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HomotopyContinuation.jl Demo for ICMS 2020
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icms2020-software-demo.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<img src=\"https://www.juliahomotopycontinuation.org/images/logo_transparent_bg.png\" style=\"height:140px;\"/>\n",
"\n",
"\n",
"[HomotopyContinuation.jl](https://www.juliahomotopycontinuation.org) is package for the numerical solution of systems of polynomial equations.\n",
"In this notebook we give an overview over some of the features of **HomotopyContinuation.jl** with examples coming from a range of applications in nonlinear algebra.\n",
"\n",
"To find out more about **HomotopyContinuation.jl** (including documentation, guides and more examples) take a look at our homepage <a href=\"https://www.juliahomotopycontinuation.org\">JuliaHomotopyContinuation.org</a>."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The notebook contains several examples. Feel free to take a look at any subset of them.\n",
"* **[A simple example](#A-simple-example)**\n",
"* **[Tritangents to a quartic surface](#Tritangents-to-a-quartic-surface)**\n",
"* **[The bottlenecks of an algebraic variety](#The-bottlenecks-of-an-algebraic-variety)**\n",
"* **[Steiner's conic problem](#Steiner's-conic-problem)**\n",
"* **[Gaussian mixtures and the method of moments](#Gaussian-mixtures-and-the-method-of-moments)**\n",
"* **[The degree of Stiefel manifolds](#The-degree-of-Stiefel-manifolds)**\n",
"* **[Tritangent planes to a quartic surface](#Tritangent-planes-to-a-quartic-surface)**\n",
"* **[Witness sets](#Witness-sets)**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Installation of notebook"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Requirements\n",
"\n",
"You need to have **Julia 1.4** installed and internet access (to install the packages).\n",
"[Here](https://www.juliahomotopycontinuation.org/guides/installation/) is an installation guide.\n",
"To install the Julia Jupyter notebook kernel you have to install the `IJulia.jl` package.\n",
"This can be done as follows in running Julia session:\n",
"```julia\n",
"using Pkg\n",
"Pkg.add(\"IJulia\")\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Instantiating\n",
"\n",
"This notebook is accompanied by two files: `Project.toml` and `Manifest.toml`. The `Project.toml` lists all packages necessary to run this notebook and the `Manifest.toml` records a working set of all (transitive) dependcies for which this notebook worked for me.\n",
"You can install an identical copy of my environment by executing"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"import Pkg\n",
"Pkg.instantiate()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Note:** This notebook is written for HomotopyContinuation.jl **2.0** which was only recently released."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# A simple example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We are interested in the numerical solution of systems of polynomial equations like\n",
"$$\\begin{array}{rl} -x^5y &= (x^4 + y^4 - 1)(x^2 + y^2 - 2) \\\\\\\\ \\frac12 &= x^2+2xy^2 - 2y^2 . \\end{array}$$\n",
"\n",
"Equivalently, we can see the solutions of this system of equations as the common zero set $V(f_1,f_2)$ of the polynomials\n",
"$$\n",
" f_1(x,y) = (x^4 + y^4 - 1)(x^2 + y^2 - 2) + x^5y \\quad \\text{ and } \\quad f_2(x,y) = x^2+2xy^2 - 2y^2 - \\frac12 .\n",
"$$\n",
"The common zero set $V(f_1,f_2)$ is also called a *variety*.\n",
"\n",
"<figure>\n",
"<p style=\"text-align:center;\">\n",
"<img alt=\"simple-example\" src=\"https://www.juliahomotopycontinuation.org/images/simple-example.png\" style=\"height:400px\"/>\n",
"</p>\n",
" <figcaption style=\"text-align: center;\" >The zero set of $f_1$ in <span style=\"color:steelblue\">blue</span> and the zero set of $f_2$ in <span style=\"color:indianred\">red</span>. Their common zero set is depicted in <span style=\"color:black\">black</span>.</figcaption>\n",
"\n",
"</figure>\n",
"\n",
"\n",
"From the figure we can see that $f_1$ and $f_2$ have 4 common zeros.\n",
"\n",
"Let's compute them."
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"System of length 2\n",
" 2 variables: x, y\n",
"\n",
" x^5*y + (-2 + x^2 + y^2)*(-1 + x^4 + y^4)\n",
" -0.5 + 2*x*y^2 + x^2 - 2*y^2"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using HomotopyContinuation\n",
"\n",
"# declare variables x and y\n",
"@var x y\n",
"\n",
"# define the polynomials\n",
"f₁ = (x^4 + y^4 - 1) * (x^2 + y^2 - 2) + x^5 * y\n",
"f₂ = x^2+2x*y^2 - 2y^2 - 1/2\n",
"# declare the system\n",
"F = System([f₁, f₂])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can use the blackbox `solve` routine to compute the isolated solutions of $F(x,y) = 0$."
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Result with 18 solutions\n",
"========================\n",
"• 18 paths tracked\n",
"• 18 non-singular solutions (4 real)\n",
"• random_seed: 0x8671c247\n",
"• start_system: :polyhedral\n"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"result = solve(F)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, the result reports that we found **18** solutions and **4 real** solutions. Why do we have 14 solutions more than expected? The reason is that we do not compute the common zero set of $f_1$ and $f_2$ over the real numbers, but over the **complex numbers**. Although there are usually more complex solutions than real solutions, this makes the problem of computing all real solutions *much* easier.\n",
"\n",
"> The shortest path between two truths in the real domain passes through the complex domain.\n",
"> -- <cite>J. Hadamard</cite>\n",
"\n",
"We can extract the coordinates of the four real solutions from the `result`"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4-element Array{Array{Float64,1},1}:\n",
" [-1.671421392838003, 0.6552051858720408]\n",
" [-0.9368979667963298, 0.31228408173860095]\n",
" [0.8209788924342627, -0.6971326459489458]\n",
" [0.8999179208471728, -1.2441827613422727]"
]
},
"execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"real_solutions(result)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that the \"realness\" of these 4 solutions is here only determined heuristically. However, we could also certify this."
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"CertificationResult\n",
"===================\n",
"• 18 solutions given\n",
"• 18 certified solutions (4 real)\n",
"• 18 distinct certified solutions (4 real)"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"certify(F, result)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The bottlenecks of an algebraic variety"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<figure>\n",
"<img src=\"https://www.juliahomotopycontinuation.org/images/curve_reach_bottlenecks.png\" style=\"width:400px;display: block;margin-left: auto;margin-right: auto;\">\n",
"<figcaption style=\"text-align:center; color:gray;\">\n",
"A place curve and its bottlenecks dashed in gray. The narrowest bottleneck is colored <span style=\"color:tomato\">red</span>.\n",
"</figcaption>\n",
"</figure>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Bottlenecks of of an algebraic variety $X = V(f)$ are pairs of points $(x,y) \\in X\\times X$, $x \\ne y$, such that $x-y$ is in the normal space $\\mathrm{N}_x X$ and in the normal space $\\mathrm{N}_y X$.\n",
"[Eklund](https://arxiv.org/abs/1804.01015) and [di Rocco et. al.](https://arxiv.org/abs/1904.04502) discuss the algebraic equations of bottlenecks. For an hypersurface given by $f=0$ the equations are\n",
"\n",
"$$f(x) = 0, \\quad \\nabla f(x) \\lambda = x - y, \\quad f(y) = 0,\\quad \\nabla f(y) \\mu = x - y$$\n",
"\n",
"where $\\nabla f(x)$ denotes the [gradient](https://en.wikipedia.org/wiki/Gradient) of $f$ at $x$ and $\\lambda$ and $\\mu$ are additional variables. The first equation defines $x\\in X$ and the second equation defines $x-y \\in \\mathrm{N}_x X$. The third equation defines $y\\in X$ and the fourth equation defines $x-y \\in \\mathrm{N}_y X$.\n",
"\n",
"Additionally, we want to enforce the condition $x \\ne y$. Algebraically, this corresponds to saturating the ideal with respect to $\\prod_i (x_i -y_i)$. We can accomplish this by having the additional equation\n",
"$$s \\prod_i (x_i -y_i) - 1\\,.$$\n",
"\n",
"Here is the code to formulate the system in HomotopyContinuation.jl."
]
},
{
"cell_type": "code",
"execution_count": 124,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"System of length 7\n",
" 7 variables: x₁, x₂, y₁, y₂, λ, μ, s\n",
"\n",
" -5 + (-1 + x₁^2 + x₂^2)*(1 + x₂ - x₂^2*x₁ + x₁^3)^2 + x₂^2\n",
" λ*(2*x₁*(1 + x₂ - x₂^2*x₁ + x₁^3)^2 + 2*(3*x₁^2 - x₂^2)*(-1 + x₁^2 + x₂^2)*(1 + x₂ - x₂^2*x₁ + x₁^3)) - (x₁ - y₁)\n",
" λ*(2*x₂ + 2*x₂*(1 + x₂ - x₂^2*x₁ + x₁^3)^2 + 2*(-1 + x₁^2 + x₂^2)*(1 + x₂ - x₂^2*x₁ + x₁^3)*(1 - 2*x₂*x₁)) - (x₂ - y₂)\n",
" -5 + (-1 + y₁^2 + y₂^2)*(1 + y₂ - y₁*y₂^2 + y₁^3)^2 + y₂^2\n",
" μ*(2*y₁*(1 + y₂ - y₁*y₂^2 + y₁^3)^2 + 2*(-1 + y₁^2 + y₂^2)*(3*y₁^2 - y₂^2)*(1 + y₂ - y₁*y₂^2 + y₁^3)) - (x₁ - y₁)\n",
" μ*(2*y₂ + 2*y₂*(1 + y₂ - y₁*y₂^2 + y₁^3)^2 + 2*(-1 + y₁^2 + y₂^2)*(1 + y₂ - y₁*y₂^2 + y₁^3)*(1 - 2*y₁*y₂)) - (x₂ - y₂)\n",
" -1 + s*(x₁ - y₁)*(x₂ - y₂)"
]
},
"execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using HomotopyContinuation\n",
"@var x[1:2]\n",
"f = (x[1]^3 - x[1]*x[2]^2 + x[2] + 1)^2 * (x[1]^2 + x[2]^2 - 1) + x[2]^2 - 5\n",
"\n",
"bn_system = let\n",
" @var y[1:2] λ μ s v[1:2]\n",
" ∇f = differentiate(f, x)\n",
" System([\n",
" f\n",
" ∇f * λ - (x - y)\n",
" f(x => y)\n",
" ∇f(x => y) * μ - (x - y)\n",
" s * (prod(x - y)) - 1\n",
" ]; variables = [x; y; λ; μ; s])\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, we can pass the system to `solve` to compute it's solutions."
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\r",
"\u001b[32mTracking 3600 paths... 100%|████████████████████████████| Time: 0:00:03\u001b[39m\n",
"\u001b[34m # paths tracked: 3600\u001b[39m\n",
"\u001b[34m # non-singular solutions (real): 1726 (104)\u001b[39m\n",
"\u001b[34m # singular endpoints (real): 0 (0)\u001b[39m\n",
"\u001b[34m # total solutions (real): 1726 (104)\u001b[39m\n"
]
},
{
"data": {
"text/plain": [
"Result with 1726 solutions\n",
"==========================\n",
"• 3600 paths tracked\n",
"• 1726 non-singular solutions (104 real)\n",
"• random_seed: 0xd4926db1\n",
"• start_system: :polyhedral\n"
]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bottleneck_result = solve(bn_system)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see that we find $1726$ solutions of which $104$ are real. Let's extract the x and y coordinates of those 104 solutions."
]
},
{
"cell_type": "code",
"execution_count": 114,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"104-element Array{Array{Float64,1},1}:\n",
" [0.7825029522761754, 1.489379650889989, 0.8380646154550541, 1.6160838973346432]\n",
" [1.1727790894167083, 2.079548084019588, 0.9785760124466275, -1.3767262787162167]\n",
" [-0.6851397617457625, -1.6953936841180695, 1.7889119199177603, 2.2353101573649563]\n",
" [0.8051038425888838, 1.6493132196797127, 1.7897879294281294, 2.234430974012365]\n",
" [0.8879431408206833, -1.3763863660933688, 0.7510886854338582, 1.4942776321034474]\n",
" [-0.8752973020567416, 1.0128271793465617, -0.24577322542521296, -2.2356250461941105]\n",
" [-0.7450302592028445, -1.6620936795974433, 0.7844050532614113, 1.488501299376511]\n",
" [0.8090380570229553, 1.4673626566954847, 1.7896119628546143, 2.2346881132415044]\n",
" [1.249273301132837, 0.034496023798154876, -1.4349379845123864, -0.23066233904808647]\n",
" [1.7884992969474989, 2.2355321564460655, -0.26164294961659484, -2.233441781967787]\n",
" [0.8624201877461506, 1.611647850351416, 0.9148150341636954, -1.375508721873141]\n",
" [0.8443605974629239, 1.6138173164101104, -0.25728006214521437, -2.235069554213805]\n",
" [1.002067622601686, 2.1111696300581775, 0.8428392881077604, 1.6142785846338044]\n",
" ⋮\n",
" [0.7778198623620384, 1.7105900399285934, -1.6293890276038359, 0.83577503958792]\n",
" [-0.24577322542521304, -2.235625046194111, -0.8752973020567415, 1.0128271793465617]\n",
" [0.6774464378123398, -1.4093566274472473, -0.015019103577389885, 1.278422162025404]\n",
" [-0.2616429496165948, -2.233441781967787, 1.7884992969474953, 2.2355321564460633]\n",
" [-2.1011362415553583, -2.2359688037139565, -1.1488357570360546, 1.0214171129679994]\n",
" [1.0183579849477273, 2.1061658652352877, -0.2574702287550515, -2.2350143949687458]\n",
" [-2.0999408233730934, -2.2360480889624266, -2.508134201914099, 2.2360639480106883]\n",
" [1.789276191067831, 2.235040748059079, -2.101735284149737, -2.2356525790884763]\n",
" [0.7525444080543093, 2.228955728598188, -0.785213792929956, -1.6443817832836278]\n",
" [-2.50830133053216, 2.236030137324172, -1.2072829155001181, -1.6423296744810023]\n",
" [1.7899679456461044, 2.234061374567233, -1.6502646297133257, 0.8911663896744955]\n",
" [-0.7462144344691877, -1.661520421101156, 0.8364325052512863, 1.6168354283844606]"
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bn_real_solutions = map(x -> x[1:4], real_solutions(bottleneck_result))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You may have noticed that our polynomial system actually overcounts the number of bottlenecks by a factor of 2. Namely, for each bottleneck pair $(x,y) \\in X \\times X$ also $(y, x) \\in X \\times X$ is a solution, but they represent the same bottleneck. We can eliminate them by using the `unique_points` function"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"52-element Array{Array{Float64,1},1}:\n",
" [0.7825029522761754, 1.489379650889989, 0.8380646154550541, 1.6160838973346432]\n",
" [1.1727790894167083, 2.079548084019588, 0.9785760124466275, -1.3767262787162167]\n",
" [-0.6851397617457625, -1.6953936841180695, 1.7889119199177603, 2.2353101573649563]\n",
" [0.8051038425888838, 1.6493132196797127, 1.7897879294281294, 2.234430974012365]\n",
" [0.8879431408206833, -1.3763863660933688, 0.7510886854338582, 1.4942776321034474]\n",
" [-0.8752973020567416, 1.0128271793465617, -0.24577322542521296, -2.2356250461941105]\n",
" [-0.7450302592028445, -1.6620936795974433, 0.7844050532614113, 1.488501299376511]\n",
" [0.8090380570229553, 1.4673626566954847, 1.7896119628546143, 2.2346881132415044]\n",
" [1.249273301132837, 0.034496023798154876, -1.4349379845123864, -0.23066233904808647]\n",
" [1.7884992969474989, 2.2355321564460655, -0.26164294961659484, -2.233441781967787]\n",
" [0.8624201877461506, 1.611647850351416, 0.9148150341636954, -1.375508721873141]\n",
" [0.8443605974629239, 1.6138173164101104, -0.25728006214521437, -2.235069554213805]\n",
" [1.002067622601686, 2.1111696300581775, 0.8428392881077604, 1.6142785846338044]\n",
" ⋮\n",
" [-2.1022747595061637, -2.2341790872009333, 2.3515877311240057, -2.235182629731724]\n",
" [1.3751177924505709, -0.9073020100028683, -2.1021315295268033, -2.2350721977383485]\n",
" [-1.1488357570360546, 1.0214171129679994, -2.101136241555358, -2.235968803713956]\n",
" [-1.2072829155001181, -1.6423296744810023, -2.50830133053216, 2.236030137324172]\n",
" [-0.785213792929956, -1.6443817832836278, 0.7525444080543093, 2.228955728598188]\n",
" [-1.6293890276038359, 0.83577503958792, 0.7778198623620384, 1.7105900399285934]\n",
" [-0.1800526800404879, -2.1490348361484717, -1.556557542552337, -1.4655206169516484]\n",
" [-2.50851579551267, 2.235894015761669, 2.3513843748697214, -2.235829427171197]\n",
" [-0.25747022875505154, -2.2350143949687453, 1.0183579849477278, 2.106165865235289]\n",
" [-1.45316313826534, -1.1966621073057155, 2.3515468028200743, -2.235523358809711]\n",
" [-1.6502646297133257, 0.8911663896744955, 1.7899679456461044, 2.234061374567233]\n",
" [-2.5084403937643804, 2.2359604167721443, 0.2254076390340674, -1.617492692422356]"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# declare that a solution x,y is identical to a solution y,x\n",
"switch = xy -> [xy[3:4];xy[1:2]]\n",
"bottlenecks = unique_points(bn_real_solutions, group_action = switch)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's visualize our bottlenecks in order to see whether we didn't make any mistakes.\n",
"But for this, we also need to visualize our plane curve $X = V(f)$. While we could use some plotting package for this, let's simply *sample* points of $X$ by intersecting $X$ with randomly drawn lines."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For this, we first intersect $X$ with one random complex linear space."
]
},
{
"cell_type": "code",
"execution_count": 118,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8-element Array{Array{Complex{Float64},1},1}:\n",
" [0.5380468817793169 + 0.6335195631612976im, 1.3398812440632095 + 0.20302845766159375im]\n",
" [0.4192351373601692 + 0.9886114321808671im, -1.4467035942883621 + 0.33069651052917204im]\n",
" [0.1818277085935789 + 0.5481972185802627im, 0.9764616069833343 - 2.5014811283850196im]\n",
" [0.5366231506492747 + 0.8590636933739537im, -0.23129615193133934 + 0.7986539225676544im]\n",
" [0.43874736956528665 + 0.4890123452596868im, 2.0775034352493122 - 0.8749845383558725im]\n",
" [0.3247959068402389 + 0.8512860834632762im, -0.745941351525879 - 0.6942601388110293im]\n",
" [0.06928077395818273 + 0.17280993861528515im, 3.28295403431093 - 4.291398017491945im]\n",
" [0.18409272357220804 + 0.17536540161437555im, 3.573431694749905 - 3.48668420560232im]"
]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L₀ = rand_subspace(2, codim = 1)\n",
"X_L₀ = solutions(solve(System([f]), target_subspace = L₀))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"These *8* solutions we can then use to sample points. We have a compact notation for performing a parameter homotopy for many different target subspaces, which also allows to directly transform the results."
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\r",
"\u001b[32mSolving for 1000 parameters... 100%|████████████████████| Time: 0:00:01\u001b[39m\n",
"\u001b[34m # parameters solved: 1000\u001b[39m\n",
"\u001b[34m # paths tracked: 8000\u001b[39m\n",
"\u001b[34m # results: 2274\u001b[39m\n"
]
},
{
"data": {
"text/plain": [
"2274"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# sample 1000 random (real) linear subspaces\n",
"target_subspaces = [rand_subspace(2, codim = 1, real = true) for k in 1:1000]\n",
"\n",
"samples = solve(System([f]), X_L₀;\n",
" start_subspace = L₀,\n",
" target_subspaces = target_subspaces,\n",
" transform_result = (R,L) -> real_solutions(R),\n",
" flatten = true\n",
")\n",
"length(samples)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"No we can plot the sample points together with our computed bottlenecks."
]
},
{
"cell_type": "code",
"execution_count": 128,
"metadata": {},
"outputs": [
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"<circle clip-path=\"url(#clip120)\" cx=\"2181.76\" cy=\"95.2176\" r=\"23\" fill=\"#1e90ff\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"0\"/>\n",
"<circle clip-path=\"url(#clip120)\" cx=\"2181.76\" cy=\"95.2176\" r=\"23\" fill=\"#1e90ff\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"0\"/>\n",
"<circle clip-path=\"url(#clip120)\" cx=\"2181.76\" cy=\"95.2176\" r=\"23\" fill=\"#1e90ff\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"0\"/>\n",
"<circle clip-path=\"url(#clip120)\" cx=\"2181.76\" cy=\"95.2176\" r=\"23\" fill=\"#1e90ff\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"0\"/>\n",
"<circle clip-path=\"url(#clip120)\" cx=\"2181.76\" cy=\"95.2176\" r=\"23\" fill=\"#1e90ff\" fill-rule=\"evenodd\" fill-opacity=\"1\" strok
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