SURFACE

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SURFACE
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Taxonomy of surfaces for CAD and CG

• Plane surface
• - the most elementary of the surface type
• - defined by four curves/ lines or by three
  points or a line and a point

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Taxonomy of surfaces for CAD and CG
2. Simple basic surface - Sphere, Cube, Cone,
and Cylinder
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Taxonomy of surfaces for CAD and CG

• 3. Ruled surface
• produced by linear interpolation between two
  bounding geometric elements. (curves, c1 and c2)
• Bounding curves must both be either geometrically
  open (line, arc) or closed (circle, ellipse).
• a surface is generated by moving a straight line
  with its end points resting on the curves.

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Taxonomy of surfaces for CAD and CG
3. Ruled surface (cont)
C2
C2
C1
C1
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Taxonomy of surfaces for CAD and CG

• 3. Tabulated cylinder
• Defined by projecting a shape curve along a line
  or a vector

Shape curve
Vector
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Taxonomy of surfaces for CAD and CG

• 4. Surface of revolution
• Generated when a curve is rotated about an axis
• Requires
• a shape curve (must be continuous)
• a specified angle
• an axis defined in 3D modelspace.
• The angle of rotation can be controlled
• Useful when modelling turned parts or parts which
  possess axial symmetry

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Taxonomy of surfaces for CAD and CG
4. Surface of revolution (cont)
axis
?
curve
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Taxonomy of surfaces for CAD and CG

• 5. Swept surface
• A shape curve is swept along a path defined by an
  arbitrary curve.
• Extension of the surface of revolution (path a
  single curve) and tabulated surface (path a
  vector)

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Taxonomy of surfaces for CAD and CG
5. Swept surface (cont)
Path- an arbitrary curve

Shape curve
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Taxonomy of surfaces for CAD and CG

• 6. Sculptured surface
• Sometimes referred to as a curve mesh surface.
• coons patch
• among the most general of the surface types
• unrestricted geometric
• Generated by interpolation across a set of
  defining shape curves

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Taxonomy of surfaces for CAD and CG

• 6. Sculptured surface (cont)
• Or
• A set of cross-sections curves are established.
  The system will interpolate the crosssections to
  define a smooth surface geometry.
• This technique called lofting or blending

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NURBS Surface
n
m

• P(u,v) ? ? wi,jNi,k(u) jNj,l(v) pi,j
• ? ? wi,jNi,k(u) jNj,l(v)
• u, v knot values in u and v direction (u k-1
  ?u? un1 ,v k-1 ?v ? vn1)
• pi,j - control points (2D graph)
• Degree k-1 (u direction) and l1 (v direction)
• wi,j weights (homogenous coordinates of the
  control points)

i0
j0
n
m
i0
j0
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NURBS Surface
P0,3
P3,3
P0,2
P0,1
P0,0
v
P1,0
P2,0
P3,0
u
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Normal Vector
N

• Perpendicular to the surface
• a?Tangent vector in u direction.
• b? tangent vector in v direction.
• Normal vector, n a x b (cross product)
• a dP(u,v) b dP(u, v)
• du dv

a
b
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NURBS surface generated by sweeping a curve

• Example sweep along a vector/ line
• NURB curve, P has degree l-1, knot value (0,1,m)
  and control points Pj
• Sweep along a line ? translate the curve in u
  direction.
• direction ? linear ? degree 1? 2 control points
  ? knot value 0,0,1,1

Pj
v
a
u
d
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NURBS surface generated by sweeping a curve

• Example sweep along a vector/ line
• P0, j P j , P1, j P j da, h0, j h1, j
  h j
• NURBS equation
• P(u,v) ? ? wi,jNi,2(u) jNj,l(v) pi,j
• ? ? wi,jNi,2(u) jNj,l(v)

Pj
v
a
u
d
1 i0
m j0
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NURBS surface generated by revolving a curve

• v direction ?NURBS curve, P has degree l-1,
  control points Pj,
• Revolution axis z axis
• u direction ? circle ?9 control points ? degree
  2 ? knot vector (0,0,0,1,1,2,2,3,3,4,4,4)
• P0, j P j , h0, j h j
• P1, j P0,j x j j, h1, j h j .1/?2
• P2, j P1,j- x ji, h2, j h j
• P3, j P2,j- x j i, h3, j h j .1/?2

u
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NURBS surface generated by revolving a curve

• P4, j P3,j- x j j, h4, j h j
• P5, j P4,j- x jj, h5, j h j 1/?2
• P6, j P5,j- x j i, h6, j h j
• P7, j P6,j- x ji, h7, j h j 1/?2
• P8, j P0,j, h8, j h j
• NURBS equation
• P(u,v) ? ? wi,jNi,3(u) jNj,l(v) pi,j
• ? ? wi,jNi,3(u) jNj,l(v)

u
8 i0
m i0
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NURBS surface display

• Use simple basic surface
• Mesh polygon flat faces ? triangle / rectangle
• Patches
• A patch is a curve-bounded collection of points
  whose coordinates are given by continuous, two
  parameter, single-valued mathematical functions
  of the form
• x x(u,v) y y(u,v) z z(u,v)

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Idea of subdivision

• Subdivision defines a smooth curve or surface as
  the limit of a sequence of successive
  refinements.
• The geometric domain is piecewise linear objects,
  usually polygons or polyhedra.
• .

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Example- curve

• subdivision for curve(bezier) in the plane

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Example - surface
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Benefit of subdivision

• The benefit simplicity and power
• Simple only polyhedral modeling needed, can be
  produced to any desired tolerance, topology
  correct
• Power produce a hierarchy of polyhedra that
  approximate the final limit object