Archivo:Nested Ellipses.svg
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Sumario
Resumen
DescripciónNested Ellipses.svg |
English: Nested Ellipses , Parameters: a=5, b=4 theta=0.2617993877991494 r=0.9434598957108945 number of ellipses=61. "The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." [1] |
Fecha | |
Fuente | Trabajo propio |
Autor | Adam majewski |
Otras versiones |
|
SVG desarrollo InfoField | El código fuente de esta imagen SVG es válido. Este archivo fue creado con un editor de texto. Su código fuente puede contener información adicional. Please do not replace the simplified code of this file with a version created with Inkscape or any other vector graphics editor |
Algorithm
Ellipse centered at origin and not rotated
the equation of a ellipse:
- centered at the origin
- with width = 2a and height = 2b
So explicit equations :
The parameter t :
- is called the eccentric anomaly in astronomy
- is not the angle of 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:
- .
This rotates column vectors by means of the following matrix multiplication,
- .
Thus, the new coordinates (x′, y′) of a point (x, y) after rotation are
- .
result
Center is in the origin ( not shifted or not moved) and rotated:
- center is the origin z = (0, 0)
- is the angle measured from x axis
- The parameter t (called the eccentric anomaly in astronomy) is not the angle of with the x-axis
- a,b are the semi-axis in the x and y directions
Here
- is fixed ( constant value)
- t is a parameter = independent variable used to parametrise the ellipse
So
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)
Fix x, then find y:
scaled
Second is scaled by factor r[5]
where:
- is the tilt angle
Python source code
import math, io
def make_svg(x_offset, y_offset):
outs = []
n = 61
a = 6035
b = 4828
theta = 15
delta = (1.0 * a / b - 1.0 * b / a) * math.sin(math.radians(theta))
r = (1 + delta * delta / 4) ** 0.5 - delta / 2
# print(delta, r)
for i in range(n):
a_i = a * r ** i
b_i = b * r ** i
deg = (-theta * i) % 180
rad = math.radians(deg)
t = math.pi * 1.5 if deg == 0 else math.pi + math.atan(b_i * math.cos(rad) / (a_i * math.sin(rad)))
x = a_i * math.cos(rad) * math.cos(t) - b_i * math.sin(rad) * math.sin(t) + x_offset - 65
y = a_i * math.sin(rad) * math.cos(t) + b_i * math.cos(rad) * math.sin(t) + y_offset + 8
## formulae from http://math.stackexchange.com/questions/1889450/extrema-of-ellipse-from-parametric-form
# print(i, deg, t)
outs.append('M%.0f%s%.0fa%.0f,%.0f %.0f 1 0 1,0' % (x, '' if y < 0 else ',', y, a_i, b_i, deg))
return '''<?xml version="1.0"?>
<svg xmlns="http://www.w3.org/2000/svg" width="1500" height="1000" viewBox="%d %d 15000 10000">
<path d="%s" fill="none" stroke="#f00" stroke-width="9"/>
</svg>''' % (x_offset - 7500, y_offset - 5000, ''.join(outs))
# <path d="%s" fill="none" stroke="#f00" stroke-width="9" marker-mid="url(#m)"/>
# <marker id="m"><circle r="9"/></marker>
## Find shortest output and write to file
(x_offset_min, length_min) = (0, 99999)
for x_offset in range(-9999, 9999, 1):
length = len(make_svg(x_offset, 0))
if length_min > length: (x_offset_min, length_min) = (x_offset, length)
# print(x_offset, length)
print(x_offset_min, length_min)
(y_offset_min, length_min) = (0, 99999)
for y_offset in range(-9999, 9999, 1):
length = len(make_svg(0, y_offset))
if length_min > length: (y_offset_min, length_min) = (y_offset, length)
# print(y_offset, length)
print(y_offset_min, length_min)
with io.open(__file__[:__file__.rfind('.')] + '.svg', 'w', newline='\n') as f: ## *.* -> *.svg
f.write(make_svg(x_offset_min, y_offset_min))
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 : 500$ 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 = 'svg, 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 )$
Licencia
- Eres libre:
- de compartir – de copiar, distribuir y transmitir el trabajo
- de remezclar – de adaptar el trabajo
- Bajo las siguientes condiciones:
- atribución – Debes otorgar el crédito correspondiente, proporcionar un enlace a la licencia e indicar si realizaste algún cambio. Puedes hacerlo de cualquier manera razonable pero no de manera que sugiera que el licenciante te respalda a ti o al uso que hagas del trabajo.
- compartir igual – En caso de mezclar, transformar o modificar este trabajo, deberás distribuir el trabajo resultante bajo la misma licencia o una compatible como el original.
Postprocessing
File size was reduced -29% with https://svgoptimizer.com/
references
- ↑ Osculating curves: around the Tait-Kneser Theoremby E. Ghys, S. Tabachnikov, V. Timorin
- ↑ Nested Ellipses (Ellipse Whirl) by benice (C. J. Chen)
- ↑ math.stackexchange question: given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
- ↑ texample : rotated-polygons
- ↑ math.stackexchange question : given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
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15 dic 2020
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Fecha y hora | Miniatura | Dimensiones | Usuario | Comentario | |
---|---|---|---|---|---|
actual | 16:26 24 feb 2023 | 1500 × 1000 (2 kB) | Cmglee | Minimise by using <path> and searching for offsets minimising file size | |
12:08 24 feb 2023 | 1500 × 1000 (7 kB) | Cmglee | Use actual SVG ellipses | ||
18:51 23 feb 2023 | 1500 × 1000 (22 kB) | Mrmw | lower filesize | ||
17:15 15 dic 2020 | 1500 × 1000 (14,85 MB) | Soul windsurfer | Uploaded own work with UploadWizard |
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