BeschreibungNested 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]
Datum	15. Dezember 2020
Quelle	Eigenes Werk
Urheber	Adam majewski
Andere Versionen	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‑Erstellung InfoField	Der SVG-Code ist valide.   Dieses Diagramm wurde mit einem Texteditor erstellt. Die Validierung hat sie für syntaktisch korrekt befunden. 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

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

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

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

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 = common points

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:

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

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

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

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

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))

/*

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 
  )$
  

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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