Synthetics surfaces unit ii

Parametric representation of
synthetic surfaces
1.Hermite Bi-cubic Surface:

1.The parametric bi - cubic surface patch connects four
corner data points and eight tangent vector at the corner
points .
2. Therefore, 16 vectors or 16×3=48 scalars are required to
determine the unknown coefficients in the equation. How?
3. Corner points=4 (P 00 ,P10,P01,P11), corner tangent
vectors=4×2=8, (PU00 , PV00, PU10, PV10 , PU01 , PV01, PU11, PV11)
And corner twist vectors=4. (PUV00 , PUV10, PUV01, PUV11)
4. This surface useful in FEA.
The parametric equation of Hermite bi- cubic equation is
3 3
P(u, v) = ∑∑ai j u i v j , 0 ≤ u ≤ 1, 0≤v ≤1
i = 0j =0
a i j is POIYNOMIALS CO – EFFICIENTS
u i v j is PARAMETERS

The equation can be expanded similar to Hermite cubic curve
where[P] , [P U] , [P V] and [P UV] are the sub –
matrices of the corner points , corner u – tangent
vectors , corner v – tangent vectors and corner
Twist.
The normal vector at N00 is: N00 = Pu00 × P v00
M is Hermite matrix
recall from Hermite curve
U AND V are parameters
of surface patch
B is corner points , corner
uv – Tangent vectors and
corner Twist.

where [MH] is the Hermite matrix and V is the geometry (or
boundary conditions) vector.

Bezier curve
• ABezier Curve is obtained by adefining polygon.
• First and last points on the curve are coincident with the firstand last
points of the polygon.
• Degree of polynomial is one lessthan the number ofpoints
• Tangent vectors at the ends of the curve have the samedirections asthe
respective spans
• Thecurve is contained within the convex hull of the definingpolygon

Bezier curve:
1. A Bezier curves is defined by approximating a set of data points.
2. Given n + 1 points (control points) P0+ P1+P2+.....Pn in space,
3. The Bezier curve defined by these control points is:
4. Where p(u ) is any point on the curve
5. Pi is a control points.
6. B n , i are the Bernstein polynomials.
7. Where as coefficient are defined as follows :

Characteristics of Bezier cure
1. The degree of a Bezier curve defined by n + 1 control points
is n : In each basis function the exponent of u is an i + (n-i)= n
therefore the degree of the curve is n.
1. P (u) passes through p o & p n :The curve passes through the
first & last control points as shown in below figure.
1. Non- negativity: All the basic functions are non- negative.
2. Convex hull property: This means that the Bezier curve
defined by the given n+1 control points lies completely in the
convex hull of the given control points.

1. The Variation diminishing property of Bezier curve is that they are
smoother than the polygon formed by their control points.
2. If a line is drawn through the curve, the number of intersections of
the curve will be less than or equal to the number of intersections
with the control polygon.
3. Line 1 intersect the curve 3 times & Polyline 7 time
5.Variation diminishing property

6.Affine Invariance
1. If any affine transformation is applied to a Bezier curve, the result can
be constructed from the affine images of its control points.
2. When we want to apply a geometric transformation to Bezier curve .
3. This property states that we can apply the transformation to control
points , which is quite easy, & once the transformed the control points
are obtained the transformed Bezier curve is one defined by these
new points.

7.Partition of unity
1. The sum of the basic functions at a fixed u is 1.
2. It is not difficult to verify that the basis functions are
the coefficients in the binomial expansion of the
expression 1= [ u + ( 1- u)]n . Hence there sum is 1.

8.Moving control points
1. Changing the position of a control point will change the
shape of the defined Bezier curve.
2. Suppose a control point PK is moved to a new position
PK + V, where vector v gives both direction & length of
this move as shown
9.Bezier curves are tangent to their first & last legs.

2.Bezier surface:
1.This is synthetic surface similar to the Bezier curve and it is obtained
by the transformation of Bezier curve.
2.It permits twists and kinks or (wrap or loop) in the surface.
3.surface does not pass through all the data points.
Definition of Bezier:
1. A two dimensional set of control points P i , j where i is the range of
0 and m , and j is the range of 0 and n .
Thus in this case , we have m+1 rows and n+1 columns of control
points and the control point on the i th row and jth columns are
denoted by P i , j
Note that we have (m + 1) (n + 1) control points in total.

The following is the equation of the Bezier surface defined by m+1 rows
and n+ 1 columns of control points:
1. Where B m,i (u) and B n,j (v) are the i th and j th
basic functions in the u and v directions
2. Recall from the Bezier curves , these basic
functions are defined as follows

1. Figure show Bezier surface defined by the three rows
and three columns and nine control points and hence
is a Bezier surface of degree ( 2 , 2 ).

1. The basic functions of a Bezier surface are the coefficients of control
points
2 From the definition, it is clear that these 2-dimensional basis functions are
the product of two one – dimensional Bezier basic functions
3.Basic functions for a Bezier surface are parametric surfaces of two
variables u and v defined on the unit square.
4.Figure shows the basic function for control points P 0, 0 (left) and P 1, 1
(right)
5. For control point P 0, 0 , its basis function is the product of two one –
dimensional Bezier basis functions B 2, 0 (U) in the U- direction and
B 2, 0 (V) in the V - direction as shown in figure (a).
6. Figure (b) shows the basic functions for P 1, 1 , which is the product of B 2,
1 (U) in the U- direction and B 2, 1 (V) in the V – direction.

3. Partition of unity : The sum of all B m, i (U) B n, j (v) is 1 for all u and v
in the range of 0 and 1. this mean that any pair of u and v in the range of
0 and 1 , the following holds :
4.Convex hull property: surface p(u, v) lies in the convex
hull defined by its control net.
5.Affine invariance: This means that to apply an affine
transformation to a Bezier surface, one can apply the
transformation to all control points & the surface is defined by the
transformed control points.
6. Variation diminishing property. No such think exist for
surfaces.

Synthetics surfaces unit ii

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Synthetics surfaces unit ii