Parametric Design

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Parametric Design

• Curve / Surface Model

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Curve
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Point representation
D.F. Rogers, J. A. Adams, Mathematical elements
for computer graphics, 2nd ed.
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Analytical representation

• Nonparametric and parametric
• Precision over point representation
• Compact storage
• Center of circle and radius vs. points
• Intermediate point
• Any points on curve can be calcuate
• Easier to generate drawings
• Easier to change curvature

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Analytical representation of point defined curve

• Interpolation
• Analytically defining a curve from a known set of
  points
• Fit
• A curve that passes all known points
• Fair
• A curve that lies close to known points

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Nonparametric vs parametric

• Mathematical classification
• Nonparametric
• Explicit y f(x)
• Implicit f(x, y) 0
• General second degree implicit equation

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Implicit representation

• Curve in 2D f(x,y) 0
• Line ax by c 0
• Circle x2 y2 r2 0
• Surface in 3d f(x,y,z) 0
• Plane ax by cz d 0
• Sphere x2 y2 z2 r2 0

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Non parametric curves

• Line
• Circle
• Parabola
• Ellipse
• Hyperbola

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Parametric curves

• Points on a curve is represented with a function
  of a single parameter
• x f(u), y g(u), z h(u)
• u parametric variable
• Parametric representation of a straight line

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Spline

• DUCKS SPRINGING A LOFTING BATTEN FOR A HACKER
  RUNABOUT Courtesy J.D. Ball, NW School of
  WoodenBoat.

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Parametric cubic curves

• Mathematical spline
• Was defined using cubic polynomials
• Eulers equation for bending moment along the
  length of the beam (spline)
• Cubic Polynomials
• f(u) axu3 bxu2 cxu dx
• u (0 lt u lt 1)

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Specifying curves

• Control Points
• A set of points that influence the curves shape
• Knots
• Control points that lie on the curve
• Interpolating Splines
• Curves that pass through the control points
  (knots)
• Approximating Splines
• Control points merely influence shape

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Continuity

• Two or more curves at knots(connecting points) to
  form a continuous curve
• Types of continuity
• Point continuity
• Tangent continuity
• Curvature continuity

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Bezier curves

• Developed by Pierre Bezier to describe ab initio
  design of free form curves and surfaces
• defining polygon
• First and last points
• Tangent vector

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B-Spline curves

• A generalization of Bezier curve
• Additional control points for knot allow local
  changes to the curves

K degree of B-Spline k 2 linear k 3
touches midpoint of the intermediate segment
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Rational B-Spline curves

• Provide a single precise mathematical form
  capable of representing the common analytical
  shapes
• Lines, planes, conic curves including circles,
  free-form curves, quadric and sculptured surfaces.

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NURBS

• Non-Uniform Rational B-Spline
• the weighting of the control points is the
  difference

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Surface
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Two approaches

• Coons
• Create a mathematical surface from known data
• Bezier
• Create a mathematical surface ab initio
• Cohen
• General surface design showing mixture of Coons
  and Bezier approach

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Surface of revolution

• surface created by rotating a curve lying on some
  plane around a straight line (the axis of
  rotation) that lies on the same plane.

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Sweep surface

• Surface created by traversing a curve along a
  path in space

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Quadric surface

• surface of an equation of second degree in
  three-dimensional Cartesian Coordinates

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Bilinear surfaces

• Linear interpolation between four points which do
  not lie in the same plane

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Ruled surfaces

• Constructed by joining 2 curves by straight lines
  between points

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Developable Surface

• a surface with a metric that can be flattened
  onto a plane without distortion (i.e. stretching,
  compressing, tearing). Inversely, it is a surface
  that can be made by transforming a plane (i.e.
  folding, bending, rolling, cutting, and gluing)

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Linear Coons surface

• Interpolation between four boundary curves
• Similar to ruled surface in the two directions

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Parametric cubic patches/Bicubic surfaces

• Parametric cubic curves as four boundary curves

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Bezier surfaces

• Bezier curves as four boundary curves

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B-Spline surfaces

• B-Spline curves as four boundary curves

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NURBS surface
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Polygon mesh

• collection of vertices and polygons that defines
  the shape of an polyhedral object
• Meshes of triangles or quadrilaterals
• triangulation
• good for boxes, cabinets, building exteriors
• bad for curved surfaces

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Questions?