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DescriçãoOsculating circles of the Archimedean spiral.svg	English: Osculating circles of the Archimedean spiral. "The spiral itself is not not drawn: we see it as the locus of points where the circles are especially close to each other." [1]
Data	27 de maio de 2019
Origem	Obra do próprio
Autor	Adam majewski
Outras versões	Nested Ellipses.png Osculating circle.svg Log spiral with osculating circles.jpg
SVG desenvolvimento InfoField	O código-fonte desta imagem SVG é válido.   Este(a) desenho vetorial foi criado com o Gnuplot    Este arquivo SVG utiliza texto incorporado que pode ser facilmente traduzido usando um editor de texto.

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Math equations

Point ${\displaystyle z_{t}}$ of an Archimedean spiral for angle t

  ${\displaystyle x_{t}=f(t)=t*cos(t)}$

  ${\displaystyle y_{t}=g(t)=t*sin(t)}$

 ${\displaystyle z_{t}=x_{t}+y_{t}*i}$

The curvature of Archimedes' spiral is

${\displaystyle K(t)={\frac {2+t^{2}}{(1+t^{2})^{3/2}}}}$

Radius of osculating circle is^[2]

${\displaystyle R(t)={\frac {1}{|K(t)|}}}$

Center of osculating circle is

 ${\displaystyle z_{c,t}=x_{c,t}+y_{c,t}*i}$

 ${\displaystyle x_{c,t}=f-{\frac {(f'^{2}+g'^{2})g'}{f'^{2}g''-f''g'}}}$

 ${\displaystyle y_{c,t}=g+{\frac {(f'^{2}+g'^{2})f'}{f'^{2}g''-f''g'}}}$

where

• ${\displaystyle f=f(t)}$
• ${\displaystyle f'={\frac {df}{dt}}(t)}$ is first derivative
• ${\displaystyle f'^{2}=f'*f'}$
• ${\displaystyle f''}$ is a second derivative

notes

Program computes 130 values of angle ( list tt) from 1/5 to 26:

 [1/5,2/5,3/5,4/5,1,6/5,7/5,8/5,9/5,2,11/5,12/5,13/5,14/5,3,16/5,17/5,18/5,19/5,4,21/5,22/5,23/5,24/5,5,26/5,27/5,28/5,29/5,6,31/5,32/5,
        33/5,34/5,7,36/5,37/5,38/5,39/5,8,41/5,42/5,43/5,44/5,9,46/5,47/5,48/5,49/5,10,51/5,52/5,53/5,54/5,11,56/5,57/5,58/5,59/5,12,61/5,62/5,
        63/5,64/5,13,66/5,67/5,68/5,69/5,14,71/5,72/5,73/5,74/5,15,76/5,77/5,78/5,79/5,16,81/5,82/5,83/5,84/5,17,86/5,87/5,88/5,89/5,18,91/5,92/5,
        93/5,94/5,19,96/5,97/5,98/5,99/5,20,101/5,102/5,103/5,104/5,21,106/5,107/5,108/5,109/5,22,111/5,112/5,113/5,114/5,23,116/5,117/5,118/5,
        119/5,24,121/5,122/5,123/5,124/5,25,126/5,127/5,128/5,129/5,26]

For each angle t computes circle ( list for draw2d). It gives a new list Circles

 Circles : map (GiveCircle, tt)$ 

Command draw2d takes list Circles and draw all circles. Commands from draw package accepts list as an input.

Algorithm

• compute a list of angles
• For each angle t from list tt compute a point ${\displaystyle z_{t}}$
• for each point ${\displaystyle z_{t}}$ compute and draw osculating circle

Maxima CAS src code

/*


http://mathworld.wolfram.com/OsculatingCircle.html
The osculating circle of a curve C at a given point  P 
is the circle that has the same tangent as C at point P as well as the same curvature. 



https://en.wikipedia.org/wiki/Archimedean_spiral
https://www.mathcurve.com/courbes2d.gb/archimede/archimede.shtml

https://www.mathcurve.com/courbes2d.gb/enveloppe/enveloppe.shtml

the osculating circles of an Archimedean spiral. There is no need to trace the envelope...

http://xahlee.info/SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html

The tangent circles of Archimedes's spiral are all nested. need to proof that archimedes spiral's osculating circles are nested inside each other.

https://arxiv.org/abs/math/0602317
https://www.researchgate.net/publication/236899971_Osculating_Curves_Around_the_Tait-Kneser_Theorem



Osculating Curves: Around the Tait-Kneser Theorem
March 2013The Mathematical Intelligencer 35(1):61-66
DOI: 10.1007/s00283-012-9336-6
Elody GhysElody GhysSerge TabachnikovSerge TabachnikovVladlen TimorinVladlen Timorin

Osculating circles of a spiral. The spiral itself is not not drawn:
we see it as the locus of points where the circles are especially close to each
other.




https://math.stackexchange.com/questions/568752/curvature-of-the-archimedean-spiral-in-polar-coordinates

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




*/


kill(all);
remvalue(all);
ratprint:false;


/* ---------- functions ---------------------------------------------------- */




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

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

ToPoints(myList):= points(map(draw_f , myList))$








f(t):= t*cos(t)$
g(t) :=t*sin(t)$


define(fp(t), diff(f(t),t,1));
define(fpp(t),	diff(f(t),t,2));
define(gp(t), diff(g(t),t,1));
define(gpp(t), diff(g(t),t,2));


/* 
 point of the Archimedean spiral
 
 
 
 t is angle in turns 
 1 turn = 360 degree = 2*Pi radians 
 
 
*/
give_spiral_point(t):= f(t)+ %i*g(t)$


/* The curvature of Archimedes' spiral is
http://mathworld.wolfram.com/ArchimedesSpiral.html

 */
GiveCurvature(t) := (2+t*t)/sqrt((1+t*t)*(1+t*t)*(1+t*t)) $


GiveRadius(t):= float(1/GiveCurvature(t));
/*
center of The osculating circle of a curve C at a given point  P = give_spiral_point(t)
*/
GiveCenter(T):= block(
	[x, y,f_, f_p, f_pp, g_, g_p, g_pp, n, d ],
	f_ : f(T),
	f_p : fp(T),
	f_pp : fpp(T),
	g_ : g(T),
	g_p : gp(T),
	g_pp : gpp(T),
	n : f_p*f_p + g_p*g_p, 
	d : f_p*g_pp - f_pp*g_p,
	x: f_ - g_p*n/d,
	y: g_ + f_p* n/d,
	return ( x+y*%i)
	
)$


GiveCircle(T):= block(
	[Center, Radius],
	Center : GiveCenter(T),
	Radius : GiveRadius(T),
	return(ellipse (float(realpart(Center)), float(imagpart(Center)), Radius, Radius, 0, 360))

)$ 





/* compute */

iMin:1;
iMax:130;
id:5;

tt: makelist(i/id, i, iMin, iMax)$

zz: map(give_spiral_point, tt)$ /* points of the spiral */

Circles : map (GiveCircle, tt)$

/* convert lists  to draw format */
points: ToPoints(zz )$



/* draw lists using draw package */

path:"~/maxima/batch/spiral/ARCHIMEDEAN_SPIRAL/a2/"$ /*  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,"spiral_rc13_", string(iMin),"_", string(iMax)),
  font_size = 13,
  font = "Liberation Sans", /* https://commons.wikimedia.org/wiki/Help:SVG#Font_substitution_and_fallback_fonts */
  title= "Osculating circles of the Archimedean spiral.\ The spiral itself is not not drawn: we see it as the locus of points where the circles are especially close to each other.",
    
  dimensions = [1000, 1000],
  /* points  of the spiral, if you want to check 
  point_type    = filled_circle,
  point_size    = 1,
  points_joined = true,
  points,*/
  /* circles */
  key = "",
  line_width = 1,
  line_type = solid,
  border = true, 
  nticks = 100, 
  color = red,
  fill_color = white,
  transparent = true,
  Circles
  
  
  
  )$
  

Licenciamento

Eu, titular dos direitos de autor desta obra, publico-a com a seguinte licença:

A utilização deste ficheiro é regulada nos termos da licença Creative Commons Atribuição-CompartilhaIgual 4.0 Internacional.

Pode:

• partilhar – copiar, distribuir e transmitir a obra
• recombinar – criar obras derivadas

De acordo com as seguintes condições:

• atribuição – Tem de fazer a devida atribuição da autoria, fornecer uma hiperligação para a licença e indicar se foram feitas alterações. Pode fazê-lo de qualquer forma razoável, mas não de forma a sugerir que o licenciador o apoia ou subscreve o seu uso da obra.
• partilha nos termos da mesma licença – Se remisturar, transformar ou ampliar o conteúdo, tem de distribuir as suas contribuições com a mesma licença ou uma licença compatível com a original.

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

see also

• benice-equation : nested-ellipses

references

1. ↑ Osculating curves: around the Tait-Kneser Theoremby E. Ghys, S. Tabachnikov, V. Timorin
2. ↑ mathworld.wolfram : OsculatingCircle