Modeling Curved Lines and Surfaces

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Lecture 13

• Modeling Curved Lines and Surfaces

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Types of Surfaces

• Ruled Surfaces
• B-Splines and Bezier Curves
• Surfaces of Revolution

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Ruled Surfaces

• A ruled surface is defined by two end curves
  P0(u) and P1(u), that are connected by a straight
  line at each different value of u.
• Formula P(u, v) (1-v) P0(u) vP1(u)

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Bezier Curves

• A Bezier curve was originally developed in the
  1960s by French engineer, Pierre Bezier, who
  used them for the body design of the Renault car.
• Bezier curves are used in computer graphics to
  produce curves which appear reasonably smooth at
  all scales.

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Bezier Curves

• Bezier curves are constructed as a sequence of
  cubic segment in which the interpolating
  polynomials depend on certain control point.
• This means to each set of four point (P0, P1, P2,
  P3) we associate a curve with three main
  properties.

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Bezier Curve

• 1. The curve starts at P0 and ends at P3.
• 2. When the curve starts from P0 is heads
  directly towards P1, and when it arrives at P3 it
  is coming from direction P2.
• 3. The entire curve is contained in a
  quadrilateral whose corners are the four given
  points (their convex hull).

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Bezier Curves
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Bezier Curves
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Bezier Curves

• If there is only one control point P0 then B(u)
  P0 for all u.
• If there are only two control points P0 and P1
  then the formula reduces to a line segment
  between the two control points.

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Bezier Curves

• Adding multiple control points at a single
  position in space will add more weight to that
  point pulling the curve towards it.
• Bezier curves have wide applications because they
  are easy to compute and very stable. There are
  similar formulations, also called Bezier curves,
  which may behave differently.

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Bezier Curves

• The degree of the curve is one less than the
  number of control points, so it is a quadratic
  for 3 control point.

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Bezier Curves

• The curve always passes through the end points
  and is tangent to the line between the last two
  and first two control points.

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Bezier Curves

• The curve always list with the convex hull of the
  control points.

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Bezier Curves

• Closed curves are generated by specifying the
  first point the same as the last point. If the
  tangent at the first point and last point match
  the the curve is closed with first order
  continuity.

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Bezier Curves

• Bezier curves are used almost exclusively for
  creating curvilinear shapes in all fields of
  design, from purely technical plans and
  blueprints to the most creative artistic genres.

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Bezier Splines

• Linear Bezier spline is obtained by linear
  interpolation between two control points P0 and
  P1.
• Quadratic Bezier spline is obtained by de
  Casteljau algorithm as a linear interpolation
  between control points P0, P1, and P2.

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Bezier Splines

• Cubic Bezier spline can also be determined by
  deCasteljau algorithm to interpolate a curve
  between (n 1) control points P0 to P(n).

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Bezier Splines

• Linear Bezier spline P(t) (1-t)P0 tP1 ,    
• 0 lt t lt 1

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Bezier Splines

• Quadratic Bezier spline.    
• P01 (1-t)P0 tP1 ,    
• P11 (1-t)P1 tP2
• P(t) (1-t)P01 tP11
• (1-t)(1-t)P0 tP1
• t(1-t)P1 tP2
• (1-t)2P0 2(1-t)tP1 t2P2 ,
•  or P(t) Si0,2 Bi2(t) Pi ,
• where Bin(t) are Bernstein polynomials

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Bezier Splines

• Cubic Bezier spline.

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Bezier Splines

• To plot a Bezier Spline use the DeCasteljau
  iterations Pij (1-t)Pij-1 tPi1j-1,     j
  1, n     i 0, n-j
• for n 3

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Bezier Curves
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Surfaces Of Revolution

• This is done with relation to a B-spline curve
  that represents a profile of the object you are
  modeling.
• The surface is formed when a profile is swept
  about the z-axis. The resulting surface has the
  parametric form
• P(u, v) (X (v) cos(u), X(v) sin(u), Z(v)).

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Modeling a Teapot

• This model was done by Martin Newell.
• He decided to break the teapot body down into
  three parts, each a separate Bezier curve based
  on 10 points.
• 1st part is points 0,1,2,3
• 2nd part is 3, 3, 4, 5, 6
• 3rd part 6, 7, 8, 9.
• Last segment of each curve is collinear with the
  first segment of next curve to ensure the Bezier
  curves blend together.

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Body of a Teapot
i X Z
0 1.4 2.25
1 1.33 2.38
2 1.43 2.38
3 1.5 2.25
4 1.75 1.725
5 2 1.2
6 2 .75
7 2 .3
8 1.5 .075
9 1.5 0
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Bezier Surface Patches

• More complex, replace the u from the original
  Bezier equation with another Bezier equation,
  that has four specified control points which
  define the control polyhedron (which determines
  the shape of the patch).
• This is a tensor product form surface.

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Bezier Patch
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Matching Bezier Patches

• Since the boundary Bezier curve is determined
  by the boundary polygon of the control
  polyhedron, it can be simple to make two patches
  meet at points along a common boundary.
• Each pair of polyhedron edges that meet at the
  boundary must be collinear.

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Modeling the rest of the Pot

• Both the handle and the spout are composed of
  four Bezier patches.
• Handles surface is symmetrical about the
  xz-plane.
• The handle is then designed like another object,
  with an upper positive y section being composed
  of 16 control points, and a lower positive y
  section also with 16 control points.

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Other Modeling types

• B-spline Patches
• Alternate technique to be used in the tensor form
  surfaces
• NURBS surfaces
• Nonuniform rational B-splines

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NURBS

• A rational spline curve is similar to the normal
  B-spline counterpart, but adds a slightly
  different set of blending functions, weights, to
  add shape and control.
• Advantage to NURBS is that with properly chosen
  points and weights you can get a exact conic
  section. (non-rational can only approximate)

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NURBS

• NURBS are invariant among more classes of
  transformations unlike normal B-spline curves.
• This means you can draw a perspective projection
  of the NURB curve much easier and more
  efficiently.