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BezierSurface[{{p1,p2,…},…}]

表示由控制点 pi 定义的贝塞尔曲面.

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
范围  
基本用法  
规格说明  
图形渲染  
区域属性  
选项  
SplineDegree  
SplineClosed  
应用  
基本应用  
建模  
属性和关系  
互动范例  
参见
相关指南
历史
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BezierSurface

BezierSurface[{{p1,p2,…},…}]

表示由控制点 pi 定义的贝塞尔曲面.

更多信息和选项

  • BezierSurface 也称为贝塞尔曲面片.
  • BezierSurface 通常用于计算机辅助设计 (CAD) 和计算机图形学中，对自由形态的曲面进行建模.
  • 控制点 pi 是普通的三维坐标，例如 {x,y,z}.
  • BezierSurface[{{p1,p2,p3,p4},…,{p13,p14,p15,p16}}] 表示一个双三次贝塞尔曲面，其控制点为 4×4 阵列 p1,p2,…,p16.
  • 一个控制点数量超过 4×4 的 BezierSurface 将表示一个复合三次贝塞尔曲面.
  • 组合曲面可以使用多个贝塞尔曲面片对复杂几何形状进行建模.
  • BezierSurface 既可用作几何区域，也可用作图形基元.
  • 在图形中，点 pi 可以是 Scaled 和 Dynamic 表达式.
  • 图形渲染会受到诸如 FaceForm、EdgeForm、Texture、Specularity、Opacity 以及颜色等指令的影响.
  • FaceForm[front,back] 可用于为 3D 曲面的正面和背面指定不同的样式.
  • 可给出以下选项：
  • SplineDegree Automatic多项式基的次数
    SplineClosed False是否使表面闭合
  • 选项 SplineDegree->d 用于指定每个方向上的最大次数 d. SplineDegree->{d1,d2} 用于指定曲面中两个方向上的不同最大次数.
  • 当 SplineDegree->{d1,d2} 时，BezierSurface 若具有 (d1+1)×(d2+1) 个控制点，则会生成一个次数分别为 d1 和 d2 的简单贝塞尔曲面. 若控制点较少，则生成较低次数的曲面. 若控制点更多，则生成复合贝塞尔曲面.

范例

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基本范例  (3)

单个贝塞尔曲面片：

Wolfram Language code: Graphics3D[BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}]]

区域与边界：

Wolfram Language code: reg = BezierSurface[IconizedObject[«[image]»]];
Wolfram Language code: Area[reg]
Wolfram Language code: RegionBounds[reg]

制作一个国际象棋兵的模型：

Wolfram Language code: Graphics3D[{MaterialShading["Pewter"], BezierSurface[IconizedObject[«[image]»]]}, ...]

范围  (26)

基本用法  (5)

构建平坦的矩形曲面：

Wolfram Language code: Graphics3D[BezierSurface[{{{0, 0, 0}, {2, 0, 0}}, {{0, 1, 0}, {2, 1, 0}}}]]

构建弯曲的矩形曲面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»]]]

构建扭转的矩形曲面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»]]]

构建笔直的管状曲面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {False, True}]]

构建弯曲的管状曲面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {False, True}]]

构建闭合的管状曲面（环形）：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, True}]]

构建不规则的管状曲面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {False, True}]]

构建旋转曲面：

Wolfram Language code: Graphics3D /@ {BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}], BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}]}

BezierSurface 可以表示拓扑圆盘：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»]]]

表示圆柱面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}]]

表示球面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> {False, True}]]

表示圆环面：

Wolfram Language code: Graphics3D[BezierSurface[IconizedObject[«[image]»], SplineClosed -> True]]

摘要框：

Wolfram Language code: BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}]

规格说明  (3)

二次贝塞尔曲面片需要 3×3 的控制点阵列：

Wolfram Language code: Graphics3D[BezierSurface[(⁠| | | | | --------- | --------- | --------- | | {0, 0, 0} | {0, 4, 4} | {0, 8, 3} | | {4, 0, 4} | {4, 4, 0} | {4, 8, 0} | | {8, 0, 1} | {8, 4, 0} | {8, 8, 3} |⁠)]]

三次贝塞尔曲面片需要 4×4 的控制点阵列：

Wolfram Language code: Graphics3D[BezierSurface[(⁠| | | | | | --------- | --------- | --------- | --------- | | {0, 0, 0} | {3, 0, 2} | {5, 0, 2} | {8, 0, 1} | | {0, 3, 2} | {3, 3, 2} | {5, 3, 1} | {8, 3, 1} | | {0, 5, 3} | {3, 5, 2} | {5, 5, 1} | {8, 5, 1} | | {0, 8, 3} | {3, 8, 2} | {5, 8, 2} | {8, 8, 3} |⁠)]]

一般来说，次数为 d 的简单贝塞尔曲面片需要 (d-1)×(d-1) 的控制点阵列：

Wolfram Language code: pts = (⁠| | | | | | | --------- | --------- | --------- | --------- | --------- | | {0, 0, 0} | {1, 0, 1} | {2, 0, 0} | {3, 0, 1} | {4, 0, 0} | | {0, 1, 1} | {1, 1, 2} | {2, 1, 1} | {3, 1, 2} | {4, 1, 1} | | {0, 2, 0} | {1, 2, 1} | {2, 2, 0} | {3, 2, 1} | {4, 2, 0} | | {0, 3, 1} | {1, 3, 2} | {2, 3, 1} | {3, 3, 2} | {4, 3, 1} | | {0, 4, 0} | {1, 4, 1} | {2, 4, 0} | {3, 4, 1} | {4, 4, 0} |⁠);
Wolfram Language code: deg = Take[Dimensions[pts], 2] - 1;
Wolfram Language code: BezierSurface[pts, SplineDegree -> deg]
Wolfram Language code: Graphics3D[%]

当控制点较少时，生成较低次数的曲面片：

Wolfram Language code: BezierSurface[Take[pts, 1 ;; 3, 1 ;; 3], SplineDegree -> deg]
Wolfram Language code: Graphics3D[%]

当控制点较多时，生成复合贝塞尔曲面：

Wolfram Language code: BezierSurface[pts, SplineDegree -> 2]
Wolfram Language code: Graphics3D[%]

开放贝塞尔曲面：

Wolfram Language code: Graphics3D[{BezierSurface[IconizedObject[«[image]»]]}]

闭合贝塞尔曲面：

Wolfram Language code: Graphics3D[{BezierSurface[IconizedObject[«[image]»], SplineClosed -> True]}]

混合开闭方向：

Wolfram Language code: Graphics3D[{BezierSurface[IconizedObject[«[image]»], SplineClosed -> #]}]& /@ {{True, False}, {False, True}}

图形渲染  (12)

着色曲面：

Wolfram Language code: Table[Graphics3D[{c, BezierSurface[IconizedObject[«[image]»]]}], {c, {RGBColor[0.93, 0.27, 0.27], RGBColor[0.14, 0.8, 0.14], RGBColor[0.4, 0.6, 1]}}]

使用 FaceForm 可以为曲面的正面和背面指定不同的属性：

Wolfram Language code: Graphics3D[{FaceForm[RGBColor[0.93, 0.27, 0.27], RGBColor[0.14, 0.8, 0.14]], BezierSurface[IconizedObject[«[image]»]]}]

EdgeForm 可用于指定曲面边界边缘的样式：

Wolfram Language code: Graphics3D[{EdgeForm[{Red, Thick, Dashed}], BezierSurface[IconizedObject[«[image]»]]}]

对于闭合边界，边缘不绘制：

Wolfram Language code: Table[Graphics3D[{EdgeForm[{StandardYellow, Thick}], BezierSurface[IconizedObject[«[image]»], SplineClosed -> closed]}, Lighting -> "Accent"], {closed, {False, True}}]

具有不同反射指数（高光）的曲面：

Wolfram Language code: Table[Graphics3D[{Black, Specularity[White, n], BezierSurface[IconizedObject[«[image]»]]}, Lighting -> "Accent"], {n, {5, 20, 100}}]

Opacity 指定面的透明程度：

Wolfram Language code: Table[Graphics3D[{Opacity[o], BezierSurface[IconizedObject[«[image]»]]}], {o, {0.1, 0.5, 0.9}}]

Glow 指定曲面的颜色，独立于任何 Lighting：

Wolfram Language code: Table[Graphics3D[{Glow[c], BezierSurface[IconizedObject[«[image]»]]}, Lighting -> None], {c, {RGBColor[0.93, 0.27, 0.27], RGBColor[0.14, 0.8, 0.14], RGBColor[0.4, 0.6, 1]}}]

将其与 Lighting 结合，以创建微妙的阴影效果：

Wolfram Language code: Table[Graphics3D[{Glow[glow], GrayLevel[0.4], BezierSurface[IconizedObject[«[image]»]]}, Lighting -> "Accent"], {glow, {RGBColor[0.93, 0.27, 0.27], RGBColor[0.14, 0.8, 0.14], RGBColor[0.4, 0.6, 1]}}]

将 Texture 应用于曲面：

Wolfram Language code: Graphics3D[{Texture[[image]], BezierSurface[IconizedObject[«[image]»]]}, Lighting -> "Neutral"]

使用非真实感着色绘制曲面：

Wolfram Language code: Table[Graphics3D[{s, BezierSurface[IconizedObject[«[image]»]]}, Lighting -> "Accent"], {s, {StippleShading[0.6], ToonShading[Red], GoochShading[]}}]

使用 MaterialShading 绘制基于物理材质的曲面：

Wolfram Language code: Table[Graphics3D[{MaterialShading[mat], BezierSurface[IconizedObject[«[image]»]]}, Lighting -> "ThreePoint"], {mat, {"Gold", "Satin", "Velvet"}}]

BezierSurface 可以在 GraphicsComplex 中使用：

Wolfram Language code: Graphics3D[{GraphicsComplex[{{0, 0, 0}, {0, 10, 12}, {0, 20, 9}, {10, 0, 12}, {10, 10, 0}, {10, 20, 0}, {20, 0, 3}, {20, 10, 0}, {20, 20, 9}}, BezierSurface[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]]}]

默认情况下，控制点被视为在世界坐标系中：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Table[Graphics3D[{BezierSurface[pts]}, PlotRange -> {{0, pr}, {0, pr}, {0, pr}}], {pr, {1, 2, 4}}]

使用 Scaled 指定相对于 PlotRange 的控制点：

Wolfram Language code: scaled = Map[Scaled, pts, {2}];
Wolfram Language code: Table[Graphics3D[{BezierSurface[scaled]}, PlotRange -> {{0, pr}, {0, pr}, {0, pr}}], {pr, {1, 2, 4}}]

坐标可以是 Dynamic：

Wolfram Language code: DynamicModule[{z = 0, pts}, pts = {{Dynamic[{0, 0, z}], {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}; {Slider[Dynamic[z], {0, 4}], Graphics3D[{BezierSurface[pts]}]}]

区域属性  (6)

嵌入维度：

Wolfram Language code: reg = BezierSurface[{{{Indexed[a, {1}], Indexed[a, {2}], Indexed[a, {3}]}, {Indexed[a, {4}], Indexed[a, {5}], Indexed[a, {6}]}, {Indexed[a, {7}], Indexed[a, {8}], Indexed[a, {9}]}}, {{Indexed[b, {1}], Indexed[b, {2}], Indexed[b, {3}]}, {Indexed[b, {4}], Indexed[b, {5}], Indexed[b, {6}]}, {Indexed[b, {7}], Indexed[b, {8}], Indexed[b, {9}]}}, {{Indexed[c, {1}], Indexed[c, {2}], Indexed[c, {3}]}, {Indexed[c, {4}], Indexed[c, {5}], Indexed[c, {6}]}, {Indexed[c, {7}], Indexed[c, {8}], Indexed[c, {9}]}}}];
Wolfram Language code: RegionEmbeddingDimension[reg]

几何维度：

Wolfram Language code: RegionDimension[reg]

点隶属测试：

Wolfram Language code: reg = BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}];
Wolfram Language code: {RegionMember[reg, {0, 0, 0}], RegionMember[reg, {1, 2, 3}]}

面积：

Wolfram Language code: reg = BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}];
Wolfram Language code: {Area[reg], RegionMeasure[reg]}

形心：

Wolfram Language code: c = RegionCentroid[reg]
Wolfram Language code: Show[Region[reg], Graphics3D[{Black, Point[c]}]]

点到曲面的距离：

Wolfram Language code: reg = BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}];
Wolfram Language code: {RegionDistance[reg, {0, 0, 0}], RegionDistance[reg, {1, 1, 0}]}

可视化等距轮廓：

Wolfram Language code: Show[Region[reg], ContourPlot3D[Evaluate@RegionDistance[reg, {x, y, z}], {x, 0, 8}, {y, 0, 8}, {z, -4, 8}, Contours -> {1, 2, 3, 4}, ...], Boxed -> True, Axes -> True]

点到曲面的有符号距离：

Wolfram Language code: reg = BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}];
Wolfram Language code: {SignedRegionDistance[reg, {0, 0, 0}], SignedRegionDistance[reg, {1, 1, 0}]}

贝塞尔曲面是有界的：

Wolfram Language code: reg = BezierSurface[{{{0, 0, 0}, {0, 4, 4}, {0, 8, 3}}, {{4, 0, 4}, {4, 4, 0}, {4, 8, 0}}, {{8, 0, 1}, {8, 4, 0}, {8, 8, 3}}}];
Wolfram Language code: BoundedRegionQ[reg]

计算其边界：

Wolfram Language code: bb = CoordinateBoundingBox[reg]
Wolfram Language code: Show[Region[reg], Graphics3D[{Opacity[0.1], EdgeForm[Dashed], AmbientLight[White], Cuboid@@bb}]]

选项  (5)

SplineDegree  (3)

BezierSurface 默认情况下，具有 4×4 或更多控制点的 BezierSurface 将使用次数 3：

Wolfram Language code: pts = Table[{i, j, (-1) ^ (i * j)}, {i, 7}, {j, 7}];
Wolfram Language code: {BezierSurface[Take[pts, 4, 4]], BezierSurface[Take[pts, 7, 7]]}

当控制点较少时，使用较低次数：

Wolfram Language code: {BezierSurface[Take[pts, 2, 2]], BezierSurface[Take[pts, 3, 3]]}

使用 SplineDegree 来指定使用较低的次数：

Wolfram Language code: BezierSurface[pts, SplineDegree -> 1]

指定较高度数：

Wolfram Language code: BezierSurface[pts, SplineDegree -> 6]

对于一个具有 n×n 控制点的 BezierSurface，指定次数 d 且满足 d<n-1 时，将得到一个复合贝塞尔曲面：

Wolfram Language code: pts = Table[{i, j, (-1) ^ (i * j)}, {i, 9}, {j, 9}];
Wolfram Language code: Region[BezierSurface[pts, SplineDegree -> #]]& /@ {2, 4, 8}

分别为每个方向指定次数：

Wolfram Language code: pts = Table[{i, j, (-1) ^ (i * j)}, {i, 9}, {j, 9}];
Wolfram Language code: Region[BezierSurface[pts, SplineDegree -> #]]& /@ {2, {2, 8}, {8, 2}}

SplineClosed  (2)

默认情况下，曲面的边界是开放的：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Graphics3D[{BezierSurface[pts]}]

沿单个方向闭合曲面：

Wolfram Language code: Graphics3D /@ {BezierSurface[pts, SplineClosed -> {True, False}], BezierSurface[pts, SplineClosed -> {False, True}]}

在两个方向均闭合曲面：

Wolfram Language code: Graphics3D[{BezierSurface[pts, SplineClosed -> True]}]

边界闭合是通过将控制点阵列的第一行和/或第一列复制到阵列末尾实现的：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Graphics3D[{BezierSurface[pts, SplineDegree -> 10]}]
Wolfram Language code: Graphics3D /@ { BezierSurface[Append[pts, First[pts]], SplineDegree -> 10], BezierSurface[pts, SplineDegree -> 10, SplineClosed -> {True, False}] }

应用  (4)

基本应用  (1)

复合贝塞尔曲面由多个贝塞尔曲面片组成：

Wolfram Language code: composite = BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}];
Wolfram Language code: composite//Region

显示构成曲面的各个曲面片：

Wolfram Language code: decompose[reg : BezierSurface[cpts_, ___]] := Block[...]
Wolfram Language code: Graphics3D[{Table[{RandomColor[Hue[_, 0.7, 1]], surface}, {surface, decompose[composite]}]}, ...]

建模  (3)

为圣杯建模离散控制笼：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Graphics3D[{Map[Line, Join[pts, Transpose[pts]]], Point[Join@@pts]}, Boxed -> False]

在控制笼上定义光滑的 BezierSurface：

Wolfram Language code: surface = BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}]

渲染曲面：

Wolfram Language code: Graphics3D[{MaterialShading["Gold"], surface}, ...]

使用复合 BezierSurface 为具有尖锐边缘的物体建模：

Wolfram Language code: Graphics3D[{MaterialShading["Glazed"], BezierSurface[IconizedObject[«[image]»], SplineClosed -> {False, True}]}, ...]

为国际象棋棋子（兵）建模：

Wolfram Language code: Graphics3D[{MaterialShading["Pewter"], BezierSurface[IconizedObject[«[image]»], SplineClosed -> {True, False}]}, ...]

属性和关系  (11)

一个简单的 BezierSurface（次数为 1）等价于 Polygon：

Wolfram Language code: pts = {{{0, 0, 1}, {2, 0, 0}}, {{0, 2, 2}, {2, 2, 1}}};
Wolfram Language code: Graphics3D /@ {BezierSurface[pts], Polygon[Join[First[pts], Reverse[Last[pts]]]]}

BezierSurface 是 BSplineSurface 的一种特殊情况：

Wolfram Language code: Graphics3D /@ {BezierSurface[IconizedObject[«[image]»]], BSplineSurface[IconizedObject[«[image]»]]}

由次数为 d 的贝塞尔曲面片组成的复合 BezierSurface，可以通过一个次数为 d 且具有特定选择的 SplineKnots 的 BSplineSurface 来表示：

Wolfram Language code: knots = {{0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2}, {0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2}};
Wolfram Language code: Graphics3D /@ {BezierSurface[IconizedObject[«[image]»]], BSplineSurface[IconizedObject[«[image]»], SplineKnots -> knots]}

使用 RegionConvert 将任意的 BezierSurface 转换为等价的 BSplineSurface：

Wolfram Language code: bezier = BezierSurface[IconizedObject[«[image]»]]; bspline = RegionConvert[bezier, "Spline"];
Wolfram Language code: Graphics3D /@ {bezier, bspline}

BezierCurve 是 BezierSurface 的一种低维形式，它接受一个控制点列表：

Wolfram Language code: pts = {{-3, 0}, {6, 5}, {-6, -5}, {3, 0}};
Wolfram Language code: Graphics[BezierCurve[pts]]

BezierSurface 的边界边由四条贝塞尔曲线构成：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: bc = Map[BezierCurve, {pts[[1]], pts[[-1]], pts[[All, 1]], pts[[All, -1]]}];
Wolfram Language code: Graphics3D[{BezierSurface[pts], Thick, StandardRed, bc}]

BezierSurface 上的所有等参数曲线都是有效的贝塞尔曲线：

Wolfram Language code: uc = Table[BezierCurve[Map[#[u]&, Map[BezierFunction, pts]]], {u, 0, 1, 1 / 5}]; vc = Table[BezierCurve[Map[#[v]&, Map[BezierFunction, Transpose[pts]]]], {v, 0, 1, 1 / 5}];
Wolfram Language code: Graphics3D[{BezierSurface[pts], Thick, RGBColor[0.14, 0.8, 0.14], uc, RGBColor[0.4, 0.6, 1], vc}]

复合 BezierSurface 可用于近似诸如 Sphere、Torus 和 Cone 等基本图形：

Wolfram Language code: Graphics3D /@ {Torus[], Sphere[], Cone[]}
Wolfram Language code: Graphics3D /@ {BezierSurface[...], BezierSurface[...], BezierSurface[...]}

BezierSurface 始终插值其四个角点处的控制点：

Wolfram Language code: pts = IconizedObject[«[image]»]; corners = {pts[[1, 1]], pts[[-1, 1]], pts[[-1, -1]], pts[[1, -1]]};
Wolfram Language code: RegionMember[BezierSurface[pts], corners]
Wolfram Language code: Graphics3D[{BezierSurface[pts], Red, PointSize[Medium], Point[corners]}]

对于其他控制点，通常并非如此：

Wolfram Language code: opts = Complement[Join@@pts, corners];
Wolfram Language code: RegionMember[BezierSurface[pts], opts]//Counts

贝塞尔曲面位于其控制点的凸包内：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Show[Region[BezierSurface[pts]], Graphics3D[{Opacity[0.1], EdgeForm[Directive[Dashed]], ConvexHullRegion[Join@@pts]}]]

Bézier 曲面可以通过使用 Bernstein 多项式对其控制点进行加权求和来构建：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: bezier[u_, v_] := Sum[BernsteinBasis[3, j, v] * BernsteinBasis[3, i, u] * pts[[i + 1, j + 1]], {i, 0, 3}, {j, 0, 3}]
Wolfram Language code: GraphicsRow[{ParametricPlot3D[bezier[u, v], {u, 0, 1}, {v, 0, 1}], Graphics3D[BezierSurface[pts], Axes -> True]}, ImageSize -> Medium]

BezierFunction 给出在 BezierSurface 上与两个参数的特定取值相对应的位置：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: bfunc = BezierFunction[pts]
Wolfram Language code: positions = Join@@Table[bfunc[u, v], {u, 0, 1, 1 / 10}, {v, 0, 1, 1 / 10}];
Wolfram Language code: Graphics3D[{BezierSurface[pts], StandardRed, PointSize[0.02], Point[positions]}]

贝塞尔曲面具有仿射不变性：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: A = AffineTransform[{(⁠| | | | | -- | - | - | | 0 | 1 | 0 | | -1 | 0 | 0 | | 0 | 0 | 2 |⁠), {1, 2, 3}}];
Wolfram Language code: Graphics3D[#, Axes -> True]& /@ {GeometricTransformation[BezierSurface[pts], A], BezierSurface[A[pts]]}

默认情况下，复合贝塞尔曲面在两个曲面片之间的接缝处并不光滑：

Wolfram Language code: pts = IconizedObject[«[image]»];
Wolfram Language code: Graphics3D[{BezierSurface[pts, SplineDegree -> 4], {Map[Point, pts], Line[pts], Line[Transpose[pts]]}}]

设置跨越接缝的点共线以保证光滑接缝：

Wolfram Language code: pts[[4, All, 3]] = pts[[5, All, 3]]; pts[[6, All, 3]] = pts[[5, All, 3]];
Wolfram Language code: Graphics3D[{BezierSurface[pts, SplineDegree -> 4], {Map[Point, pts], Line[pts], Line[Transpose[pts]]}}]

SubdivisionRegion 从控制网格而非控制面片生成光滑曲面：

Wolfram Language code: Graphics3D /@ {SubdivisionRegion[[image]], BezierSurface[IconizedObject[«[image]»]]}

互动范例  (2)

德卡斯特利亚乌算法通过递归线性插值生成贝塞尔曲线上的点：

Wolfram Language code: Manipulate[ DynamicModule[{pts = {...}}, Graphics[{{Thick, LightGray, BezierCurve[pts]}, {StandardOrange, PointSize[Medium], casteljau[pts, u]}, PointSize[Large], Point[BezierFunction[pts][u]]}] ], {u, 0, 1}, Initialization :> {...} ]

它同样可以扩展到贝塞尔曲面：

Wolfram Language code: Manipulate[ DynamicModule[{pts = {...}, prims}, prims = casteljau[pts, uv[[2]], uv[[1]]]; Graphics3D[{Map[Line, Join[pts, Transpose[pts]]], {Opacity[0.8], BezierSurface[pts]}, Thickness[0.005], PointSize[0.02], RGBColor[0.4, 0.6, 1], prims[[1]], RGBColor[0.98, 0.56, 0.17], prims[[2]], LightDarkSwitched[GrayLevel[0], GrayLevel[1]], Ball[BezierFunction[pts]@@uv, Scaled[0.02]]}, Boxed -> False, Lighting -> "ThreePoint"] ], {{uv, {0.5, 0.5}}, {0, 0}, {1, 1}}, ControlPlacement -> Left, Initialization :> {...} ]

展示圆环如何展开为矩形贝塞尔曲面：

Wolfram Language code: Manipulate[ DynamicModule[{pts}, pts = unrolledTorusControlPoints[t, {21, 10}, 5, 10, 7]; Graphics3D[{EdgeForm[Thick], BezierSurface[pts]}, ...] ], {t, 0, 1}, Initialization :> {...} ]
Wolfram Research (2026)，BezierSurface，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BezierSurface.html.

文本

Wolfram Research (2026)，BezierSurface，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BezierSurface.html.

CMS

Wolfram 语言. 2026. "BezierSurface." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/BezierSurface.html.

APA

Wolfram 语言. (2026). BezierSurface. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/BezierSurface.html 年

BibTeX

@misc{reference.wolfram_2026_beziersurface, author="Wolfram Research", title="{BezierSurface}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/BezierSurface.html}", note=[Accessed: 23-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_beziersurface, organization={Wolfram Research}, title={BezierSurface}, year={2026}, url={https://reference.wolfram.com/language/ref/BezierSurface.html}, note=[Accessed: 23-August-2026]}