Summary

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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

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

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 : 700$
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  = 'png,
  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 
  )$
  

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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

Category:CC-BY-SA-4.0#Nested%20Ellipses.pngCategory:Self-published work

references

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