Skeda:Osculating circles of the Archimedean spiral.svg
Dokument origjinal (skedë SVG, fillimisht 1.000 × 1.000 pixel, madhësia e skedës: 108 KB)
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Lëndë
Përmbledhje
PërshkrimiOsculating 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 | |
Burimi | Punë e juaja |
Autori | Adam majewski |
Versione tjera |
|
SVG genesis InfoField | This plot was created with Gnuplot. This plot uses embedded text that can be easily translated using a text editor. |
Përmbledhje
Math equations
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
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
- for each point 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 )$
Licencim
- Je i lirë të:
- ta shpërndani – ta kopjoni, rishpërndani dhe përcillni punën
- t’i bëni “remix” – të përshtatni punën
- Sipas kushteve të mëposhtme:
- atribuim – Duhet t’i jepni meritat e duhura, të siguroni një lidhje për tek licenca dhe të tregoni nëse janë bërë ndryshime. Këtë mund ta bëni në ndonjë mënyrë të arsyeshme, por jo në ndonjë mënyrë që sugjeron se licencuesi ju del zot juve apo përdorimit tuaj.
- share alike – Nëse bëni një “remix”, e shndërroni, ose ndërtoni duke u bazuar te materiali, duhet t’i shpërndani kontributet tuaja sipas të njëjtës licencë ose një të tille të përputhshme me origjinalen.
see also
references
- ↑ Osculating curves: around the Tait-Kneser Theoremby E. Ghys, S. Tabachnikov, V. Timorin
- ↑ mathworld.wolfram : OsculatingCircle
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osculating circle anglisht
27 maj 2019
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