Summary

DescriptionNormal lines to the ellipse.svg	English: Normal lines to the ellipse.[1] The evolute of the ellipse ( = Stretched astroid) can be seen. The streched astroid itself is not not drawn: we see it as the locus of points where the lines are especially close to each other. Stretched = "squashed" astroid = dilated in one direction only= the Lamé curve. Is it generalization of the astroid curve)
Date	22 June 2019
Source	Own work
Author	Adam majewski
Other versions	Evolute1.gif Ellipse evolute 00.svg Evolute-elli.svg Nested Ellipses.svg Weisstein, Eric W. "Ellipse Evolute." From MathWorld--A Wolfram Web Resource
SVG development InfoField	The SVG code is valid.   This plot was created with Gnuplot.    This plot uses embedded text that can be easily translated using a text editor.

Licensing

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

Maxima CAS src code

 /*

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

------------text output =--------------
maxima

Maxima 5.41.0 http://maxima.sourceforge.net
using Lisp GNU Common Lisp (GCL) GCL 2.6.12
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) batch("e.mac");

read and interpret file: #p/home/a/maxima/batch/curves/ellipse/e2/e.mac
(%i2) kill(all)
(%o0)                                done
(%i1) remvalue(all)
(%o1)                                 []
(%i2) ratprint:false
(%o2)                                false
(%i3) display2d:false
(%o3) false
(%i4) ellipse:x^2/a^2+y^2/b^2-1
(%o4) y^2/b^2+x^2/a^2-1
(%i5) GiveSlopeEquation(e):=block([deriv,m],depends(y,x),deriv:diff(e,x),
                        m:solve(deriv,'diff(y,x)),m:m[1],m:rhs(m),return(m))
(%i6) define(m(x,y),GiveSlopeEquation(ellipse))
(%o6) m(x,y):=-(b^2*x)/(a^2*y)
(%i7) t:0.33
(%o7) 0.33
(%i8) a:2
(%o8) 2
(%i9) b:1
(%o9) 1
(%i10) x0:a*cos(t)
(%o10) 1.892084687056774
(%i11) y0:b*sin(t)
(%o11) 0.3240430283948683
(%i12) m0:m(x0,y0)
(%o12) -1.459748028239586
(%i13) tangent:y = m0*(x-x0)+y0
(%o13) y = 0.3240430283948683-1.459748028239586*(x-1.892084687056774)
(%i14) normal:y = (-(x-x0))/m0+y0
(%o14) y = 0.3240430283948683-0.6850497350600779*(1.892084687056774-x)
(%i15) load(draw)
(%o15) "/usr/share/maxima/5.41.0/share/draw/draw.lisp"
(%i16) path:"~/maxima/batch/curves/ellipse/e2/"
(%i17) draw2d(user_preamble = "set key top right; unset mouse; ",
              terminal = 'svg,file_name = sconcat(path,"ell3"),font_size = 13,
              font = "Liberation Sans",
              title = "Normal and tangent line to the ellipse ",
              proportional_axes = 'xy,xrange = [-3,3],yrange = [-3,3],
              ip_grid = [100,100],color = blue,key = "circle",
              implicit(ellipse,x,-4,4,y,-4,4),key = "tangent",color = red,
              implicit(tangent,x,-4,4,y,-4,4),key = "normal",color = green,
              implicit(normal,x,-4,4,y,-4,4),key = "point",
              point_type = filled_circle,color = black,points([[x0,y0]]))
(%o17) "e.mac"


*/


kill(all);
remvalue(all);
ratprint:false; /* a message informing the user of the conversion of floating point numbers to rational numbers is displayed. */
display2d:false;


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


/* ellipse in implicit form */
ellipse : x^2/a^2 + y^2/b^2 - 1;


 
/* 
https://www.linuxjournal.com/content/maximum-calculus-maxima

implicit differentiation
 compute m equation 
 find dy/dx of implicit equation e 
 
*/ 
GiveSlopeEquation(e):=block( 
	[deriv,m],
   	depends(y,x),
   	deriv:diff(e,x),
	m:solve(deriv,'diff(y,x)),
	m:m[1],
	m:rhs(m),
	return(m)
	)$
/*

m(x,y):=-x/y

*/

define(
	m(x,y), 
	GiveSlopeEquation(ellipse));
	
	
	
	
	
	
	
	
give_line(t):=block(
	[x0, y0, m0],
	t:float(t),
	
	/* compute point z = x+y*i of the ellipse */
	x0: a*cos(t),
	y0: b*sin(t),
	/* compute slope */
	m0 : m(x0,y0),
	
	return(explicit(-(x-x0)/m0 + y0,x,-4,4)) /* normal */
)$
	
	
	







/* ellipse parameters */
a:2;
b:1;
iMin:1; /* cot: argument 0.0 isn't in the domain of cot. #0: give_line(t=0) */ 

iMax:200;


/* --------------------computations  -------------------------------*/

tt: makelist(2*%pi*i/iMax, i, iMin, iMax)$ /* list of angles from 0 to 2 Pi */

lines: map(give_line,tt)$




/* ------- draw --------------------------------------- */
load(draw);
path:"~/maxima/batch/curves/ellipse/en/"$ /*  pwd, if empty then file is in a home dir , path should end with "/" */


draw2d(
  user_preamble="set key top right; unset mouse; ",
  terminal  = 'svg,
  file_name = sconcat(path,"en200_2000_e"),
  font_size = 50,
  font = "Liberation Sans", /* https://commons.wikimedia.org/wiki/Help:SVG#Font_substitution_and_fallback_fonts */
  title= "Normal lines to the ellipse ",
  proportional_axes = 'xy, 
  dimensions = [2000,2000],
  xrange = [-5,5],
  yrange = [-5,5],
  ip_grid=[200,200],
  color= blue,
  key = "",
  lines,
  color=black,
  implicit(ellipse, x,-4,4, y,-4,4) 
  
  
  )$
  

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