curves and surfaces representation and application

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Curves and SurfacesRepresentation, and Applications

Diaa F. ElKott, Ph.D.

Carl Zeiss Industrial MetrologyBrighton, Michigan

August 11, 2011

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

VitaeDiaa ElKott

Work: Software Engineer, Carl Zeiss Industrial Metrology.

� Development of coordinate metrology software componentsa.

� Research of sculptured surfaces measurement methods.

Education:

� B.Sc., Production Engineering - Mansoura University.

� M.A.Sc., Industrial Engineering - University of Windsor.

� Ph.D., Mechanical Engineering - McMaster University.

aExamples: offline simulation of the measurement process, blade section re-sampling.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

A Math Tale

The Problem

Once upon a time, at a school in Germany, the teacher askedhis pupils to calculate:∑

81297 + 81495 + 81693. . . + 100701 + 100899

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

A Math Tale

The solution

Solution by Carl Friederich Gaussa

A =∑

81297 + 81495 + 81693. . . + 100701 + 100899

Ar =∑

100899 + 100701 + . . . + 81693 + 81495 + 81297

ai + ar i = 182196

A = ( ai+ar i2

)× 100

Final answer:

A = ( 81297+1008992

) × 100

A = ( 1821962

) × 100 = 91098 × 100 = 9109800.

aCarl Gauss was 10 years old when he solved this problem!

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

ManufactureSculptured vs primitive surfaces

Part with primitive surfaces Part with sculptured surfaces

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

ManufactureThe manufacturing process

The Manufacturing Process

Design Process

●Conceptual design.●FEA.●Assembly design.●Design for manufacture.....

Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)

ManufacturePart(CNC)


Design Intent

Material Requirement

Design Constraints

CAD Model(with tolerances)


Physical Object(with manufacturing errors)


Inspection Planning

●Where to measure.●Sample size.....

Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

AnalyseMeasurement

Results

AnalyseMeasurement

Results

AcceptAcceptRejectReject

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



Design Process


Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)



Design Intent


Design Constraints





Inspection Planning


Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

AnalyseMeasurement

Results

AnalyseMeasurement

Results


ErrorsErrorsErrorsErrors

ErrorsErrors

ErrorsErrors ErrorsErrors

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



Design Process


Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)



Design Intent


Design Constraints





Inspection Planning


Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

AnalyseMeasurement

Results

AnalyseMeasurement

Results


CADCADCADCAD

CADCAD

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

BackgroundTheory, and experience.

“Experience without theory is blind, but theory withoutexperience is mere intellectual play”.

Immanuel Kant (1724 1804)

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

I. Curve and Surface Representation Methods

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane CurvesPoints

Points

A point is a 0-dimensional mathematical object. A point is the buildingelement of higher-dimensional geometric objects (line, plane, etc.)

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane CurvesPoints

Two-Dimensional Cartesian Coordinate System

x

y

Origin

A coordinate system is a system which is used to define the location of apoint uniquely with respect to a point of origin.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane CurvesPoints

Cartesian Coordinates of Co-Planar Points

x

y

(5,4)

(-3,-3)

Origin

A coordinate system is a system which is used to define the location of apoint uniquely with respect to a point of origin.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves

Definition of a Curve - 1

x

y

A curve is the path of a point in motion.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves


Curve

x

y


Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves


Curve equation

x

y


Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves

Mathematical Representation of Curves

F ( x , y )=Constant

x

y

Implicit curve equation.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves


y=F ( x)

x

y

Explicit curve equation.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves


x=x (u)y= y (u)u1⩽u⩽u2

x

y

Parametric curve representation.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves

Curve Properties - 1

t⃗ in⃗i

p i

xi

y i

x

y

Tangent, and normal vectors of planar curve at point pi .

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane Curves

Curve Properties - 2

t⃗ in⃗i

p i

xi

y i

x

y

r i

Osculatingcircle

Osculating circle of planar curve at point pi . Curve curvature, κi = 1.0ri

.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Plane CurvesMathematical representation of a circle

The Circle

x

y

(x i , y i)

A circle is the path of a point that moves such that it is at a constantdistance from a given co-planar point.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Circle

x

y

r y i

x i

r=√ x i2+ y i2

A circle is the path of a point that moves such that it is at a constantdistance from a given co-planar point.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Circle

x

y

x2+ y2=r 2

Implicit equation of a circle.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Circle

x

y

y=±√r2−x2

Explicit equation of a circle.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Circle

x

y

ry i

x i

cos (θi)=x ir

sin(θi)=y ir

θ i

Derivation of parametric representation.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Circle

x

yx=r cos (θ)y=r sin (θ)0⩽θ⩽2π

θ

Parametric representation.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Space CurvesRepresentation, and important properties.

Space Curve

Space curve

x=x (u)y= y (u)z=z (u)u1⩽u⩽u2

( x(ui) , y (ui) , z(ui))

x

y

z

A space curve is the path of a 3-dimensional point in motion.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Space Curve

Space curve

x=x (u)y= y (u)z=z (u)u1⩽u⩽u2

( x(ui) , y (ui) , z(ui))

x

y

z

uu1 u2u i

Mapping from the parametric space to the Cartesian space.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Space Curve

Space curve

t⃗ i

n⃗i

p i=C (ui)

b⃗i

x

y

z

C (u)

Tangent, principal normal, and binormal vectors at point pi.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Space Curve

Space curve

t⃗ i

n⃗i

b⃗i

Osculatingplane

Normal plane

Rectifying plane

p i=C (ui)

x

y

z

C (u)

Moving trihedron (Frenet Frame) of space curve.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

SurfacesRepresentation, and important properties.

Parametric Representation of Surfaces

x

y

z

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

u1⩽u⩽u2v1⩽v⩽v2

Surface( x(ui , v j) , y (ui , v j ) , z (ui , v j ))

A surface is a subset of R3 that looks like a piece of R2 in the vicinity ofany given point.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



x

y

z

Surface( x(ui , v j) , y (ui , v j ) , z (ui , v j ))

Isoparametriclines v

uu1 u2u i

v1

v2

v j

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

u1⩽u⩽u2v1⩽v⩽v2

Mapping from the parametric space to the Cartesian space.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



x

y

z

Surface

S (u , v)

p i , j=S (ui , v j)n⃗i , j Sui , j

S v i , j

Tangentplane

Tangent plane, and normal vector at point pi,j.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

II. Application of Curve and Surface Methods

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

BackgroundTheory, and experience.



Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

ManufactureThe manufacturing process, revisited


Design Process


Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)



Design Intent


Design Constraints





Inspection Planning


Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

AnalyseMeasurement

Results

AnalyseMeasurement

Results


Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Manufacturing Process - Inspection

Design Process


Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)



Design Intent


Design Constraints





AnalyseMeasurement

Results

AnalyseMeasurement

Results


Inspection Planning


Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Manufacturing Process - Inspection Feedback

Design Process


Design Process


GenerateMachining

Instructions(CAM)

GenerateMachining

Instructions(CAM)



Design Intent


Design Constraints





Inspection Planning


Inspection Planning


MeasurePart(CMM)

MeasurePart(CMM)

AnalyseMeasurement

Results

AnalyseMeasurement

Results


Feedback

Feedback

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

MetrologyThe science of measurement

Metrology

Metrology is derived from the Greek words: Metron -Measure, and Logos - calculation, study, reason.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Metrology

Metrology is derived from the Greek words: Metron -Measure, and Logos - calculation, study, reason.

Metrology is the science of measurement.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Coordinate Metrology

Coordinate metrology is concerned with measuring the spatialcoordinates, mostly of physical objects.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Coordinate MetrologyMeasuring physical objects

The Coordinate Measuring Machine

The CMM is a Cartesian robot which is utilised to measure the coordinatesof surface points.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Coordinate Measuring Machine Sensor

The CMM is a Cartesian robot which is utilised to measure the coordinatesof surface points.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Coordinate Measurement

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Micro-Measuring Technology

Micro-measuring using hybrid sensors.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



TactileSensor

OpticalSensor

Workpiece


Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY




Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY



Micro-components

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Coordinate MetrologyChallenge

Measurement Inspection

CAD geometry

Upper tolerance limit

Lower tolerance limit

Physical model

Test conformity of the geometry of physical model to the design intent.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Inspection Process, Revisited

Inspection ProcessInspection Process

CAD Model

Physical Object

AcceptAccept

RejectReject

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Inspection Process - Sources of Error

Inspection ProcessInspection Process

CAD Model

Physical Object

AcceptAccept

RejectReject

AlgorithmicErrors

SoftwareErrors

HardwareErrors

HumanErrors

The inspection process is influenced by several sources of error.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Measurement Uncertainty

x

y

z

ϵTrue point(unknown)

Measured point

Measurement Uncertainty

Uncertainty cloud - all the possible readings of a particular point.

The form, and size of the uncertainty cloud may vary with time.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyProbing direction

Effect of Probing Direction

rcp

n

probe

workpiece

probing in surface normal direction

p

(a) Correct contact point, p = cp +r ×~n

rcp

n '

probe

workpiece

probing in vertical direction

pp '

(b) Erroneous contact point, p′ =cp + r ×~n′

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyProbing direction

Turbine Blade Measurement Path

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyMeasurement Path

Measurement path

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyMeasurement Path

Measurement path

Use the CAD geometry to define the CMM measurement path.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyData localisation

The Problem - 1

xCAD

yCAD

zCADCAD model

Nominalmeasurementpoints

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


The Problem - 2

xCAD

yCAD

zCADCAD model

Nominalmeasurementpoints

xActual

yActual

zActual

Physical model Actualmeasurementpoints

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Problem formulation

Align scanned data points P with the CAD model of the physical objectsuch that:

P = {P0, . . . ,Pn−1}Pi = {pi

0, . . . , pimi−1}

using the objective function:

MinimizeFunction

1n−1∑i=0

mi

×∑∀pl∈P

d2l

dl = GetShortestDistance(S,R× pl + T)

pnewl =

xnewl

ynewl

znewl

= R(O, θx)×R(O, θy )×R(O, θz)×

xoldl

y oldl

zoldl

+

δxδyδz

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY


Example

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyDeviation calculation

Solid Model, and Physical Object

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Application of Geometry in Coordinate MetrologyDeviation calculation

Measurement Data, Object Assessment Results

0.080 mm

0.030 mm

0.005 mm

0.000 mm

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Summary

Summary

1 Geometric elements are utilised to represent physical objects virtually.

2 The mathematical representation of geometric elements is utilised tocalculate important properties of the physical objects.

3 The properties of the physical objects’ geometry are utilised in thedesign, analysis, manufacture, and measurement of the physicalobjects.

Curves andSurfaces

D. ElKott

PREAMBLE

MATH TALE

MANUFACTURE

CURVES &SURFACES

Curves

Surfaces

APPLICATION

Manufacture

Metrology

Inspection

SUMMARY

Summary

