Surface representation

SURFACE REPRESENTATION
MalakanagoudaB B
SUNITHKUMAR H G
Mtech 1st sem CIM

Bezier Surfaces:
Introduction
 Bezier surfaces were first described in 1962 by the French engineer Pierre Bezier who used
them to design automobile bodies.
 Bezier surfaces can be of any degree, but bicubic Bezier surfaces generally provide
enough degrees of freedom for most applications.
 This is a synthetic surface similar to the Bezier curve and is obtained by transformation of
a Bezier curve.
 It permits twists and kinks in the surface. The surface does not pass through all the data
points.
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Bezier Surface
 A tensor product Bezier surface is an
extension of the Bezier curve in two
parametric dimensions u and v.
 An orderly set of data or control points is
used to build a topologically rectangular
surface as shown in Fig.
 The surface equation can be written as,
3
 
m
i
m
j
mjniij vBuBP
0 0
,, )()(P(u, v) = 0<=u<=1, 0<=v<=1

 In addition, the Bezier surface possesses the convex hull property.
 The convex hull in this case is the polyhedron formed by connecting
the furthest control points on the control polyhedron.
 The convex hull includes the control polyhedron of the surface as it
includes the control polygon in the case of the Bezier curve.
 The shape of the Bezier surface can be modified by either changing
some vertices of its polyhedron or by keeping the polyhedron fixed
and specifying multiple coincident points of some vertices.
 Moreover, a closed Bezier surface can be generated by closing its
polyhedron or choosing coincident corner points as illustrated in Fig.
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 In the design environment, the Bezier surface is superior to a
bicubic surface in that it does not require tangent or twist
vectors to define the surface.
 However, its main disadvantage is the lack of local control.
 Changing one or more control point affects the shape of the
whole surface.
 Therefore, the user cannot selectively change the shape of part
of the surface.
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 A composite Bezier curve can have C0 (positional) and/or C1 (tangent)
continuity.
 A positional continuity between two patches requires that the common
boundary curve between the two patches must have a common
boundary polygon between the two characteristic polyhedrons, see Fig
a.
 For tangent continuity across the boundary, the segments, attached to
the common boundary polygon, of one patch polyhedron must be
collinear with the corresponding segments of the other patch
polyhedron, as shown in Fig b.
 This implies that the tangent planes of the patches at the common
boundary curve are coincident.
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In Bezier surface:
 The surface takes the general shape of the control points. set of
control points is usually referred to as the Bezier net or control net.
 The Basis functions of a Bezier surface are the coefficients of control
points.
 The corner of the surface and the corner control points are
coincident. Affine transformation can be applied to Bezier surfaces.
 Affine transformation is a linear mapping method that preserves
points, straight lines, and planes. Sets of parallel lines remain parallel
after an affine transformation. The affine
transformation technique is typically used to correct for geometric
distortions or deformations that occur with non-ideal camera angles.
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 When the number of control points increases.The degree of
Bezier surface increases. This can be compensated by making
large surfaces as a combination of small surface patches. This
will help reduce the degree of the Bezier surface patch to a
manageable value. However ,care has to be taken to see that
appropriate continuity is maintained between surface patch
boundaries.
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General Equation of the Bezier surface is
given as,
 P(u,v) = Pi,j Bi,m (u) Bj,n(v)
 0 ≤ u ≤1, 0≤ v ≤1
 P(u,v) is any point on the surface
 Pi,j defines the rectangular array of(m+1)*(n+1) control points
 Bi,m (u) and Bj,n (v) are the i and j Bezier basis functions in the
u and v directions.
 B i, m(u)= ui (1-u)m-i
 C(m,i) is the binomial coefficient
C(m, i)=
10

B-Spline Surface
 This is a synthetic surface that can approximate or
interpolate the given input data. The surface is capable of
giving very smooth contours, and can be reshaped with
local controls.
13

B-Spline Surface
 The tensor product method used
with Bezier curves can extend B-
splines to describe B-spline
surfaces.
 A rectangular set of data (control)
points creates the surface.
 This set forms the vertices of the
characteristic polyhedron that
approximates and controls the
shape of the resulting surface.
 A B-spline surface can
approximate or interpolate the
vertices of the polyhedron as
shown in Fig.
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 The degree of the surface is independent of the number of control
points and continuity is automatically maintained throughout the
surface by virtue of the form of blending functions. As a result,
surface intersection can easily be managed.
 B-spline surfaces have the same characteristics as B-spline curves.
 Their major advantage over Bezier surface is the local control.
 Composite B-spline surfaces can be generated with C0 and/or C1
continuity in the same way as composite Bezier surfaces.
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 A rectangular set of data points creates the surface.
 This set forms the vertices of the characteristic polyhedron that
approximates and controls the shape of the resulting surface.
 Non negativity the product of Ni,k (u) and Nj,l (v) is non
negative for all k,l,i,jand u,v the range of 0 and 1.
 Convex hull property A B-spline defined by its control points
pij
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 Affine transformation can be applied to B-spline surface.
 The weakness of B-spline surface is that primitive surfaces
such as cylinders, spheres and cones cannot be represented
precisely. Since it approximates these surfaces ,dimensional
error occur when machining the surfaces.
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 A B-spline surface patch defined by an (n+1) *(m+1)array of
control points is given by extending eqn into dimensions
 P(u,v) = Pi,j Ni,k (u) Nj,l (v)
 0 ≤ u ≤ Umax , 0≤ v ≤ Vmax
19

Coons Surface
 The above surfaces are used with either open boundaries
or given data points. The Coons patch is used to create a
surface using curves that form closed boundaries.
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Interpolated Surfaces – Coons Patch
 A linear interpolation between four bounded curves is used to
generate a Coons surface, also called as Coons patch. The
method is credited to S. Coons who developed this concept for
generating a surface.
 Linear interpolation between the boundary curves P(0,v),
P(u,0), P(1,v) , and P(u,1) gives the equation
 Q(u,v) = (1-v) P(u,0) + u P(1,v) + v P(u,1) + (1-u) P(0,v)
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 It is a form of “transfinite interpolation” which indicates that the
Coons scheme interpolates to an infinite number of data points, that
is, to all points of a curve segment, to generate the surface.
 The Coons patch is particularly useful in blending four prescribed
intersecting curves which form a closed boundary as shown in Fig.
 The figure shows the given four boundary curves P(u, 0), P(u, 0),
P(u, 0) and P(u, 0).
 It is assumed that u and v range from 0 to 1 along these boundaries
and that each pair of opposite boundary curves are identically
parameterized.
22

application
 One of the important application of the coon surface is the auto body
styling. The first step in auto body styling is the production of clay or
wooden model of external shape of the car. This profile as envisaged
by the artist is to be communicated to the cad database for further
refining.
 The car body style is digitalized using a Coordinate Measuring
Machine (CMM) where a probe of the appropriate tip touches the
model surface to record a number of points. The probe is moved over
the model along certain predefined lines, called feature lines.CMM
digitizes these feature lines into a sequence of points and feeds into
the cad data base. The cad system fits these points into a network of
curves from which a surface description of the model is generated.
For generating the surface from network of curves, Coons surface
methods are used.
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Tabulated Cylinder
 A Tabulated cylinder has been defined as a surface that
results from translating a space planar curve along a given
direction.
 It can also be defined as a surface that is generated by
moving a straight line (called generatrix) along a given
planar curve (called directrix).
 The straight line always stays parallel to a fixed given
vector that defines the v direction of the cylinder as
shown in fig.
24

Fig. Parametric Representation of a Tabulated Cylinder
25

Sculptured Surface
 Sculptured surfaces are used in geometric modelling to describe
all sorts of bendy things like aeroplane wings, car bodies, gas-
turbine blades, ship's hulls and so on that can't be described by
simple curved surfaces such as cylinders and cones.
 A Sculptured surface is defined as a collection or sum of
interconnected and bounded parametric patches together with
blending and interpolation formulas . OR
 A Sculptured surface can be defined as a complex surface formed
as a sum of different types of parametric surfaces and blending
surfaces to get the smooth transition across the surfaces.
28

 These are often called sculptured surfaces or free–form
surfaces.
 The Analytic and Synthetic patches can be used to create a
sculptured surface.
 The surface must be susceptible to APT (Automatically
Programming Tool), or other machining languages, processing
for NC machine tools.
29

Surface Manipulation
 Manipulating the surface during the design phase is
important to achieve the desired result.
 Surface manipulation provides the designer with the
capabilities to use surfaces effectively in design
applications.
Some useful features of surface manipulations are:
Displaying
Segmentation
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Surface Displaying
 A surface is normally displayed in a CAD system as a
mesh of curves on the surface.
 For a parametric surface p(u, v) if u is fixed to a value, say
0.05, and v is varied from 0 to 1. This generates a curve on
the surface whose u coordinate is a constant.
 This gives the isoparametric curve in the v direction with
u=0.05.
 This is a very inefficient type of display, as depending
upon the parameter increments used for u and v.
 This is sometimes called wireframe display.
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 Another good method to use to display surfaces is shading.
Various surfaces can be shaded with various colors. Surface
curvatures and other related properties can also be displayed
via shading.
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Segmentation
 Segmentation is a process of splitting a curve or a surface into
a number of parts such that the composite curve or surface of
all the segments is identical to the parent curve or surface.
 Segmentation is essentially a reparamterising transformation of
a surface while keeping the degree of the surface in u and v
parameter space remains unchanged.
 For example, let a surface patch be defined in the range of
u0<u<um and v0<v<vm
 Let this surface patch be divided at a point (u1 , v1) into four
segments , with two divisions along the u direction and two
along the v direction.
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 A new variable set is introduced for each of the surface patch
segments as (u1 , v1 ) whose range is (0 , 1) for each of the
segments.
 The parametric transformation for the first segment is
u1=u0+(u1-u0)u
v1=v0+(v1-v0)v
34Fig .Reparametrization of a Segmented Surface Patch

References
 CAD/CAM by Ibrahim Zeid & R Sivasubramanian 2nd edition
 CAD/CAM by P.N Rao
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Surface representation

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