Curves and surfaces by il4Fo7Y


									Curves and surfaces

 Advanced surface modeling
Two main principles:
• Many polygons to approximate curved
• Subdivide surfaces into subsurfaces,
  patches (curved)
          With polygons
• increasing number of polygons
• decreasing sizes of the polygons
• can be tens of thousands of polygons
  (cp. saxophone)
• much storage
           With patches
• boundaries are mathematically defined
• more advanced algorithms are required
• more processing/patch
• fewer patches
• no patch representation possible for
  very complex objects (cp. saxophone)
  Different curve approaches
General curves are usually defined from some
  type of control points (methods given in
  increasing complexity order):
• Connect points with straight lines
• Interpolation polynomial
• Least square approximation
• Splines, approximating (usual) or
  – Bézier splines
  – B-splines
Interpolation vs approximation
Interpolation; curve
  goes through the
  control points

Approximation; curve
  close to (or through)
  the control points
Example spline curve
     Example spline surface
A surface is a direct
  extension of curves,
  actually a set of
  orthogonal curves
            Curve properties
•   Control points (knots)
•   Multiple valued
•   Axis independance
•   Global or local control
•   Variation-diminishing
•   Versatility
•   Order of continuity, C0, C1, C2
    Function representation
Parametric (or vector-valued) functions in
• P(u) = (x(u) y(u) z(u)), curve
• P(u,v) = (x(u,v) y(u,v) z(u,v)), surface

Ex: P(u)=(cos u, sin u), 0≤u≤2(unit
             Spline curves
Piecewise continuous polynomial sections
Interpolating or approximating (or a
A control graph (control polygon) connects the
  control points
Different representations:
• set of boundary conditions
• matrix
• set of blending (weight) functions
     Spline curve approach
Usually, a third degree polynomial in each
  x(u)=axu3 + bxu2 + cxu + dx
  y(u)=ayu3 + byu2 + cyu + dy
  z(u)=azu3 + bzu2 + czu + dz
Plotting: (x(u),y(u),z(u)) is then written
  using u:=0 step u until n, u small
       Cubic Bézier Curves
Assume C1 (1st order continuity) and four
   control points (knots) in each section
   (polynomial), pk=(xk,yk,zk), k=0,1,2,3
a) Boundary conditions: (cp. f(x)=x3 =>
   x(0) = x0
   x(1) = x3
   x’(0) = 3(x1 - x0)
   x’(1) = 3(x3 - x2)
Similar for y and z
            Cubic spline,cont’d
b) Matrix
        Cubic spline,cont’d
c) Blending functions (Bernstein polynomials)
   x(u) = x0.BEZ0,3(u) + … + x3.BEZ3,3(u)
Example circle as a spline
Example spline curves

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