File:Nested Ellipses.png
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Summary
DescriptionNested Ellipses.png |
English: Nested Ellipses or spiral galaxy or Ellipse Whirl: "a sequence of concentric and similar ellipses, so that each ellipse lies inside the previous ellipse and is tangent to it"[1]. Parameters: a=5, b=4 theta=0.2617993877991494 r=0.9434598957108945 number of spirals=70. "The spiral itself is not not drawn: we see it as the locus of points where the circles are especially close to each other." [2] |
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Adam majewski |
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Ellipse centered at origin and not rotated
![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ellipse-def0.svg/220px-Ellipse-def0.svg.png)
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
![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Counterclockwise_rotation.png/220px-Counterclockwise_rotation.png)
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
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 : 700$ 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 = 'png, 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 )$
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references
- ↑ Nested Ellipses (Ellipse Whirl) by benice (C. J. Chen)
- ↑ 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
- ↑ math.stackexchange question : given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
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