分段函数与泊松分布的可视化几何构造任务
几何画板及绘图内容加载中,请稍等……
该范例以几何作图的方式展示“分段函数/相关性度量”与“泊松分布曲线”在同一动态结构中的对照关系。核心数学概念包括:分段函数(piecewise functions)、幂函数(power functions)形状刻画、统计相关性(correlation)的可视化思路,以及泊松分布(Poisson distribution)概率质量随离散次数变化的表达。作图要点是把关键点、线与圆弧作为“参数化几何部件”,再通过旋转、平移与缩放把局部构型拼接成整体:使用 Segment、Ray、Line、Polygon 等构造基准几何元素;用 Point、Intersect、OrthogonalLine 提取交点与垂线关系;用 Circle、CircleArc、CircleSector 与 Min/Angle 等获得弧段与角度约束;再用 Rotate、Translate、Dilate、Vector、Rotate/Translate 等实现从一个构型到另一个构型的变换映射。适合高中到大学的数学建模与信息化作图教学,用于函数可视化、概率统计直观理解与GeoGebra几何-代数联动实验。
GeoGebra 命令及参数
Segment(A, B) → aRotate(B, (a_{2} * pi / 180), A) → B'Ray(A, B') → fPoint(f) → CSegment(A, C) → bOrthogonalLine(C, b) → rCircle(C, 1) → tIntersect(t, r, 2) → KOrthogonalLine(K, r) → i_1Point(i_1) → WRotate(E, (a_{2} * pi / 180), D) → E'Ray(D, E') → jSegment(D, E) → a'Circle(D, (a' / a * b)) → dIntersect(d, j, 1) → FSegment(D, F) → b'Line(K, b') → k_1Point(k_1) → ULine(K, a') → j_1Point(j_1) → VCircle(B, 1) → sOrthogonalLine(B, a) → qPoint(q) → MIntersect(s, q, 1) → JOrthogonalLine(J, q) → h_1Point(h_1) → PPolygon(A, B, C) → t1, c_{2}, a_1, b_{1}Polygon(D, F, E) → t2, e, d_1, f_1Min({a, a', b, b'}) → gCircle(A, (0.35 * g)) → hIntersect(h, a, 1) → GIntersect(h, b, 1) → HCircleSector(A, G, H) → kLine(A, a') → iDilate(A, If[-1 < l ≤ 0, (-l), 0], D) → A'Vector(A, A') → uTranslate(k, u) → k'Point(i) → IAngle(I, A, B) → αCircleArc(A, G, H) → pDilate(D, If[1 < l ≤ 2, 2 - l, 0], J) → D'Vector(D, D') → vTranslate(D, v) → D'_{1}Translate(E, v) → E'_{1}Segment(D'_{1}, E'_{1}) → a''Line(J, a') → g_1Point(g_1) → LAngle(P, J, L) → βRotate(D'_{1}, If[1 < l ≤ 2, ((l - 1) * β), β], D'_{1}) → NRotate(E'_{1}, If[1 < l ≤ 2, ((-(l - 1)) * β), (-β)], D'_{1}) → OSegment(N, O) → a'''Dilate(D, If[2 < l ≤ 3, 3 - l, 0], K) → D'_{2}Vector(D, D'_{2}) → wTranslate(D, w) → D'_{3}Translate(E, w) → E'_{2}Point(j_1) → QPoint(i_1) → RAngle(W, K, U) → γRotate(D'_{3}, If[2 < l ≤ 3, ((l - 2) * γ), γ], D'_{3}) → SRotate(E'_{2}, If[2 < l ≤ 3, ((l - 2) * γ), γ], D'_{3}) → TTranslate(D, w) → D'_{4}Translate(F, w) → F'Segment(D'_{4}, F') → b''Rotate(D'_{4}, If[2 < l ≤ 3, ((-(l - 2)) * γ), (-γ)], K) → ZRotate(F', If[2 < l ≤ 3, ((-(l - 2)) * γ), (-γ)], K) → F''Segment(Z, F'') → b'''Angle(A, C, B) → δIntersect(t, b, 1) → A_1Rotate(A_1, If[-3 < l ≤ -2, ((l + 3) * δ), δ], C) → B_1CircleSector(C, A_1, B_1) → e_1Dilate(C, If[-1 < l ≤ 0, (-l), 0], F) → C'Vector(C, C') → u_1Translate(e_1, u_1) → p_1Line(C, b') → l_1Point(l_1) → C_1Angle(C_1, C, A) → εSegment(F, E) → m_1Segment(B, C) → cOrthogonalLine(B, c) → r_1Circle(B, 1) → s_1Intersect(s_1, r_1, 1) → D_1Dilate(E, If[3 < l ≤ 4, 4 - l, 0], D_1) → E'_{3}Vector(E, E'_{3}) → v_1Translate(F, v_1) → F'_{1}Translate(E, v_1) → E'_{4}Segment(F'_{1}, E'_{4}) → t_1Line(E, c) → f_2Point(f_2) → E_1Angle(F, E, E_1) → ζRotate(F'_{1}, If[3 < l ≤ 4, ((l - 3) * ζ), ζ], E'_{3}) → F_1Rotate(E'_{4}, If[3 < l ≤ 4, ((l - 3) * ζ), ζ], E'_{3}) → G_1Segment(F_1, G_1) → g_2Angle(B, A, C) → ηRotate(G, If[-3 < l ≤ -2, ((l + 3) * η), η], A) → G'CircleSector(A, G, G') → d_2CircleArc(C, A_1, B_1) → e_2Rotate(p_1, If[-1 < l ≤ 0, ((-(l + 1)) * ε), (-ε)], C') → q_1Rotate(k', If[-1 < l ≤ 0, ((-(l + 1)) * α), (-α)], A') → k''
AI 绘图提示语参考
绘制一个包含分段函数与泊松分布对比的几何构造图。先画两条射线,分别以两个点为起点,并旋转一定角度;在射线上取点,构造垂线、圆及交点,形成辅助线。用多边形连接关键点,构造圆弧与扇形,并通过平移、缩放等变换复制图形。添加角度标注和长度限制,展示函数曲线与离散概率点的关联。
以上提示语为根据原始绘图命令整理而来,仅供参考。