Explicit representation of geometry - curves and surfaces

• Point cloud: list of points (x, y, z)
  turns the point cloud into a polygonal surface, which can be output in the computer
• Polygonal mesh: Polygon Mesh
  is split into small triangles
  
• Bezier curve
  defines the curve as long as it meets the starting and ending points, P1 and P2 determine which direction it will bend
  
• The de Casteljau algorithm
  defines a time t ranging from 0 to 1,
  finds the time t interpolation between (b0, b1), (b1, b2) respectively, and
  then calculates the time t interpolation between these two points. Get the time t interpolation from b0 to b2
  
  How to calculate multiple points?
  
  Use recursion to gradually reduce the size of the operation
• The Bernstein polynomial
  
  is actually its own nth-order expansion.
  Example:
  
  
  Note: The Bezier curve property is mostly used for affine transformation, and it does not take effect for projection
• Convex hull properties:
  the Bezier curve will not exceed the blue range
  , similar to wrapping a rubber band on several nails:
  
• (Piecewise) Segment-by-segment Bezier curve
  
  
  Form a point set of four points by dragging points to generate a smooth curve
• C0 and C1 are continuous
  C0: The end point is geometrically continuous
  C1: The left and right points form a straight line with equal proportions
  Example:
  
• Bezier surface
  Similar to Bezier curve, expand four points to 16 points to represent a surface. Apply Bezier curve
  
  on two planes
  respectively: