advanced-topics

# SymPy 高级主题 本文档涵盖了 SymPy 的高级数学功能，包括几何、数论、组合数学、逻辑与集合、统计学、多项式和特殊函数。 ## 几何 ### 二维几何 python from sympy.geometry import Point, Line, Circle, Triangle, Polygon # 点 p1 = Point(0, 0) p2 = Point(1, 1) p3 = Point(1, 0) # 点间距离 dist = p1.distance(p2) # 直线 line = Line(p1, p2) line_from_eq = Line(Point(0, 0), slope=2) # 直线属性 line.slope # 斜率 line.equation() # 直线方程 line.length # oo (直线长度无限) # 线段 from sympy.geometry import Segment seg = Segment(p1, p2) seg.length # 有限长度 seg.midpoint # 中点 # 交点 line2 = Line(Point(0, 1), Point(1, 0)) intersection = line.intersection(line2) # [Point(1/2, 1/2)] # 圆 circle = Circle(Point(0, 0), 5) # 圆心，半径 circle.area # 25*pi circle.circumference # 10*pi # 三角形 tri = Triangle(p1, p2, p3) tri.area # 面积 tri.perimeter # 周长 tri.angles # 角度字典 tri.vertices # 顶点元组 # 多边形 poly = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1)) poly.area poly.perimeter poly.vertices ### 几何查询 python # 检查点是否在直线/曲线上 point = Point(0.5, 0.5) line.contains(point) # 检查是否平行/垂直 line1 = Line(Point(0, 0), Point(1, 1)) line2 = Line(Point(0, 1), Point(1, 2)) line1.is_parallel(line2) # True line1.is_perpendicular(line2) # False # 切线 from sympy.geometry import Circle, Point circle = Circle(Point(0, 0), 5) point = Point(5, 0) tangents = circle.tangent_lines(point) ### 三维几何 python from sympy.geometry import Point3D, Line3D, Plane # 3D 点 p1 = Point3D(0, 0, 0) p2 = Point3D(1, 1, 1) p3 = Point3D(1, 0, 0) # 3D 直线 line = Line3D(p1, p2) # 平面 plane = Plane(p1, p2, p3) # 通过 3 点定义 plane = Plane(Point3D(0, 0, 0), normal_vector=(1, 0, 0)) # 通过点和法向量定义 # 平面方程 plane.equation() # 点到平面的距离 point = Point3D(2, 3, 4) dist = plane.distance(point) # 平面与直线的交点 intersection = plane.intersection(line) ### 曲线与椭圆 python from sympy.geometry import Ellipse, Curve from sympy import sin, cos, pi # 椭圆 ellipse = Ellipse(Point(0, 0), hradius=3, vradius=2) ellipse.area # 6*pi ellipse.eccentricity # 离心率 # 参数化 c