The mathematical beauty of epicycles

Juan Carlos Ponce Campuzano

What are epicycles?

🤔

A circle moving on another circle.

What are epicycles?

\left\{ \begin{array}{l} x(t) \\ y(t) \end{array} \right. 0\leq t \leq 2\pi

\left\{ \begin{array}{l} x(t) \\ y(t) \end{array} \right. 0\leq t \leq 2\pi

\big(x(t) , y(t) \big)

\big(x(t) , y(t) \big)

🤔

How can we represent them mathematically?

A parametric function!

\big(x , y \big)

\big(x , y \big)

Double rotations

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t)\\ R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) \end{array} \right.

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t)\\ R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) \end{array} \right.

Triple rotations

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

Triple rotations

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

\left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \,\sin(\omega_1 t) + R_2 \,\sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

R_1 = 1, R_2 = \dfrac{1}{2}, R_3 = \dfrac{1}{3}

R_1 = 1, R_2 = \dfrac{1}{2}, R_3 = \dfrac{1}{3}

\omega_1 = 1,

\omega_1 = 1,

\omega_2 =6,

\omega_2 =6,

\omega_3 = -14

\omega_3 = -14

👉

Complete last task!

\left\{ \begin{array}{l} \displaystyle R_1\cos(\omega_1 t + \phi_1) + \cdots +R_N\cos(\omega_N t + \phi_N) \\ \displaystyle R_1\sin(\omega_1 t + \phi_1) + \cdots +R_N\sin(\omega_N t + \phi_N) \end{array} \right.

\left\{ \begin{array}{l} \displaystyle R_1\cos(\omega_1 t + \phi_1) + \cdots +R_N\cos(\omega_N t + \phi_N) \\ \displaystyle R_1\sin(\omega_1 t + \phi_1) + \cdots +R_N\sin(\omega_N t + \phi_N) \end{array} \right.

In general we have:

0\leq t\leq 2\pi

0\leq t\leq 2\pi

What are epicycles?

\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi

\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi

In general we have:

This new symbol $\phi_k$ indicates how much the disk $k$ is initially rotated at time $t = 0$ , and we called a phase offset.

If we do not specify a phase offset, we can only describe epicycles where the circles start aligned.

What are epicycles?

\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi

\left\{ \begin{array}{l} \displaystyle \sum_{k=1}^{N} R_k\cos(\omega_k t + \phi_k) \\ \displaystyle \sum_{k=1}^{N}R_k\sin(\omega_k t + \phi_k) \end{array} \right. \; 0\leq t\leq 2\pi

This new symbol $\phi_k$ indicates how much the disk $k$ is initially rotated at time $t = 0$ , and we called a phase offset.

If we do not specify a phase offset, we can only describe epicycles where the circles start aligned.

Line segment

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(-\omega t)\\ R \sin(\omega t ) + R \sin(-\omega t ) \end{array} \right.

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(-\omega t)\\ R \sin(\omega t ) + R \sin(-\omega t ) \end{array} \right.

\begin{array}{l} = 2R \cos(\omega t) \\ = 0 \end{array}

\begin{array}{l} = 2R \cos(\omega t) \\ = 0 \end{array}

Q $=(2R\cos(\omega t),0)$

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(\omega t)\\ R \sin(\omega t ) - R \sin(\omega t ) \end{array} \right.

\left\{ \begin{array}{l} R \cos(\omega t) + R \cos(\omega t)\\ R \sin(\omega t ) - R \sin(\omega t ) \end{array} \right.

c(t) = \left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

c(t) = \left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

(0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z)

(0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z)

$R_1, R_2, R_3$ are positive real numbers and $\omega_1, \omega_2, \omega_3$ are also real numbers.

How can we determine the epicycles for our mysterious curve?

It is going to get complex! 🙃

(\cos t, \sin t)

(\cos t, \sin t)

=\cos t + i \sin t

=\cos t + i \sin t

= e^{i t}

= e^{i t}

\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)

\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)

= R_1e^{i \omega_1 t}

= R_1e^{i \omega_1 t}

i = \sqrt{-1}

i = \sqrt{-1}

c(t) = \left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

c(t) = \left\{ \begin{array}{l} R_1 \cos(\omega_1 t) + R_2 \cos(\omega_2 t) + R_3 \cos(\omega_3 t) \\ R_1 \sin(\omega_1 t) + R_2 \sin(\omega_2 t) + R_3 \sin(\omega_3 t) \end{array} \right.

0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z

0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z

It is going to get complex! 🙃

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

C_k \text{ is a complex number}, \;\omega_k\in \mathbb Z

C_k \text{ is a complex number}, \;\omega_k\in \mathbb Z

It is going to get complex! 🙃

C_k = a + i b = (a,b)

C_k = a + i b = (a,b)

The mysterious curve

c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}

c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

C_1=1,\,C_2=\dfrac{1}{2},\, C_3= \dfrac{i}{3}

C_1=1,\,C_2=\dfrac{1}{2},\, C_3= \dfrac{i}{3}

\omega_1=1,\,\omega_2=6,\, \omega_3=-14

\omega_1=1,\,\omega_2=6,\, \omega_3=-14

The mysterious curve

c(t) = \left\{ \begin{array}{l} \displaystyle \cos( t) +\frac{1}{2}\cos(6 t) +\frac{1}{3}\sin(14 t) \\ \\ \displaystyle \sin( t) +\frac{1}{2} \sin(6 t)\, +\frac{1}{3} \cos(14 t) \end{array} \right.

c(t) = \left\{ \begin{array}{l} \displaystyle \cos( t) +\frac{1}{2}\cos(6 t) +\frac{1}{3}\sin(14 t) \\ \\ \displaystyle \sin( t) +\frac{1}{2} \sin(6 t)\, +\frac{1}{3} \cos(14 t) \end{array} \right.

Using properties of complex numbers:

$i\cdot i = -1$

More mystery curves

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

In fact we can use this equation to create more complex epicycles!

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

🤯

c(t)= \displaystyle \sum_{k=1}^{N} C_ke^{ \omega_k i t }

c(t)= \displaystyle \sum_{k=1}^{N} C_ke^{ \omega_k i t }

Discrete Fourier Transform

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

In fact we can use this equation to create more complex epicycles!

🤔

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

$\Bigg\{\Bigg.$

Discrete Fourier Transform (DFT)

👈 DFT

The inverse of the DFT

👈

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}

They are very similar to this:

$\Bigg\{\Bigg.$

Discrete Fourier Transform (DFT)

🤔

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

They are very similar to this:

$\Bigg\{\Bigg.$

Discrete Fourier Transform (DFT)

c(t) = \left\{ \begin{array}{l} \text{Sum of a bunch sines and cosines} \\ \text{Sum of a bunch sines and cosines} \end{array} \right.

c(t) = \left\{ \begin{array}{l} \text{Sum of a bunch sines and cosines} \\ \text{Sum of a bunch sines and cosines} \end{array} \right.

😃

A nice methaphor 🧐

What does the Discrete Fourier Transform do?

Given a smoothie, it finds the recipe.

How?

Run the smoothie through filters to extract each ingredient.

Why?

Recipes are easier to analyze, compare, and modify than the smoothie itself.

How do we get the smoothie back?

Blend the ingredients.

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Applications of the DFT

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Modern digital media

Applications of the DFT

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Video

Applications of the DFT

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Images

Applications of the DFT

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Sound

Applications of the DFT

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }

Magnetic Resonance Imaging

The mathematical beauty of epicycles Juan Carlos Ponce Campuzano 🔗 jcponce.com