Mystery curves

Goal:

Explore mystery curves

Mystery curves

Intuitive definition of mystery curves.
How to construct these curves
Explore some of its mathematical properties.
Propose a challenge for you.

Route

1. Definition

\left\{ \begin{array}{l} x(t) \\ y(t) \end{array} \right. a\leq t \leq b

Epicycles

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

Simple rotations

\left\{ \begin{array}{l} R \cos( \omega t) \\ R\, \text{sen}(\omega t) \end{array} \right. 0\leq t \leq 2\pi

1. Definition

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

1. Definition

Double 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 \,\text{sen}(\omega_1 t) + R_2 \,\text{sen}(\omega_2 t) + R_3 \,\text{sen}(\omega_3 t) \end{array} \right.

1. Definition

🤔

\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.

2. Construction in GeoGebra

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

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

Mystery curve = ?

Steps to contruct Epicycles

A = (0, 0)
c = Circle(A, 1)
B = Point(c)
t = Slider(0, 2 pi, 0.001, 0.4)
B' = Rotate(B, t)
v1 = Vector(A, B')
d = Circle(B', 1/2)
C = Point(d)
C' = Rotate(C, 6 * t, B')
e = Circle(C', 1/3)
v2 = Vector(B', C')
D = Point(e)
D' = Rotate(D, -14 * t, C')
v3 = Vector(C', D')
locus = Locus(D', t)

a.k.a. GeoGebra Script

2. Construction in GeoGebra

Steps to construct mystery curve

R1 = 1
R2 = 1/2
R3 = 1/3
w1 = 1
w2 = 6
w3 = -14
fx(x) = R1 * cos(w1 * x) + R2 * cos(w2 * x) + R3 * cos(w3 * x)
fy(x) = R1 * sin(w1 * x) + R2 * sin(w2 * x) + R3 * sin(w3 * x)
a = Curva(fx(t), fy(t), t, 0, 2 pi)

a.k.a. GeoGebra Script

2. Construction in GeoGebra

3. Rotational symmetry exploration

{1, 6, -14}

3. Rotational symmetry exploration

What is the relationship between the frequencies \(\{1, 6, -14\}\) and the rotational symmetry of order \(5\)?

🤔

3. Rotational symmetry exploration

\(1-6=-5\)

\(6-(-14)=20\)

\(1-(-14)=15\)

The greatest common divisor of \(-5, 20,\) y \(15\) is:

\(1, 6\) y \(-14\) congruent 1 modulus 5

\(1,6,-14 =1 \left(\text{mod} \, 5\right)\)

3. Rotational symmetry exploration

1. Look for rotational symmetries of order 3 and 4.

2. What is the order of symmetry of the numbers 2, 8, -10?

Note: The common factor for the numbers 2, 8 y -10 is 2.

3. Rotational symmetry exploration

Activity

Define the values of \(R_1\), \(R_2\) and \(R_3\) using the command
- RandomUniform( <Min>, <Max> )
  - Da por resultado un número real aleatorio a partir de una distribución uniforme en el intervalo [Mínimo, Máximo].
Define the values of \(\omega_1\), \(\omega_2\) and \(\omega_3\) using the command
- RandomBetween( <Min> , <Max> )
  - Genera un número entero aleatorio entre el mínimo y el máximo (inclusive).

Example:

R1 = RandomUniform(0.5, 4.5)
w1 =  RandomBetween(0, 9)

3. Create a boton with the command UpdateConstruction()

3. Rotational symmetry exploration

Example:

R1 = RanformUniform(0.5, 4.5)
w1 =  RandomBetween(0, 9)

3. Create a boton with the command UpdateConstruction()

3. Rotational symmetry exploration

Activity

Share your work in GeoGebra Classroom

Join in with the code:

YHEG X7HK

or use the link:

https://www.geogebra.org/classroom/yhegx7hk

In Twitter/Mathstodon

@jcponcemath

Actividad

By email

jcponcemath@gmail.com

3. Rotational symmetry exploration

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.

4. More mystery curves

0\lt R_i\in \mathbb R, \;\omega_i\in \mathbb Z

(\cos t, \sin\, t)

=\cos t + i \,\sin \,t

= e^{i t}

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

= R_1e^{i \omega_1 t}

i = \sqrt{-1}

4. More mystery curves

c(t)= R_1e^{ \omega_1 i t}+R_2e^{ \omega_2 i t}+R_3e^{ \omega_3 i t}

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_i\in \mathbb R, \;\omega_i\in \mathbb Z

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

C_i\in \mathbb C, \;\omega_i\in \mathbb Z

4. More mystery curves

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

4. More mystery curves

c(t) = \left\{ \begin{array}{l} \cos( t) +1/2\cos(6 t) +1/3\,\text{sen}(14 t) \\ \,\text{sen}( t) +1/2 \,\text{sen}(6 t) +1/3 \cos(14 t) \end{array} \right.