Resumen

DescripciónNested Ellipses.svg	English: Nested Ellipses , Parameters: a=5, b=4 theta=0.2617993877991494 r=0.9434598957108945 number of ellipses=61. "The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." [1]
Fecha	15 de diciembre de 2020
Fuente	Trabajo propio
Autor	Adam majewski
Otras versiones	Nested Ellipses (Ellipse Whirl) by (C. J. Chen)[2] tangential and concentric ellipse [3] rotated polygons[4] "Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the n mice in the mice problem, and form n logarithmic spirals."Weisstein, Eric W. "Whirl." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Whirl.html math.stackexchange question: scaling-and-rotating-a-square-so-that-it-is-inscribed-in-the-original-square nested polygons - mathworld.wolfram : NestedPolygon nested circles nested squares Normal lines to the ellipse.svg on A diagram of how arms form in spiral galaxis.
SVG desarrollo InfoField	El código fuente de esta imagen SVG es válido.   Este archivo fue creado con un editor de texto. Su código fuente puede contener información adicional. Previous version had been created with Gnuplot (15 566 434 bytes)  d  now 0.01% of previous size   Please do not replace the simplified code of this file with a version created with Inkscape or any other vector graphics editor

Algorithm

Ellipse centered at origin and not rotated

the equation of a ellipse:

• centered at the origin
• with width = 2a and height = 2b

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}$

the parametric equation is:

${\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .}$

So explicit equations :

${\displaystyle x(t)=a\cos(t)}$

${\displaystyle y(t)=b\sin(t)}$

The parameter t :

• is called the eccentric anomaly in astronomy
• is not the angle of ${\displaystyle (x(t),y(t))}$ with the x-axis
• can be called internal angle of the ellipse

ellipse rotated and not moved

Rotation In two dimensions

In two dimensions, the standard rotation matrix has the following form:

${\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}$.

This rotates column vectors by means of the following matrix multiplication,

${\displaystyle {\begin{bmatrix}x'\\y'\\\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}}$.

Thus, the new coordinates (x′, y′) of a point (x, y) after rotation are

${\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \,\\y'&=x\sin \theta +y\cos \theta \,\end{aligned}}}$.

result

Center is in the origin ( not shifted or not moved) and rotated:

• center is the origin z = (0, 0)
• ${\displaystyle \theta }$ is the angle measured from x axis
• The parameter t (called the eccentric anomaly in astronomy) is not the angle of ${\displaystyle (x(t),y(t))}$ with the x-axis
• a,b are the semi-axis in the x and y directions

${\displaystyle \mathbf {x} =\mathbf {x} (\theta )=x\cos \theta -y\sin \theta }$

${\displaystyle \mathbf {y} =\mathbf {y} (\theta )=x\sin \theta +y\cos \theta }$

Here

• ${\displaystyle \theta }$ is fixed ( constant value)
• t is a parameter = independent variable used to parametrise the ellipse

${\displaystyle \mathbf {x} =\mathbf {x} _{\theta }(t)=a\cos \ t\cos \theta -b\sin \ t\sin \theta }$

${\displaystyle \mathbf {y} =\mathbf {y} _{\theta }(t)=a\cos \ t\sin \theta +b\sin \ t\cos \theta }$

So

${\displaystyle {\frac {\mathbf {x} ^{2}}{a^{2}}}+{\frac {\mathbf {y} ^{2}}{b^{2}}}=1.}$

${\displaystyle {\dfrac {(x\cos \theta -y\sin \theta )^{2}}{a^{2}}}+{\dfrac {(x\sin \theta +y\cos \theta )^{2}}{b^{2}}}=1}$

${\displaystyle {\dfrac {(a\cos \ t\cos \theta -b\sin \ t\sin \theta )^{2}}{a^{2}}}+{\dfrac {(a\cos \ t\sin \theta +b\sin \ t\cos \theta )^{2}}{b^{2}}}=1}$

intersection of 2 ellipses

Intersection = common points

not scaled

2 ellipses:

• both are cetered at origin
• first is not rotated, second is rotated (constant angle theta)
• with the same the aspect ratio s (the ratio of the major axis to the minor axis)

${\displaystyle s={\frac {a}{b}}={\frac {a^{\prime }}{b^{\prime }}}}$

${\displaystyle {\begin{cases}x=x^{\prime }\\y=y^{\prime }\end{cases}}}$

Fix x, then find y:

${\displaystyle x=x\cos \theta -y\sin \theta }$

${\displaystyle y={\frac {x\cos \theta -x}{\sin \theta }}}$

scaled

Second is scaled by factor r^[5]

${\displaystyle r={\frac {a^{\prime }}{a}}={\frac {b^{\prime }}{b}}}$

${\displaystyle r={\sqrt {1+{\frac {\Delta ^{2}}{4}}}}-{\frac {\Delta }{2}}}$

where:

• ${\displaystyle \theta }$ is the tilt angle

${\displaystyle \Delta =\left({\frac {a}{b}}-{\frac {b}{a}}\right)\sin \theta }$

Python source code

import math, io
def make_svg(x_offset, y_offset):
 outs  = []
 n     = 61
 a     = 6035
 b     = 4828
 theta = 15
 delta = (1.0 * a / b - 1.0 * b / a) * math.sin(math.radians(theta))
 r     = (1 + delta * delta / 4) ** 0.5 - delta / 2
 # print(delta, r)
 for i in range(n):
  a_i = a * r ** i
  b_i = b * r ** i
  deg = (-theta * i) % 180
  rad = math.radians(deg)
  t   = math.pi * 1.5 if deg == 0 else math.pi + math.atan(b_i * math.cos(rad) / (a_i * math.sin(rad)))
  x   = a_i * math.cos(rad) * math.cos(t) - b_i * math.sin(rad) * math.sin(t) + x_offset - 65
  y   = a_i * math.sin(rad) * math.cos(t) + b_i * math.cos(rad) * math.sin(t) + y_offset + 8
  ## formulae from http://math.stackexchange.com/questions/1889450/extrema-of-ellipse-from-parametric-form
  # print(i, deg, t)
  outs.append('M%.0f%s%.0fa%.0f,%.0f %.0f 1 0 1,0' % (x, '' if y < 0 else ',', y, a_i, b_i, deg))
 return '''<?xml version="1.0"?>
<svg xmlns="http://www.w3.org/2000/svg" width="1500" height="1000" viewBox="%d %d 15000 10000">
<path d="%s" fill="none" stroke="#f00" stroke-width="9"/>
</svg>''' % (x_offset - 7500, y_offset - 5000, ''.join(outs))
# <path d="%s" fill="none" stroke="#f00" stroke-width="9" marker-mid="url(#m)"/>
# <marker id="m"><circle r="9"/></marker>

## Find shortest output and write to file
(x_offset_min, length_min) = (0, 99999)
for x_offset in range(-9999, 9999, 1):
 length = len(make_svg(x_offset, 0))
 if length_min > length: (x_offset_min, length_min) = (x_offset, length)
 # print(x_offset, length)
print(x_offset_min, length_min)
(y_offset_min, length_min) = (0, 99999)
for y_offset in range(-9999, 9999, 1):
 length = len(make_svg(0, y_offset))
 if length_min > length: (y_offset_min, length_min) = (y_offset, length)
 # print(y_offset, length)
print(y_offset_min, length_min)
with io.open(__file__[:__file__.rfind('.')] + '.svg', 'w', newline='\n') as f: ## *.* -> *.svg
 f.write(make_svg(x_offset_min, y_offset_min))

Maxima CAS src code

/*

kissing ellipses



These animations are constructed by shrinking and rotating a sequence of concentric and similar ellipses,
so that each ellipse lies inside the previous ellipse and is tangent to it.

https://benice-equation.blogspot.com/2019/01/nested-ellipses.html

==================================================
https://math.stackexchange.com/questions/3773593/given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips

tangential concentric ellipse and insribed ellipses

Let’s say I have an ellipse with horizontal axis $a$ and vertical axis $b$, centered at $(0,0)$. 
I want to compute $a’$ and $b’$ of a smaller ellipse centered at $(0,0)$, 
with the axes rotated by some angle $t$, tangent to the bigger ellipse and $\frac{a’}{b’}=\frac{a}{b}$.




---------------------

The standard parametric equation is:

(x,y)->(a cos(t),b sin(t))


---------------------------

Rotation counterclockwise about the origin through an angle α carries 

(x, y) to (x cos α − ysin α, ycos α+x sin α) 

https://www.maa.org/external_archive/joma/Volume8/Kalman/General.html

=====================================
https://math.stackexchange.com/questions/2987044/how-to-find-the-equation-of-a-rotated-ellipse

===============================
https://math.stackexchange.com/questions/3773593/given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips

============================================================
intersection of 2 ellipses

the common point of 2 ellipses are not vertices ( vertex)

https://math.stackexchange.com/questions/1688449/intersection-of-two-ellipses
https://math.stackexchange.com/questions/425366/finding-intersection-of-an-ellipse-with-another-ellipse-when-both-are-rotated/425412#425412

https://math.stackexchange.com/questions/3312747/intersection-area-of-concentric-ellipses
https://math.stackexchange.com/questions/426150/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-and-rotate/434482#434482
------
xc <- 1 # center x_c or h
yc <- 2 # y_c or k
a <- 5 # major axis length
b <- 2 # minor axis length
phi <- pi/3 # angle of major axis with x axis phi or tau

t <- seq(0, 2*pi, 0.01) 
x <- xc + a*cos(t)*cos(phi) - b*sin(t)*sin(phi)
y <- yc + a*cos(t)*cos(phi) + b*sin(t)*cos(phi)
plot(x,y,pch=19, col='blue')
https://stackoverflow.com/questions/41820683/how-to-plot-ellipse-given-a-general-equation-in-r

===============
Batch file for Maxima CAS
save as a e.mac
run maxima : 
 maxima
and then : 
batch("e.mac");




*/


kill(all);
remvalue(all);
ratprint:false;
numer:true$
display2d:false$


/* 
converts complex number z = x*y*%i 
to the list in a draw format:  
[x,y] 
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$

/* give Draw List from one point*/
dl(z):=points([d(z)])$




/* trigonometric functions in Maxima CAS use radians */
deg2rad(t):= float(t*2*%pi/360)$

GiveImplicit(a,b):=implicit( x^2/(a^2) + (y^2)/(b^2) = 1, x, -4,4, y, -4,4)$

GivePointOfEllipse(a,b, t):= a*cos(t) + b*sin(t)*%i$


/*

xc <- 1 # center x_c or h
yc <- 2 # y_c or k
a <- 5 # major axis length
b <- 2 # minor axis length
phi <- pi/3 # angle of major axis with x axis phi or tau

t <- seq(0, 2*pi, 0.01) 
x <- xc + a*cos(t)*cos(phi) - b*sin(t)*sin(phi)
y <- yc + a*cos(t)*sin(phi) + b*sin(t)*cos(phi)

<math>\mathbf{x} =\mathbf{x}_{\theta}(t) = a\cos\ t\cos\theta - b\sin\ t\sin\theta</math>

<math>\mathbf{y} =\mathbf{y}_{\theta}(t) = a\cos\ t\cos\theta + b\sin\ t\cos\theta</math>


https://stackoverflow.com/questions/65278354/how-to-draw-rotated-ellipse-in-maxima-cas/65294520#65294520
*/

GiveRotatedEllipse(a,b,theta, NumberOfPoints):=block(
	[x, y, zz, t , tmin, tmax, dt, c, s],
	zz:[],
	dt : 1/NumberOfPoints, 
 	tmin: 0, 
 	tmax: 2*%pi,
 	c:float(cos(theta)),
 	s:float(sin(theta)),
 	for t:tmin thru tmax step dt do(
 		x: a*cos(t)*c - b*sin(t)*s,
 		x: float(x), 
 		y: a*cos(t)*s + b*sin(t)*c,
 		y:float(y),
 		zz: cons([x,y],zz)
 	),
 	return (points(zz))
)$

GiveScaledRotatedEllipse(a,b, r,theta, NumberOfPoints):= GiveRotatedEllipse(r*a,r*b,theta, NumberOfPoints)$

GiveEllipseN(a,b,r,n,theta, NumberOfPoints):=GiveRotatedEllipse(a*(r^n),b*(r^n),n*theta, NumberOfPoints)$

Give_N(n):= GiveEllipseN(a,b,r,n,theta, NumberOfPoints)$

GiveEllipses(n):=block(
	[elipses],
	
	ellipses:makelist(i, i, 0, n, 1),
	ellipses:map(Give_N, ellipses),
	return(ellipses)
	


)$

/* 
scale ratio r = a'/a = b'/b

https://math.stackexchange.com/questions/3773593/given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
*/
GiveScaleRatio(a, b, theta):= block(
	[d, r], 
	d: (a/b - b/a)*sin(theta), 
	d:float(d),
	r: sqrt(1+d*d/4) - d/2,
	r:float(r),
	return(r)


)$


compile(all)$

/* compute */

/* angles fo trigonometric functions in radians */
angle: 15$
theta:deg2rad(angle) $  /* theta is the angle between    */
a: 5$
b: 4$
NumberOfPoints : 500$
r:GiveScaleRatio(a, b, theta)$  /* 0.942$ the (axis) scaled ratio r = a'/a = b'/b */


n:70;



ee:GiveEllipses(n)$



path:"~/Dokumenty/ellipse/scaled/s1/"$ /*  pwd, if empty then file is in a home dir , path should end with "/" */

/* draw it using draw package by */

 load(draw); 
/* if graphic  file is empty (= 0 bytes) then run draw2d command again */

 draw2d(
  user_preamble="set key top right; unset mouse",
  terminal  = 'svg,
  file_name = sconcat(path, string(a),"_",string(b), "_",string(theta), "_",string(r),"_", string(n)),
   title = "",  
  dimensions = [1500, 1000],
   axis_top         = false,
  axis_right       = false,
  axis_bottom         = false,
  axis_left       = false,
 ytics  = 'none,
 xtics  = 'none,
  proportional_axes = xy,
  line_width = 1,
  line_type = solid,
  
  fill_color = white,
  point_type=filled_circle,
  points_joined = true,
  point_size = 0.05,
  
    
  key = "",
  color = red,
  ee 
  )$
  

Licencia

Yo, el titular de los derechos de autor de esta obra, la publico en los términos de la siguiente licencia:

Este archivo está disponible bajo la licencia Creative Commons Attribution-Share Alike 4.0 International.

Eres libre:

• de compartir – de copiar, distribuir y transmitir el trabajo
• de remezclar – de adaptar el trabajo

Bajo las siguientes condiciones:

• atribución – Debes otorgar el crédito correspondiente, proporcionar un enlace a la licencia e indicar si realizaste algún cambio. Puedes hacerlo de cualquier manera razonable pero no de manera que sugiera que el licenciante te respalda a ti o al uso que hagas del trabajo.
• compartir igual – En caso de mezclar, transformar o modificar este trabajo, deberás distribuir el trabajo resultante bajo la misma licencia o una compatible como el original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Postprocessing

File size was reduced -29% with https://svgoptimizer.com/

references

1. ↑ Osculating curves: around the Tait-Kneser Theoremby E. Ghys, S. Tabachnikov, V. Timorin
2. ↑ Nested Ellipses (Ellipse Whirl) by benice (C. J. Chen)
3. ↑ math.stackexchange question: given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
4. ↑ texample : rotated-polygons
5. ↑ math.stackexchange question : given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips