Juan Carlos Ponce Campuzano
Independent Mathematics Educator
The mathematical beauty of epicycles
Juan Carlos Ponce Campuzano
The mathematical beauty of epicycles
A mysterious curve
What are epicycles?
🤔
A circle moving on another circle.
What are epicycles?
😃
A circle moving on another circle, on another circle.
What are epicycles?
😃
A circle moving on another circle, on another circle, on another circle...
What are epicycles?
{x(t)y(t)0≤t≤2π
\left\{
\begin{array}{l}
x(t) \\
y(t)
\end{array}
\right. 0\leq t \leq 2\pi
(x(t),y(t))
\big(x(t) , y(t) \big)
🤔
How can we represent them mathematically?
A parametric function!
(x,y)
\big(x , y \big)
What are epicycles?
🤔🔍
RAWK QT8A
Code:
What are epicycles?
🤔🔍
Work only with
- Simple rotation
- Double rotation
RAWK QT8A
Code:
{Rcos(ωt)Rsin(ωt)0≤t≤2π
\left\{
\begin{array}{l}
R \cos( \omega t) \\
R\, \sin(\omega t)
\end{array}
\right. 0\leq t \leq 2\pi
Simple rotations
Simple rotations
sinx
Simple rotations
sinx and cosx
Double rotations
{R1cos(ω1t)+R2cos(ω2t)R1sin(ω1t)+R2sin(ω2t)
\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
{R1cos(ω1t)+R2cos(ω2t)+R3cos(ω3t)R1sin(ω1t)+R2sin(ω2t)+R3sin(ω3t)
\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
{R1cos(ω1t)+R2cos(ω2t)+R3cos(ω3t)R1sin(ω1t)+R2sin(ω2t)+R3sin(ω3t)
\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.
R1=1,R2=21,R3=31
R_1 = 1, R_2 = \dfrac{1}{2}, R_3 = \dfrac{1}{3}
ω1=1,
\omega_1 = 1,
ω2=6,
\omega_2 =6,
ω3=−14
\omega_3 = -14
👉
Complete last task!
{R1cos(ω1t+ϕ1)+⋯+RNcos(ωNt+ϕN)R1sin(ω1t+ϕ1)+⋯+RNsin(ωNt+ϕN)
\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≤t≤2π
0\leq t\leq 2\pi
What are epicycles?
What are epicycles?
⎩⎨⎧k=1∑NRkcos(ωkt+ϕk)k=1∑NRksin(ωkt+ϕk)0≤t≤2π
\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 ϕk 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?
⎩⎨⎧k=1∑NRkcos(ωkt+ϕk)k=1∑NRksin(ωkt+ϕk)0≤t≤2π
\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 ϕk 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.
Question
It seems that the geometric shapes described
by epicycles are smooth closed curves.
But, do all epicycles have to be curvy or closed?
🤔
Line segment
{Rcos(ωt)+Rcos(−ωt)Rsin(ωt)+Rsin(−ωt)
\left\{
\begin{array}{l}
R \cos(\omega t) + R \cos(-\omega t)\\
R \sin(\omega t ) + R \sin(-\omega t )
\end{array}
\right.
=2Rcos(ωt)=0
\begin{array}{l}
= 2R \cos(\omega t) \\
= 0
\end{array}
Q =(2Rcos(ωt),0)
{Rcos(ωt)+Rcos(ωt)Rsin(ωt)−Rsin(ωt)
\left\{
\begin{array}{l}
R \cos(\omega t) + R \cos(\omega t)\\
R \sin(\omega t ) - R \sin(\omega t )
\end{array}
\right.
🤔
How can we determine the epicycles for our mysterious curve?
How can we determine the epicycles for our mysterious curve?
Before answering this question,
let's explore an interesting mathematical property.
Rotational symmetry exploration
Rotational symmetry exploration
{R1cos(ω1t)+R2cos(ω2t)+R3cos(ω3t)R1sin(ω1t)+R2sin(ω2t)+R3sin(ω3t)
\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.
R1=1,R2=21,R3=31
R_1 = 1, R_2 = \dfrac{1}{2}, R_3 = \dfrac{1}{3}
ω1=1,ω2=6,ω3=−14
\omega_1 = 1, \omega_2 =6, \omega_3 = -14
{1, 6, -14}
Rotational symmetry exploration
Rotational symmetry exploration
{1, 6, -14}
Rotational symmetry exploration
What is the relationship between the frequencies {1,6,−14} and the rotational symmetry of order 5?
🤔
Rotational symmetry exploration
What is the relationship between the frequencies {1,6,−14} and the rotational symmetry of order 5?
🤔
ZT7S CNXU
Code:
Rotational symmetry exploration
1−6=−5
6−(−14)=20
−14−1=−15
The greatest common divisor (GCD) of −5,20, and −15 is:
5
Rotational symmetry exploration
What is the order of symmetry for the values
-2, 5, 19?
Rotational symmetry exploration
What is the order of symmetry for the values
2, 8, -10?
Note that 2, 8, and -10 have a common factor.
How can we determine the epicycles for our mysterious curve?
c(t)={R1cos(ω1t)+R2cos(ω2t)+R3cos(ω3t)R1sin(ω1t)+R2sin(ω2t)+R3sin(ω3t)
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<Rk∈R,ωk∈Z)
(0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z)
R1,R2,R3 are positive real numbers and ω1,ω2,ω3 are also real numbers.
How can we determine the epicycles for our mysterious curve?
It is going to get complex! 🙃
(cost,sint)
(\cos t, \sin t)
=cost+isint
=\cos t + i \sin t
=eit
= e^{i t}
(R1cos(ω1t),R1sin(ω1t))
\big(R_1 \cos (\omega_1 t), R_1 \sin (\omega_1 t)\big)
=R1eiω1t
= R_1e^{i \omega_1 t}
i=−1
i = \sqrt{-1}
c(t)={R1cos(ω1t)+R2cos(ω2t)+R3cos(ω3t)R1sin(ω1t)+R2sin(ω2t)+R3sin(ω3t)
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<Rk∈R,ωk∈Z
0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z
It is going to get complex! 🙃
c(t)=R1eω1it+R2eω2it+R3eω3it
c(t)= R_1e^{ \omega_1 i t}+R_2e^{ \omega_2 i t}+R_3e^{ \omega_3 i t}
0<Rk∈R,ωk∈Z
0\lt R_k\in \mathbb R, \;\omega_k\in \mathbb Z
It is going to get complex! 🙃
c(t)=C1eω1it+C2eω2it+C3eω3it
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
Ck is a complex number,ωk∈Z
C_k \text{ is a complex number}, \;\omega_k\in \mathbb Z
It is going to get complex! 🙃
Ck=a+ib=(a,b)
C_k = a + i b = (a,b)
The mysterious curve
c(t)=eit+21e6it+3ie−14it
c(t) = e^{it} + \dfrac{1}{2}e^{6 i t} + \dfrac{i}{3}e^{-14 i t}
c(t)=C1eω1it+C2eω2it+C3eω3it
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
C1=1,C2=21,C3=3i
C_1=1,\,C_2=\dfrac{1}{2},\, C_3= \dfrac{i}{3}
ω1=1,ω2=6,ω3=−14
\omega_1=1,\,\omega_2=6,\, \omega_3=-14
The mysterious curve
c(t)=⎩⎨⎧cos(t)+21cos(6t)+31sin(14t)sin(t)+21sin(6t)+31cos(14t)
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⋅i=−1
More mystery curves
c(t)=C1eω1it+C2eω2it+C3eω3it
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+C_3e^{ \omega_3 i t}
c(t)=C1eω1it+C2eω2it+⋯+CNeωNit
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
More mystery curves
c(t)=C1eω1it+C2eω2it+⋯+CNeωNit
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)=C1eω1it+C2eω2it+⋯+CNeωNit
c(t)= C_1e^{ \omega_1 i t}+C_2e^{ \omega_2 i t}+\cdots + C_Ne^{ \omega_N i t}
🤯
c(t)=k=1∑NCkeωkit
c(t)= \displaystyle \sum_{k=1}^{N} C_ke^{ \omega_k i t }
Discrete Fourier Transform
xn=k=0∑N−1Xk⋅eikN2πn
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!
🤔
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
{
Discrete Fourier Transform (DFT)
👈 DFT
The inverse of the DFT
👈
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
c(t)=C1eω1it+C2eω2it+⋯+CNeωNit
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:
{
Discrete Fourier Transform (DFT)
🤔
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
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:
{
Discrete Fourier Transform (DFT)
c(t)={Sum of a bunch sines and cosinesSum of a bunch sines and cosines
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.
😃
Discrete Fourier Transform (DFT)
A nice methaphor 😃
- What does the Discrete Fourier Transform do?
Given a smoothie, it finds the recipe.
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
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.
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Applications of the DFT
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
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
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Video
Applications of the DFT
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Images
Applications of the DFT
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Sound
Applications of the DFT
Xk=N1n=0∑N−1xn⋅e−ikN2πn
X_k = \displaystyle \frac{1}{N} \sum_{n=0}^{N-1} x_n \cdot e^{- i k \frac{2 \pi }{N} n }
xn=k=0∑N−1Xk⋅eikN2πn
x_n =\displaystyle \sum_{ k=0}^{N-1} X_{k} \cdot e^{ i k \frac{2 \pi }{N} n }
Magnetic Resonance Imaging
Thanks!
dynamicmath.xyz
The mathematical beauty of epicycles Juan Carlos Ponce Campuzano 🔗 jcponce.com
The mathematical beauty of epicycles
By Juan Carlos Ponce Campuzano
The mathematical beauty of epicycles
The mathematical beauty of epicycles
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