, a plane that cuts through 2 cones.

Standard Equation

A hyperbola is kind of like an opposite ellipse. The standard equation is:

$$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1$$ Remember, in order to turn it into a function, we have to move all terms to the same side, so it will look like:

$$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}-1$$ And now we can start graphing. It’s the same process as ellipses and circles, so we need to get an equation for some $h$, $k$, $a$ and $b$.

Horizontal Hyperbola

Here’s the equation:

$$\frac{(x+4{)}^2}{1^2}-\frac{y^2}{2^2}-1$$ Assign it to a function and make the $y$ and $z$, just like all the previous times.

m <- function(x, y) {
  (x + 4)^2/1^2 - y^2/2^2 - 1     
}

hyperbola1 <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = m(x, y))

And we can even use the same template for the plot!

ggplot(hyperbola1, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "orange1") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Uh-oh…

Why are we running out of lines? They’re supposed to go on infinitely!

Making the $y$ and $z$ data takes a bit of trial and error. Once you have an equation, you should be able to get a rough idea of what your range should be. The first part of the equation $\frac{(x+4{)}^2}{1^2}$ should have been my first clue. $4$ squared is $16$, and divided by $1$ is still $16$! That’s bigger than $7$! I should adjust my range to fit that a little better.

m <- function(x, y) {
  (x + 4)^2/1^2 - y^2/2^2 - 1     
}

hyperbola2 <- expand_grid(
  x = seq(-10, 10, length.out = 100),
  y = seq(-10, 10, length.out = 100)) |> 
  mutate(z = m(x, y))

I only moved it to $-10$ and $10$ because that’s the limit of the Cartesian plane I currently have.

ggplot(hyperbola2, aes(x = x, y = y)) +
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  geom_contour(aes(z = z), breaks = 0, color = "orange1") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

That’s much better! You can make an up and down opening hyperbola pretty much the same way.

Vertical Hyperbola

Same process as last time, we need to start out with an equation. The $x$ and the $y$ terms switch places.

$$\frac{y^2}{2^2}-\frac{(x-4{)}^2}{1^2}=1$$ We will write this equation as a function, and make data for an $x$ and a $z$ this time, with a range of $10$ to $-10$.

b <- function(y, x) {
  y^2/2^2 - (x-4)^2/1^2 - 1    
}

hyperbola3 <- expand_grid(
  y = seq(-10, 10, length.out = 100),
  x = seq(-10, 10, length.out = 100)) |> 
  mutate(z = b(y, x))

And we follow the same procedure as last time for the graph as well.

ggplot(hyperbola3, aes(x = x, y = y)) +
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  geom_contour(aes(z = z), breaks = 0, color = "deeppink") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Review

The process is quite similar for all of the conic sections (except for parabola, which is easier). Take your equation with some $h$, $k$ $a$, and $b$ and create a function by moving all terms to the same side. With that function, create a $y$ and a $z$. With that finished data, you can then graph. For parabolas, you don’t necessarily have to move all terms to the same side, and don’t need to make a $y$ or $z$.

Circle

$$(x-2{)}^2+(y-1{)}^2=9$$

a <- function(x, y) {
  (x-2)^2 + (y-1)^2 - 9    
}

example_circle <- expand_grid(
  x = seq(-7, 7, length.out = 100),
  y = seq(-7, 7, length.out = 100)) |> 
  mutate(z = a(x, y))

ggplot(example_circle, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "blue") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  labs(x = "X",
       y = "Y") +
  theme_linedraw() +
  theme(panel.grid.minor = element_blank(),
        axis.title.y = element_text(angle = 0, vjust = 0.5))

Ellipse

$$\frac{(x-3{)}^2}{8^2}+\frac{(y+3{)}^2}{4^2}=1$$

j <- function(x, y) {
  (x-3)^2/6^2 + (y+3)^2/4^2 - 1     
}

example_ellipse <- expand_grid(
  x = seq(-15, 15, length.out = 100),
  y = seq(-15, 15, length.out = 100)) |> 
  mutate(z = j(x, y))

ggplot(example_ellipse, aes(x = x, y = y)) +
  geom_contour(aes(z = z), breaks = 0, color = "springgreen3") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Parabola

$$x^2+5x+3$$

k <- function(x, y) {
   x^2+5*x + 3  
}     


ggplot() +
  geom_hline(yintercept = 0) +
  geom_vline(xintercept = 0) +
  geom_function(fun = k, color = "red") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  labs(x = "X",
       y = "Y")+
  theme_linedraw()+
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Hyperbola

$$\frac{3y^2}{4^2}-\frac{(2x-7{)}^2}{3^2}=1$$

l <- function(x, y) {
  3*y^2/4^2-(2*x-7)^2/3^2-1    
}

example_hyperbola <- expand_grid(
  x = seq(-15, 15, length.out = 100),
  y = seq(-15, 15, length.out = 100)) |> 
  mutate(z = l(x, y))

ggplot(example_hyperbola, aes(x = x, y = y)) +
  geom_hline(yintercept = 0)+
  geom_vline(xintercept = 0)+
  geom_contour(aes(z = z), breaks = 0, color = "deeppink") +
  coord_fixed(xlim = c(-10, 10), ylim= c(-10, 10)) +
  scale_x_continuous(breaks = -10:10, limits = c(-10, 10))+
  scale_y_continuous(breaks = -10:10, limits = c(-10, 10))+
  theme_linedraw()+
  labs(x = "X",
       y = "Y") +
  theme(
    panel.grid.minor = element_blank(),
    axis.title.y = element_text(angle = 0, vjust = 0.5)
  )

Figure 1: Jens Vyff, Types Of Conic Sections , via Wikimedia Commons, February 2021