6. surfaces and surface modeling

6. Surfaces and Surface Modeling

Assoc.Dr. Ahmet Zafer Şenalpe-mail: [email protected]

Mechanical Engineering DepartmentGebze Technical University

ME 521Computer Aided Design

mailto:[email protected]

Analytical Surfaces Primitive surfaces Plane surface Offset surface Tabulated cylinder Surface of revolution Swept surface Ruled surface

Synthetic Surfaces Coons patches Bilinear surface Bicubic surface Bezier surface B-spline surface NURBS surface

Types od Surfaces

Dr. Ahmet Zafer Şenalp ME 521 2Mechanical Engineering Department,

GTU


A surface patch a curved bounded collection of⎯ points whose coordinates are given by continuous, two-parameter, single-valued mathematical expression.

Parametric representation:

p = p(u,v)

x=x(u,v),y=y(u,v),z=z(u,v)

p(u,v) = [x(u,v) y(u,v) z(u,v)]T

Surface Patch


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v

u

Isoparametric curves

Surface Patch


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Surface Patch

u=ui

v=vj

v=0

v=1

p(ui,vj)-

n(ui,vj)-


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Analytical Surfaces

Primitive surfaces Plane surface Offset surface Tabulated cylinder Surface of revolution Swept surface Ruled surface


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Primitive Surfaces

Plane: P(u, v) = u i + v j + 0 k

Cylinder: P(u, v) = R cos u i + R sin u j + v k


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Primitive Surfaces

• PlaneP(u, v) = u i + v j + 0 k

• CylinderP(u, v) = R cos u i + R sin u j + v k

• SphereP(u, v) = R cos u cos v i + R sin u cos v j + R sin v k

• ConeP(u, v) = m v cos u i + m v sin u j + v k

• TorusP(u, v) = (R + r cos v) cos u i + (R + r cos v) sin u j + r sin v k


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Defined by 3 points and 3 vectors

Planar Surface

)()(),( 02010 ppvppupvup 10;10 vu


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Planar Surface

)()(),( 02010 ppvppupvup 10;10 vu

)ˆ()ˆ),( 02010 sppvrppupvup 10;10 vu

normal surface ;srn

VectorsDirection Normalized ;pppps ;

ppppr

02

02

01

01


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Offset Surface

Offset yönü


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Tabulated Cylinder

• Curve is projected along a vector• In most CAD software it is called as “extrusion”

Surface generation curve

Vector


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Surface of Revolution

• Revolve curve about an axis

Axis

Curve


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Surface of Revolution

When a planar curve is revoled around the axis with an angle v a circle is constructed (if v=360 ). Center is on the revolving axis and rz(u) is variable.


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Swept Surface

• Defining curve swept along an arbitrary spine curve

Defining curve

Spine


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Ruled Surface

• Linear interpolation between two edge curves• Created by lofting through cross sections• Lines are used to connect edge curves• There is no restriction for edge curves• It is a linear surface

Edge curve 1

Edge curve 2

Linear interpolation


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Ruled Surface

Edge curves: G(u) ve Q(u)

Ruled surce only permits slope in the direction of curves in u direction. Surface has zero slope in v direction. Ruled surface cannot be used to model surfaces that have slopes in 2 directions.

C1(u)=G(u) C2(u)=Q(u)


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Synthetic Surfaces

Coons patches Bilinear surface Bicubic surface Bezier surface B-spline surface NURBS surface


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Linearly BlendedCoons Surface

p00

p11

p01

p10

v

u

D1

D0

C1

C0


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• Surface is defined by linearly interpolating between the boundary curves• Simple, but doesn’t allow adjacent patches to be joined smoothly


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• Most of the surface algorithms use finite number of points to model surface. However Coons surface patch uses interpolation method with infinite number of points.

• Coons surface seeks P(u,v) function that will fill between 4 edge curves.

• Bilineer Coons patch form:


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The form given above does not satisfy the boundary conditions as shown below.

Here below is a corrrection surface

With the application of correction surface;

elde edilir ve bu form sınır koşullarını sağlar.


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–1, 1-u, u, 1-v, and v functions are called blending functions, because they blend boundary curves to form one surface. For cubic blending functions the form given below is valid:

In the above matrix left column is P1(u,v), middle column is P2(u,v), right column is P3(u,v).


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Coons surface can be used by using ruled surfaces.


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Bilinear SurfaceA bilinear surface is derived by interpolating four data points, using linear equationsin the parameters u and v so that the resulting surface has the four points at its corners,denoted; P00, P10, P01, and P11.

P0v = (1-v)P00 + vP01

P1v = (1-v)P10 + vP11

Similarly P(u, v) can be obtained by using P0v ve P1v :P(u, v) = (1-u)P0v + uP1v

By replacing P0v and P1v into P(u, v): ]vP-v)Pu[(]vP-v)P-u)[( (P(u, v) 11100100 111

11

10

01

00

1111

PPPP

uv-u)v(-v)u(-v)-u)((P(u, v)


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Bilinear Surface

Advantage:To supply 4 corner points is enough

Limitations: Bilinear surface is flat Surfaces generally form in flat form


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Bicubic Patch

• As blending functions are not linear unlike bilinear surfaces it is possible to model nonlinear surface forms

• Extension of cubic curve• 16 unknown coefficients - 16 boundary conditions• Tangents and “twists” at corners of patch can be used• Like Lagrange and Hermite curves, difficult to work with


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Bicubic Patch


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Bicubic Patch

To find 16 coefficients in C matrix 16 boundary conditions are necessary. These are: 4 corner points 8 tangent vectors at corner points (in u and v directions at each point ) 4 twist vectors at corner points


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Bicubic Patch

The twist vector at a point on a surface measures the twist in the surface at the point. It is the rate of change of the tangent vector Pu with respect to v or Pv with respect to u or it is the cross (mixed) derivative vector at the point.

The normal to a surface is another important analytical property. The surface normal at a point is a vector which is perpendicular to both tangent vectors at the point.

And the unit normal vector is given by:


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Bicubic Patch

The Hermite bicubic surface can be written in terms of the 16 input vectors:

; Hermite matrix

; geometri ya da sınır koşulu matrisi


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Bicubic Patch

P(u,v) equation can be further expressed as:

The second order twist vectors Puv are difficult to define. The Ferguson surface(also called the F-surface patch) is a bicubic surface patch with zero twist vectors atthe patch corners. Thus, the boundary matrix for the F-surface patch becomes:


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Bicubic Patch

F-surface patch

This special surface is useful in design and machining applications.Dr. Ahmet Zafer Şenalp ME 521 33Mechanical Engineering Department,

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Bicubic Patch

• Advantages– Boundary curves are Hermite curves– Interior points can be controlled

• Disadvantages–What should be the twist factor? It is not esay to sense the effect of twist vector(Ferguson pacth twist vector is 0).– Cannot be used with high order polynomials.


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Bicubic PatchExample:

Parametric bicubic surface is defined in terms of cartesian components for u=0.5 and v=1:

11020112110112015

1),(2

3

23

vvv

uuuvux

11201201301001101

1),(2

3

23

vvv

uuuvuy

10010015000121210

1),(2

3

23

vvv

uuuvuz

u=1/2, v=1 noktasındaki teğet vektörleri nelerdir?


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Bicubic PatchExample: To find the tangent vectors it is necesary to differentiate with respect to u and v:

14

1111

1020112110112015

01175,0),(

vuxs

25,9

1111

1201201301001101

01175,0),(

vuys

12

1111

0010015000121210

01175,0),(

vuzs

(s=1/2,t=1) noktasında

1vvv

matrixrelated

01u2u3P2

3

2u

01v2v3

matrixrelated

1uuuP

2

23v

375,11

0123

1020112110112015

15,025,0125,0),(

vuxt

25,11

0123

1201201301001101

15,025,0125,0),(

vuyt

10

0123

0010015000121210

15,025,0125,0),(

vuzt

kjiPu 1225,914 kjiPv 1025,11375,11 Dr. Ahmet Zafer Şenalp ME 521 36Mechanical Engineering Department,

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Bezier Surfaces

• Bezier curves can be extended to surfaces• Same problems as for Bezier curves:

– no local modification possible– smooth transition between adjacent patches difficult to achieve

Parametric space Cartesian space


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Bezier Surfaces

Bezier Surfaces:• Two sets of orthogonal Bezier curves can be used to design an object surface.• A tensor product Bezier surface is an extension for the Bezier curve in two parametric

directions u and v:

• P(u, v) is any point on the surface and Pij are the control points. These points form the vertices of the control or characteristic polyhedron.

• Curves are formed, when u is constant v changes in [0..1]when v is constant u changes in [0..1]

• Like in Beziér curves Bin(u) and Bj

m(v) n. and m. degree Bernstein polynomials.• Generally n=m=3: cubic Beziér patch is used. (4x4=16 control points; Pi,j is necessary.)


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Bezier Surfaces

P(u, v) is a point on the surface and Pij are control points.These points form the control polygon’s vertex points.

Below figure shows cubic Bezier patch.When n=3 and m=3 is placed in Bezier equation then Bezier patch equation becomes:

Parametric space Cartesian space


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Bezier Surfaces


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Bezier Surfaces

A 3rd degree Bezier surface defined with 16 control points:


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Bezier Surfaces

Open and closed Bezier surface examples


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B-Spline Surfaces

• As with curves, B-spline surfaces are a generalization of Bezier surfaces• The surface approximates a control polygon• Open and closed surfaces can be represented


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B-Spline Surfaces

A tensor product B-spline surface is an extension for the B-spline curve in two parametric directions u and v.

For n=m=3, the equivalent bicubic formulation of an open and closed cubic B-splinesurface can be derived as below.


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B-Spline Surfaces

where [P] is an (n +1)×(m +1) matrix of the vertices of the characteristic polyhedron of the B-spline surface patch.

For a 4×4 cubic B-spline patch:


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B-Spline Surfaces

B-Spline surface example


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NURBS

NURBS surface (Non-Uniform Rational B-Spline surface) is a generilization to Bézier and B-splines surfaces. NURBS is used widely in computer graphics in CAD applications.

A NURBS surface is a parametric surface defined with its degree.


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NURBS


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Triangular Patches

Cartesian space Parametric space

In triangulation techniques, three parameters u, v and w are used and the parametric domain is defined by a symmetric unit triangle

10 ,10 ,10 wvu

The coordinates u, v and w are called “barycentric coordinates.” While the coordinate w is not independent of u and v (note that u+v+w=1 for any point inthe domain)


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Triangular Patches

A triangular Bezier patch is defined by:

For example, a cubic triangular patch is;


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Triangular Patches

For n=4, the triangular patch is defined as;


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Triangular Patches


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Free Form Surface


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Sculptured Surface

• General surface form• Composed of united surface pieces


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Subdivision Surface

New points are added between control points by interpollation to obtain a fine surface


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