Geometric model & curve

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Geometric Modelling

& Curves

Geometric Modeling

Creating symbolic models of the

physical world has long been a goal of

mathematicians, scientists, engineers,

etc.

Recently technology has advanced

sufficiently to make computer modeling

of physical geometry feasible.

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Modelers for Engineering

Modeling for engineering applications require higher accuracy of representation.

Engineering models are used in computer-based design, manufacturing and analysis.

Geometric modeling

simply means that design concepts are digitally inputted into software

subsequently displays them in either 2-D or 3-D forms.

a general term applied to 3-D computer-aided design techniques.

Geometric models are computational (symbol) structures that capture the spatial aspects of the objects of interest for an application.

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Geometric model = Geometry + Topology Feature model = Geometric model + Design intent Design intent = Constraints + Rules

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Components of geometric modeling system

B-rep, CSG

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What is Geometric modeling ?

The process of constructing a complete mathematical description (geometric database) to model a physical entity or system.

Geometric Modeling

GM is a general term applied to 3D computer-aided design techniques.

There are three main types of geometrical modelling used, namely:

line or wireframe modelling,

surface modelling,

solid modelling.

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How does geometric modeling fit into a modern design sequence?

Computer-based geometric modeling is used to:

• visualize,

• analyze,

• document,

• produce a product or process.

Geometric modeling

Design Process

Geometric modeling

Stress Analysis

EngineeringAnalysis Thermal

Analysis

After-life Analysis

Visualization

Production planning

Engineering drawing

CNC programming

Geometric modeling is a basic engineering tool.

Serves as the backbone of design

Shadows the design process

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Geometric modeling

produce an appropriate database used for input into specialized engineering software tools to perform tasks in an integrated design sequence. support the move to a seamless work environment where the flow of data is continuous and need not recreated at each stage of design development.

Geometric modeling

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Shape modeling

Re-construction

Feature Analysis

Simplification Mesh

generation

Curve

Shape

Surface

Sample Mesh Surface Volume

Medialaxis

Segmentation

Dimension detection

Geometry • A typical solid model is defined by volumes, areas, lines, and

keypoints.

– Volumes are bounded by areas. They represent solid objects.

– Areas are bounded by lines. They represent faces of solid objects, or planar or shell objects.

– Lines are bounded by keypoints. They represent edges of objects.

– Keypoints are locations in 3-D space. They represent vertices of objects.

Volumes Areas Lines & Keypoints

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...Geometry - Preprocessing

• There is a built-in hierarchy among solid model entities. Keypoints are the “foundation” entities. Lines are “built” from the keypoints, areas from lines, and volumes from areas.

• This hierarchy holds true regardless of how the solid model is created.

• not allow you to delete or modify a lower-order entity if it is attached to a higher-order entity. (Certain types of modifications are allowed… discussed later.)

Keypoints

Lines

Areas

Volumes I’ll just

change

this line Lines

Keypoints

Are

as

OOPs!

Keypoints

Lines

Areas

Volumes

Mathematical Representation of Curves

The user constructs a geometric model of an object on a CAD/CAM system by inputting the object data as required by the modeling technique via user interface.

The software converts such data into a mathematical representation.

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Geometric modeling to CAD/CAM is important; it is a mean to enable useful engineering analyses and judgment.

In computer-based modeling and analysis, geometric models determine their relevance to design, analysis and manufacturing.


Methods of defining points Explicit methods Implicit methods

Absolute cartesian coordinates

Absolute cylindrical coordinates

Y

X

Z

+ P(x,y,z) + d

A digitize d

+ P(R,,z)

R

+ E1

+ E2

+ E1, E2

+ E2 + E1 + E1

+ E2

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Methods of defining points Explicit methods Implicit methods

Incremental cartesian coordinates

Incremental cylindrical coordinates

Y

X

Z

+

P0(x,y,z)

+ C

+ P1(R+ΔR,+Δ,z)

R +

I1

+ P1(x+Δx,y+Δy,z-Δz)

Δx Δy

Δz + C

+ C

+

P0(R,,z)

Δ ΔR

+

I2


Methods of defining lines Methods Illustration

Points defined by any method

Incremental cylindrical coordinates

+

+ +

+

Yw Yw

Xw

Zw

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Methods of defining lines Methods Illustration

Parallel or perpendical to an

existing line

Tangent to existing entities


Methods of defining arcs and circles Methods Illustration

Radius or diameter and center

Three points defined by any method

+

R

+

R

1

2

+

+

+

+

+

+

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Methods of defining arcs and circles Methods Illustration

Center and a point on the circle

Tangent to line, pass through a given

point, and with a given radius

+

+

1

2

+

+

+

R


Methods of defining ellipses Methods Illustration

Center and axes lengths

Four points

Two conjugate diameters

+ Pc

A B

+

+

+

+

+

+

+

+

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Methods of defining parabolas Methods Illustration

Vertex and focus

Three points

+

Pv +

P2

+

P1


Methods of defining synthetic curves Methods Illustration

Cubic spline

Bezier curves

B-Spline curves

+

P0

+

Pn

+ + +

+

P’n P’0

+

P0

+

Pn

+ +

+ +

+

+

+ + +

+ +

+

+ +

+ +

+

Approximate a

given set of data

points

Interpolate a

given set of data

points

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Curve Representation

• All forms of geometric modeling require the ability to define curves.

• Linear curves (1st order) may be defined simply through their endpoints.

• Must have a means for the representation for curves of a higher order:

– conics – free form or space curves


• Some terms we will use:

– Tangent vector:

Vector tangent to the slope of a curve at a given point.

– Normal vector :

Vector perpendicular to the slope of a curve at a given point.

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Parametric curves; Non-parametric and parametric forms

• Analytical representations of curves – Parabola: y = b x2 + C

– Ellipse: x2/a2 + y2/b2 = 1

– Hyperbola: x y = k

• Implicit form: f (x, y) = 0

• Explicit form: y = f (x) – hard to represent multi-valued function

Analytical representations

• Analytical forms are not suitable for CAD because; – Equation is dependent on coordinate sys. – Curves are unbounded – Implicit form is inconvenient for finding

points on the curve –Difficult for transformation (rotation,

pan)

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• Curves may be defined using different equation formats.

• explicit Y = f(X), Z = g(X)

• implicit f(X,Y,Z) = 0

• parametric X = X(t), Y = Y(t), Z = Z(t)

• The explicit and implicit formats have serious disadvantages for use in computer-based modeling

Parametric form

• Equations are de-coupled

• x = f (u)

• Matrix form: p (u) = [ u3 u2 u 1 ] [ A ]

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• Parametric: X = X(t), Y = Y(t), Z = Z(t); 0 ≤ t ≤ 1 (typ) • Substituting a value for t gives a

corresponding position along curve • Overcomes problems associated with

implicit and explicit methods • Most commonly used representation

scheme in modelers

Parametric example Recall the parametric line representation Parametric representation of a line. The parameter u, is varied from 0 to 1 to define all points along the line. X = X(u) Y = Y(u)

P2

P1

u

P2

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Parametric Line

• Line defined in terms of its endpoints

• Positions along the line are based upon the parameter value

– For example, the midpoint of a line occurs at u = 0.5

Parametric Line

• This means a parametric line can be defined by:

L(u) = [x(u), y(u), z(u)] = A + (B - A)u

where A and B and the line endpoints. e.g. A line from point A = (2,4,1) to point B = (7,5,5) can be represented as:

x(u) = 2 + (7-2)u = 2 + 5u y(u) = 4 + (5-4)u = 4 + u z(u) = 1 + (5-1)u = 1 + 4u

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Parametric cubic curves

• Algebraic form

• Geometric form: blending fn * geometric (boundary) conditions

• Blending function: p (u) = [ F1 F2 F3 F4 ] [ p(0), p(1), pu(0), pu(1) ]

• Magnitude and direction of tangent vectors

• Cubic Hermite blending function

Boundary conditions

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Blending functions

Parametric definition

Expanding the 2D parametric technique we used for a line to 3D, two parameters (u and v) are used.

P4

P1

P3

u

P2

v

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Points along edge P1 P2 have the form of P(u,0), along P3 P4 , P(u,1) and so on. P4

P1

P3

u

P2

v P(0,v) P(0,v)

P(u,1) P(1,v)

P(u,0)


By varying value of u and v, any point on the surface or the edge of the face may be defined.

P4

P1

P3

u

P2

v

(u1,v1)

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Another basic example would be that of a conic (circle)

Two parameter curves are :

X = cos (u)

Y = sin (u)

with range –π/4 ≤ u ≤ π/4


Graph of X = cos (u) Graph of Y = sin (u)

–π/4 –π/4

Cos(u)

–π/4 –π/4

Sin (u)

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Combined curve is a quarter circle

1

u = –π/4

Y

X

u = –π/4

u = π/4

0


Controls for this curve

• Shape (based upon parametric equation)

• Location (based upon center point)

• Size

– arc (based upon parameter range)

– radius ( a coefficient to unit value)

• Similar list could be formed for other conics

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Curve Use in Design

Engineering design requires ability to express complex curve shapes (beyond conics).

– examples are the bounding curves for: • turbine blades • ship hulls

• automotive body panels

– also curves of intersection between surfaces

Representing Complex Curves

Typically represented

– a series of simpler curves (each defined by a single equation) pieced together at their

endpoints (piecewise construction). – Simpler curves may be linear or polynomial

– Equations for simpler curves based upon control points (data pts. used to define the curve)

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Use of control points

General curve shape may be generated using methods of:

– Interpolation (also known as Curve fitting ) curve will pass though control points

– Approximation curve will pass near control points may interpolate the start and end points

Control Points Defining Curves

The following example shows an:

• Interpolating (passes through control points)

• Piecewise linear curve

• curve defined by multiple segments, in this case linear

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Interpolating Curve

Piecewise linear

• Linear segments used to approximate smooth shape

• Segments joints known as KNOTS

• Requires too many datapoints for most shape approximations

• Representation not flexible enough to editing

Piecewise

linear

Piecewise polynomial (composite curves)

• Segments defined by polynomial functions

• Again, segments join at KNOTS

• Most common polynomial used is cubic

(3rd order) • Segment shape

controlled by two or more adjacent control

points.

Piecewise

linear

Interpolation

curve (cubic)

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Knot points

• Locations where segments join referred

to as knots

• Knots may or may not coincide with control points in interpolating curves, typically they DO NOT coincide.

Curve continuity

• concern is continuity at knots (where

curve segments join) • continuity conditions: – point continuity (no slope or curvature

restriction) – tangent continuity (same slope at knot) – curvature continuity (same slope and

curvature at knot)

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Composite curves

• Piecewise curves

• Curve segments

• Continuity conditions

Continuity is symbolically represented by capital “C” with a superscript representing level.

Curve continuity

– C0 continuity, point/position continuity continuity of endpoint only, or continuity of

position – C1 continuity, tangent continuity tangent continuity or first derivative of

position – C2continuity, curvature continuity:

Hydrodynamic character, Light reflection curvature continuity or second derivative

of position

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Composite curves: continuity

Point continuity Point continuity Tangent continuity

Curvature continuity

Interpolation curves

• Interpolating piecewise polynomial curve

• Typically possess curvature continuity

• Shape defined by:

– endpoint and control point location

– tangent vectors at knots*

– curvature at knots*

*often calculated internally by software

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Approximation techniques

Developed to permit greater design flexibility in the generation of freeform curves.

Two very common methods in modern CAD systems, Bezier and B-Spline.

Approximation techniques

• employ control points (set of vertices that approximate the curve)

• curves do not pass directly through points (except possibly at start and end)

• intermediate points affect shape as if exerting a “pull” on the curve

• allow user to to set shape by “pulling” out curve using control point location

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Dr. M. Abid ([email protected])

Curves

NURBS

B-spline

Bezier

Rational

Bezier

Both Bézier curves and B-splines are polynomial parametric curves. Polynomial parametric forms can not represent some simple curves such as circles. Bézier curves and B-splines are generalized to rational Bézier curves and Non-Uniform Rational B-splines, or NURBS for short. Rational Bézier curves are more powerful than Bézier curves since the former now can represent circles and ellipses. Similarly, NURBS are more powerful than B-splines. The relationship among these

Recursive

Subdivision

Algorithm

Continuity

Bezier curves

• P. Bezier of Renault, P. de Casteljau of Citroen

• Intuitive interaction: Direct manipulation

• Approximated curve vs. Interpolated curve

• Control points

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Bezier curve defined by 4 pts

Pull by coincident control points

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Closed Bezier curve

Influence of point position

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Bezier curve . . .

• Bernstein polynomial

• B i,n (u) = C(n,i) ui (1-u) n-i

• Binomial coefficient: Combination

• Invariant under affine transformation

• Convex hull property: Straight interval

• Variation diminishing property

Blending fn for a cubic Bezier curve

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B-spline curves

Approximating Splines Piecewise Polynomials

NURBS

B0,1 B1,1 B2,1 B3,1 B4,1 B5,1 B6,1

B0,2 B1,2 B2,2 B3,2 B4,2 B5,2

B0,3 B1,3 B2,3 B3,3 B4,3

B0,4 B1,4 B2,4 B3,4

B-Spline Synthetic Curves

Analytical curves are insufficient for designing complex machinery parts and, therefore synthetic curves are used.

Synthetic curves are commonly used when interpolation curves are needed and it is easy to modify these curves locally.

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CAD\CAM systems have got 3 types of synthetic curves such as Hermite cubic splines, Bezier and B-spline curves.

Cubic splines are interpolating curves. Bezier and B-splines are approximating curves. On some cases B-splines can be interpolating.


B-spline curves are specified by giving set of coordinates, called control points, which indicates the general shape of the curve.

B-splines can be either interpolating or approximating curves. Interpolation splines used for construction and to display the results of engineering.

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Effect of curve order

• The higher the order of a curve, the “stiffer” the curve (less dramatic curvature changes)

• Maximum curve order dependent upon the number of control points order = one less than the number of control pts

• High order curves can exhibit irregularities


With (m+1) control points, there are always (n=m+p-1) basic functions. The basis functions are 1, at the end points of the curve defined as a and b. If there’s no other definition, then a=0 and b=1. {Pi} the set of the control points forms the control polygon from Figure.

P1

P2

P3

P4

B-spline curve

Control Polygon

N0,p(a)=1

Nn,p(b)=1

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B-splines

• Generalization of Bezier curve

• Bezier: p(u) = S P B i,n (u)

• B-spline: p(u) = S P N j,k (u)

• Basis fn or blending fn is different

• Local changes

• Degree of the curve is independent of # of control points

Linear, quadratic, cubic B-spline

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Influence of control point position

Blending functions for linear B-spline

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Quadratic B-spline blending fn (k=3)

Non-Uniform Rational,B-splines

Most modern CAD systems use the NURB curve representation scheme.

• NURB stands for Non-Uniform, Rational, B-spline.

• Uniformity deals with the spacing of control points.

• Rational functions include a weighting value at each control point for effect of control point.

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Non-Uniform Rational,B-splines

• very popular due to their flexibility in

curve generation. • same mathematical form may be used to

represent entire family of curves including:

– Bezier – B-Spline – conics

NURBS & CAD

For years major CAD software has been using NURBS for good surface definition. However, the exchange formats IGES and DXF are more limited - hence, the original model can become somewhat distorted in transfering from one program to another.

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NURBS & CAD Add that to some inconsistency in how these

formats are defined and consider a task like importing the NURBS sphere into a Coons Patch based program.

With NURBS it is now possible to represent the geometry the same as the CAD packages represent it internally - so rather than using faulty exchange formats we have partnered with the major CAD vendors and now have a direct link that will pull a model from memory in the CAD program into one of our programs.

Rational parametric curves

• Normal Bezier or B-spline cannot precisely represent conics and circles

• Homogeneous coordinate • Ph = (hx, hy, hz, h)

• NURBS (non-uniform rational B-

splines)

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85 Circular arc as a rational quadratic Bezier

Geometric model & curve

Engineering

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