列表中近邻点的过滤方法：用圆域与阈值选择点

几何画板及绘图内容加载中，请稍等……

2024-02-25 23:06

圆数列概率统计几何作图

所属专题 intersection: surface-surface, curve-curve

该范例演示如何在一组点中按距离阈值筛选“彼此足够近”的近邻点，并可视化保留/剔除的结果。核心数学概念包括平面欧氏距离、用圆（半径阈值）表示邻域、距离比较与集合筛选（去重与保留满足条件的元素），必要时结合概率统计或参数化半径的直观理解。作图要点：利用 GeoGebra 命令 Append、Sequence、Zip 构造点列表与配对数据，再用 Unique 去重；用 Circle 生成以目标点为圆心、给定阈值为半径的邻域圆；通过 KeepIf 与 RemoveUndefined 按条件保留落入圆域/满足近邻判据的点，并用 Length 与 Textfield 辅助计算列表长度与展示参数；对象方面体现了输入框 InputBox1/2、文本显示 text1–text6、以及用于半径/阈值的变量（如 ε）与各步骤生成的点/集合对象（如 lPP、lälPP 及其带撇版本）。适合中学阶段的算法化几何探索、信息学与数学融合课程，或用于大学基础几何编程与可视化教学。

GeoGebra 命令及参数

Append(C, {(-7.227122352568576, 6.3), (-7.43, 6.12), (-8.4, 7.4), (-7.53, 5.45), (-8.663587255344709, 5.7560425400543656), (7.670062882970545, 9.617954642623875), (16.63748163988375, -3.794521177257613), (1.480940077731411, 6.488422425606599), (3.304980892883012, -0.754002561420931), (-3.576833723657818, 15.671132474717396), (-4.194436781028301, -8.708008440411136), (-11.083561995415032, 5.337562142971605), (7.670062882970545, 9.617954642623875), (13.339976248350501, -10.641861678089121), (-0.003912146532369, 0.003057364501322), (-5.367206613941206, 6.978426222543205), (9.423030478007094, 2.151559322647076), (-4.149501888261364, -5.203310788936281), (13.33997624835086, 10.641861678088436), (-4.149501888261364, -5.203310788936281), (-6.5, 6.6), (14.103627785207602, -3.218885485469269), (-1.481524888897013, -6.490984650735204), (4.151140492734138, 5.205365533953871), (2.650124822430615, 2.11382520338954), (-9.423383705823298, -2.150011730466254), (-3.576833723657818, 15.671132474717396), (-7.67066327274682, -9.618707507590461), (-7.67066327274682, -9.618707507590461), (-14.482304084306326, -6.974310068854369), (-11.310307488728789, 9.019450370318966), (16.637481639884076, 3.794521177257068), (-3.304834344153331, -0.754644633357172), (3.576923096002452, 15.671524040543373), (7.557320220484236, 6.025746592197113), (-10.022315800760783, -12.567586505684634), (-0.000329292006409, -3.389899475872677), (5.996203959520251, 2.887619636098174), (-6.983533852871922, 4.637938034270736), (1.471118941670167, 3.054051018632799), (14.103627785207413, 3.218885485469204), (-12.302788674135952, 0.000000000000103), (-7.4, 61), (-14.1035516291821, -3.219219146817295), (3.576923096002452, 15.671524040543373), (0.004851262974027, 0.001057173468102), (-7.435410285261154, 4.927602414007428), (12.30182572370019, -0.000000000000051), (-7.267404945013875, 4.713250773002275), (4.193006590798404, 8.708697183726047), (-16.63623403438272, -3.799987294107673), (-7.53, 6.05), (-0.000171121035375, -14.466289795346938), (9.423030478006956, -2.151559322646906), (-0.002803344115093, 17.06470563541584), (6.276842082288939, 13.033602493027322), (16.074144845895862, 0.000000000000193), (-0.002803344115093, 17.06470563541584), (4.151140492734323, -5.205365533954343), (-10.022315800760783, -12.567586505684634), (1.480940077731411, 6.488422425606599), (-1.470525577977862, -3.05433676752634), (7.406624013808355, 15.373552192001993), (4.193006590798404, 8.708697183726047), (2.737628026035893, -11.994332076089227), (-7.139952617929733, 4.91601583881796), (14.103627785207413, 3.218885485469204), (-2.737413749405578, -11.993393268831783), (-1.470525577977912, 3.05433676752622), (-12.302788674135952, 0.000000000000103), (6.276842082288856, -13.03360249302747), (-3.304834344153293, 0.754644633357136), (-9.423383705823298, -2.150011730466254), (2.65012482243087, -2.113825203389885), (-2.737413749405578, -11.993393268831783), (-1.481524888897013, -6.490984650735204), (-7.670663272746861, 9.618707507590605), (-8.698346980913112, 7.239124164306228), (0.000793695698577, 9.66554242028682), (11.310094104287712, 9.01971794594481), (4.193006590798501, -8.708697183725626), (-6.276533732837609, 13.033750986296848), (-0.002803344115354, -17.064705635415883), (4.151140492734323, -5.205365533954343), (-7.556330498136631, -6.026987664766406), (-3.304834344153331, -0.754644633357172), (-3.304834344153293, 0.754644633357136), (-9.423383705823237, 2.150011730466243), (0.000793695698577, 9.66554242028682), (12.30182572370019, -0.000000000000051), (-11.083561995415025, -5.337562142971636), (-2.650535442846633, -2.11331030167522), (-7.178513973229819, 5.725638281528233), (-8.721520131292046, 5.217266793744119), (-7.556330498136631, -6.026987664766406), (-8.605654379397372, 5.443205009938739), (14.482665945541832, 6.974484332041084), (-6.276533732837609, 13.033750986296848), (-11.310307488728789, 9.019450370318966), (2.65012482243087, -2.113825203389885), (-5.372999901535939, 6.404890750664555), (11.084429583778933, -5.337979951507175), (10.022065385016155, 12.567272494272537), (-10.02231580076091, 12.567586505684472), (-5.529418666593751, 7.1406382751957524), (-0.002803344115354, -17.064705635415883), (5.996203959520218, -2.887619636098153), (10.022065385016333, -12.56727249427238), (3.304980892883135, 0.754002561420965), (2.737628026035849, 11.994332076089428), (11.310094104287712, 9.01971794594481), (-7.32, 6), (-7.401572562260538, -15.375984842855647), (-10.02231580076091, 12.567586505684472), (-0.000329292006234, 3.389899475872126), (16.637481639884076, 3.794521177257068), (0.000793695698457, -9.665542420286704), (-4.149501888261418, 5.203310788936329), (7.406624013808278, -15.373552192001977), (-1.48152488889687, 6.490984650734765), (-8.58827451661317, 5.593830487401818), (-5.355620038751739, 6.787247731916989), (-11.083561995415025, -5.337562142971636), (-8.825799307997254, 5.912461305112179), (6.657912415501918, -0.000000000000069), (-1.48152488889687, 6.490984650734765), (-0.000171121035503, 14.466289795346976), (-16.636234034382696, 3.79998729410738), (-5.772736745572568, 7.18119128835889), (-6.655284297980534, 0.2), (11.310094104287693, -9.019717945944542), (-7.401572562260836, 15.37598484285544), (-5.998571811588356, -2.888759933555225), (14.482665945541832, 6.974484332041084), (5.996203959520251, 2.887619636098174), (7.670062882970544, -9.617954642624042), (13.33997624835086, 10.641861678088436), (5.996203959520218, -2.887619636098153), (-7.25, 5.93), (-7.48, 6.8), (-7.4, 61), (-11.083561995415032, 5.337562142971605), (0.000793695698457, -9.665542420286704), (14.103627785207602, -3.218885485469269), (-4.194436781028413, 8.70800844041119), (-2.650535442846633, -2.11331030167522), (-14.482304084306225, 6.974310068854532), (16.074144845895862, 0.000000000000193), (6.276842082288856, -13.03360249302747), (-0.000329292006234, 3.389899475872126), (-16.07454648156653, -0.000000000000145), (-14.103551629182197, 3.219219146817304), (-8.530341640665833, 7.354989916200904), (-8, 6.8), (2.737628026035893, -11.994332076089227), (11.084429583778938, 5.337979951507497), (-1.470525577977912, 3.05433676752622), (-0.004829527286543, 0.001152403735828), (-14.1035516291821, -3.219219146817295), (-7.45, 61.3), (-5.662367951715511, 5.416763221814026), (11.084429583778933, -5.337979951507175), (1.480940077731417, -6.488422425606563), (1.471118941670276, -3.054051018632759), (-6.838701663003578, 4.800150086923283), (6.657912415501918, -0.000000000000069), (6.276842082288939, 13.033602493027322), (16.63748163988375, -3.794521177257613), (-7.556330498136652, 6.026987664766561), (-7.35, 5.85), (11.310094104287693, -9.019717945944542), (-0.000171121035503, 14.466289795346976), (11.084429583778938, 5.337979951507497), (-3.576833723657824, -15.671132474717346), (-8.605654379397372, 7.557754982016589), (1.480940077731417, -6.488422425606563), (10.022065385016333, -12.56727249427238), (-7.227122352568576, 6.387338342354329), (-11.310307488728865, -9.019450370318971), (4.151140492734138, 5.205365533953871), (7.670062882970544, -9.617954642624042), (0.004851262973816, -0.001057173468078), (-0.004829527286561, -0.001152403735806), (3.304980892883012, -0.754002561420931), (-6.655284297980534, 0.000000000000006), (-5.680044144056828, 6.497583352180297), (-13.34347196128419, -10.637478192717984), (-16.636234034382696, 3.79998729410738), (7.557320220484236, 6.025746592197113), (-14.482304084306225, 6.974310068854532), (7.406624013808355, 15.373552192001993), (13.339976248350501, -10.641861678089121), (-14.103551629182197, 3.219219146817304), (7.406624013808278, -15.373552192001977), (-4.194436781028301, -8.708008440411136), (-11.310307488728865, -9.019450370318971), (-8.762073144455183, 5.4779647355071415), (-7.58, 6.2), (14.4826659455416, -6.974484332041754), (-5.998571811588356, -2.888759933555225), (-14.482304084306326, -6.974310068854369), (-8.808419445213053, 7.337610053416703), (-13.34347196128419, -10.637478192717984), (-8.860559033565655, 5.727076102080696), (2.650124822430615, 2.11382520338954), (-6.5, 5.8), (2.737628026035849, 11.994332076089428), (-9.423383705823237, 2.150011730466243), (-8.988011360649798, 7.215951013927293), (4.193006590798501, -8.708697183725626), (-2.65053544284646, 2.11331030167519), (-9, 7.5), (1.471118941670167, 3.054051018632799), (9.423030478007094, 2.151559322647076), (-0.003912146532319, -0.003057364501384), (14.4826659455416, -6.974484332041754), (-8.930078484702461, 5.402651996775601), (-7.401572562260538, -15.375984842855647), (-4.194436781028413, 8.70800844041119), (-7.3, 5.9), (-16.07454648156653, -0.000000000000145), (-1.470525577977862, -3.05433676752634), (-7.47, 5.85), (-5.737977020004165, 6.87994033343273), (-2.65053544284646, 2.11331030167519)}) → m1
Length(m1) → läm1
Sequence(RemoveUndefined[Sequence[If[Distance[m1(k), m1(s)] < ε, (s, k), NaN], k, s + 1, Length[m1]]], s, 1, Length[m1] - 1) → m2a
Unique(y(Flatten[m2a])) → m2bb
Zip(m1(k), k, (1…Length[m1]) \ m2bb) → lPP
Length(lPP) → lälPP
Textfield(ε) → InputBox1
Zip(m1(k), k, m2bb) → lPP'
Length(lPP') → lälPP'
KeepIf(Distance[A, a] < ε, a, m1) → l1
Length(l1) → läl1
Circle(m1(2), ε) → c
Circle((0, 0), ε) → c1
Circle(A, ε) → c2
RemoveUndefined(Sequence[If[Distance[A, m1(k)] < ε, k], k, 1, Length[m1]]) → l1_k
Textfield(a) → InputBox2

AI 绘图提示语参考

在平面直角坐标系中给出一组离散点集合，并提供两个可调参数：ε（距离阈值）与参考点a。用圆来表示“邻域”，先画以参考点为圆心、半径为ε的圆；同时也可画以原点为圆心、半径为ε的阈值圆，以及以另一给定点为圆心半径为ε的圆。再根据欧氏距离条件，将所有距离参考点小于ε的点从原始集合中筛选出来：用视觉保留/移除两类点来体现筛选结果。最后用文本框显示ε与筛选后点的数量。

以上提示语为根据原始绘图命令整理而来，仅供参考。