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"# 二维胶粒的耗尽力\n",
"+ slug: two-dimension-depletion-force-between-colloids\n",
"+ date: 2015-09-15\n",
"+ tags: soft matter"
]
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{
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"source": [
"## 基本数据结构\n",
"+ 曲线由分段圆弧组成\n",
"+ 直线可以看作是一种特殊的半径为无穷大的圆弧"
]
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{
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"### 优良性质\n",
"__引理__ 分段圆弧扩展之后的曲线依然是分段圆弧\n",
"+ 直线扩展后还是直线\n",
"+ 半径为$R$的圆扩展$d$后是半径为$R+d$的圆\n",
"+ 转接点扩展$d$后产生的新的曲线是圆$d$\n",
"+ 综上，无论怎么扩展，扩展前后曲线类型不变"
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],
"source": [
"from deplete.colloids import *\n",
"L, C = Line([0,2], [3,5]), Circle(3, [0,0], [0, pi/2])\n",
"subplot(121)\n",
"L.draw()\n",
"L.extend(0.1).draw()\n",
"axis('square');\n",
"subplot(122)\n",
"C.draw()\n",
"C.extend(0.1).draw()\n",
"axis('square');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 扩展曲线\n",
"### Naive的扩展方式\n",
"+ 设定好定向，对应图像轮廓由一系列矢量弧段$a_1, a_2,\\ldots a_n$组成\n",
"+ 普通段普通扩展$a_i\\rightarrow a_i'$\n",
" + 区分角度增加/减少的区别\n",
" - 角度**增加**的时候半径为正$r=|r|$\n",
" - 角度**减少**的时候半径为负$r=-|r|$\n",
" + 做变换$r\\rightarrow r'=r+d$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
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" seg.extend(0.1).draw()\n",
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 处理转折点\n",
"+ 如果顺拐，即$a_i\\times a_{i+1}>0$，转折段扩展出圆弧\n",
"+ 如果逆拐，即$a_i\\times a_{i+1}<0$，转折段可以扩展出圆弧，但是直线更适合处理，因为终究需要剪除"
]
},
{
"cell_type": "code",
"execution_count": 4,
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"### 终极扩展方式\n",
"+ __问题__：我们现在的曲线，已经包含了扩展后的边界，但是有部分面积被环绕了多次\n",
"+ __解决方法__：剪除重复的正面积，保留真正的边界\n",
"+ __简化__方法：即最大的正边界\n",
"+ __BUG__: 某些凹陷严重情形下扩展会有bug，蚀刻时也会有bug"
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"source": [
"off=linspace(0, 0.4, 5)\n",
"for o in off:\n",
" c.offset(o).draw(label='Offset {:.1f}'.format(o))\n",
"legend(loc=\"upper right\")\n",
"axis('equal');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 曲线相交\n",
"Depletion把两个当成一个整体"
]
},
{
"cell_type": "code",
"execution_count": 11,
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{
"data": {
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"<div>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
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" <th>Area</th>\n",
" <th>Offset</th>\n",
" <th>Perimeter</th>\n",
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" <tbody>\n",
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" <th>0</th>\n",
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" <th>1</th>\n",
" <td>3.170664</td>\n",
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" <th>2</th>\n",
" <td>3.926929</td>\n",
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" <th>3</th>\n",
" <td>4.699730</td>\n",
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" <tr>\n",
" <th>4</th>\n",
" <td>5.521669</td>\n",
" <td>0.4</td>\n",
" <td>8.510362</td>\n",
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"text/plain": [
" Area Offset Perimeter\n",
"0 2.423661 0.0 7.423733\n",
"1 3.170664 0.1 7.516340\n",
"2 3.926929 0.2 7.608948\n",
"3 4.699730 0.3 7.938916\n",
"4 5.521669 0.4 8.510362"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"area=[c.offset(o).area() for o in off]\n",
"peri=[c.offset(o).perimeter() for o in off]\n",
"pd.DataFrame({\"Offset\": off, \"Area\": area, \"Perimeter\": peri})"
]
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