{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 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" ] }, { "cell_type": "markdown", "metadata": {}, "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" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 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" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "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": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "c=triangloid(1, 1, 0.2, -0.4)\n", "c.draw(label='colloid')\n", "title('Naive idea')\n", "for seg in c:\n", " seg.extend(0.1).draw()\n", "axis('equal');" ] }, { "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, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "c.draw(label='colloid')\n", "title('Considering Turning points')\n", "c.offset_raw(0.1).draw_each()\n", "axis('equal');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 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" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "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, "metadata": {}, "outputs": [ { "data": { "text/html": [ "

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	Area	Offset	Perimeter
0	2.423661	0.0	7.423733
1	3.170664	0.1	7.516340
2	3.926929	0.2	7.608948
3	4.699730	0.3	7.938916
4	5.521669	0.4	8.510362

\n", "

" ], "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})" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }