File:Osculating circles of the Archimedean spiral.svg
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Summary
[edit]DescriptionOsculating 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] |
Date | |
Source | Own work |
Author | Adam majewski |
Other versions |
|
SVG development InfoField | ![]() This plot was created with Gnuplot. ![]() This plot uses embedded text that can be easily translated using a text editor. |
Summary
[edit]Math equations
[edit]Point of an Archimedean spiral for angle t
The curvature of Archimedes' spiral is
Radius of osculating circle is[2]
Center of osculating circle is
where
is first derivative
is a second derivative
notes
[edit]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
[edit]- compute a list of angles
- For each angle t from list tt compute a point
- for each point
compute and draw osculating circle
Maxima CAS src code
[edit]/* 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 )$
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