几何形体

几何的表达方式

• Implicit

  • Based on classifying points
  • Sampling Can Be Hard
  • Inside/Outside Tests Easy
  • 种类
    • algebraic surface
      • 最直接的，用数学公式
      • 不直观
    • Constructive Solid Geometry（CSG）
      • 基本形状的布尔操作组合成复杂形状
      • Combine implicit geometry via Boolean operations
    • distance functions
      • 相关：Signed Distance Field
      • 解析形式表达
    • level sets （水平集）
      • Grid方式描述distance
      • 例子：CT扫描
    • Fractals
      • 自相似
      • 递归
    • ...

  Pros:

  • compact description (e.g., a function)

  • certain queries easy (inside object, distance to surface)

  • good for ray-to-surface intersection (more later)

  • for simple shapes, exact description / no sampling error

  • easy to handle changes in topology (e.g., fluid)

  Cons:

  • difficult to model complex shapes

• Explicit

  • All points are given directly or via parameter mapping
  • Sampling Is Easy
  • Inside/Outside Test Hard
  • 种类
    • point cloud
      • 一堆点
      • 可以表示任何几何
      • Useful for LARGE datasets (>>1 point/pixel)
      • Often converted into polygon mesh
      • Difficult to draw in undersampled regions
    • triangle/polygon mesh
      • Store vertices & polygons (often triangles or quads)
      • More complicated data structures
      • Perhaps most common representation in graphics
      • 例子：.obj格式
        • Just a text file that specifies vertices, normals, texture coordinates and their connectivities
    • subdivision, NURBS
    • Bezier surfaces
    • subdivision surfaces
    • NURBS
    • ...

No “Best” Representation, Each choice best suited to a different task/type of geometry

More Implicit Representations in Computer Graphics

曲线 Curve

Bézier Curves 贝塞尔曲线

一条由四个点（其实是任意≥3个点）定义的曲线：

• p0和p3定义起点和终点
• p1和p2定义起点与终点的切线方向（与p0和p3一起）

Evaluating Bézier Curves (de Casteljau Algorithm)

例子：(quadratic Bezier)

计算方法：