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# Curves And Surfaces Representation And Application

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Lecture given at the University of Windsor on 11 August 2011.

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### Curves And Surfaces Representation And Application

1. 1. Curves and Surfaces D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves and Surfaces Curves SurfacesRepresentation, and Applications APPLICATION Manufacture Metrology Inspection SUMMARY Diaa F. ElKott, Ph.D. Carl Zeiss Industrial Metrology Brighton, Michigan August 11, 2011
2. 2. Curves andVitae SurfacesDiaa ElKott D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & Work: Software Engineer, Carl Zeiss Industrial Metrology. SURFACES Curves Surfaces Development of coordinate metrology software componentsa . APPLICATION Manufacture Research of sculptured surfaces measurement methods. Metrology Inspection SUMMARY Education: B.Sc., Production Engineering - Mansoura University. M.A.Sc., Industrial Engineering - University of Windsor. Ph.D., Mechanical Engineering - McMaster University. a Examples: oﬄine simulation of the measurement process, blade section re-sampling.
3. 3. Curves andA Math Tale Surfaces D. ElKott PREAMBLE MATH TALEThe Problem MANUFACTURE CURVES & Once upon a time, at a school in Germany, the teacher asked SURFACES his pupils to calculate: Curves Surfaces 81297 + 81495 + 81693. . . + 100701 + 100899 APPLICATION Manufacture Metrology Inspection SUMMARY
4. 4. Curves andA Math Tale Surfaces D. ElKott PREAMBLE MATH TALEThe solution MANUFACTURE CURVES & Solution by Carl Friederich Gaussa SURFACES Curves Surfaces A= 81297 + 81495 + 81693. . . + 100701 + 100899 APPLICATION Manufacture Metrology Ar = 100899 + 100701 + . . . + 81693 + 81495 + 81297 Inspection SUMMARY r ai + a i = 182196 r A = ( ai +a i ) × 100 2 Final answer: A = ( 81297+100899 ) × 100 2 A = ( 182196 ) × 100 = 91098 × 100 = 9109800. 2 a Carl Gauss was 10 years old when he solved this problem!
5. 5. Curves andManufacture SurfacesSculptured vs primitive surfaces D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Part with primitive surfaces Part with sculptured surfaces Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY
6. 6. Curves andManufacture SurfacesThe manufacturing process D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Reject Reject Accept Accept
7. 7. Curves andManufacture SurfacesThe manufacturing process D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY Errors Errors Errors Errors Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) Errors Errors Errors Errors Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Errors Errors Reject Reject Accept Accept
8. 8. Curves andManufacture SurfacesThe manufacturing process D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY CAD CAD Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) CAD CAD CAD CAD Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Reject Reject Accept Accept
9. 9. Curves andBackground SurfacesTheory, and experience. D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection “Experience without theory is blind, but theory without SUMMARY experience is mere intellectual play”. Immanuel Kant (1724 1804)
10. 10. Curves and Surfaces D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture MetrologyI. Curve and Surface Representation Methods Inspection SUMMARY
11. 11. Curves andPlane Curves SurfacesPoints D. ElKott PREAMBLE MATH TALE MANUFACTURE Points CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY A point is a 0-dimensional mathematical object. A point is the building element of higher-dimensional geometric objects (line, plane, etc.)
12. 12. Curves andPlane Curves SurfacesPoints D. ElKott PREAMBLE MATH TALE MANUFACTURE Two-Dimensional Cartesian Coordinate System CURVES & SURFACES y Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Origin x A coordinate system is a system which is used to deﬁne the location of a point uniquely with respect to a point of origin.
13. 13. Curves andPlane Curves SurfacesPoints D. ElKott PREAMBLE MATH TALE MANUFACTURE Cartesian Coordinates of Co-Planar Points CURVES & SURFACES y Curves Surfaces APPLICATION (5,4) Manufacture Metrology Inspection SUMMARY Origin x (-3,-3) A coordinate system is a system which is used to deﬁne the location of a point uniquely with respect to a point of origin.
14. 14. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Deﬁnition of a Curve - 1 MANUFACTURE CURVES & SURFACES y Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY x A curve is the path of a point in motion.
15. 15. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Deﬁnition of a Curve - 2 MANUFACTURE CURVES & SURFACES y Curves Surfaces Curve APPLICATION Manufacture Metrology Inspection SUMMARY x A curve is the path of a point in motion.
16. 16. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Deﬁnition of a Curve - 3 MANUFACTURE CURVES & SURFACES y Curves Surfaces Curve equation APPLICATION Manufacture Metrology Inspection SUMMARY x A curve is the path of a point in motion.
17. 17. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Mathematical Representation of Curves MANUFACTURE CURVES & SURFACES y Curves F ( x , y )=Constant Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY x Implicit curve equation.
18. 18. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Mathematical Representation of Curves MANUFACTURE CURVES & SURFACES y Curves y=F ( x) Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY x Explicit curve equation.
19. 19. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Mathematical Representation of Curves MANUFACTURE CURVES & SURFACES y x =x (u) Curves y= y ( u) Surfaces u 1⩽u⩽u 2 APPLICATION Manufacture Metrology Inspection SUMMARY x Parametric curve representation.
20. 20. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Curve Properties - 1 MANUFACTURE CURVES & SURFACES y Curves ni ⃗ Surfaces ⃗i t APPLICATION yi Manufacture Metrology pi Inspection SUMMARY x xi Tangent, and normal vectors of planar curve at point pi .
21. 21. Curves andPlane Curves Surfaces D. ElKott PREAMBLE MATH TALE Curve Properties - 2 MANUFACTURE CURVES & SURFACES y Curves ni ⃗ Surfaces ⃗i t APPLICATION yi Manufacture Metrology ri pi Inspection SUMMARY Osculating circle x xi 1.0 Osculating circle of planar curve at point pi . Curve curvature, κi = ri .
22. 22. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves Surfaces APPLICATION (x i , y i ) Manufacture Metrology Inspection SUMMARY x A circle is the path of a point that moves such that it is at a constant distance from a given co-planar point.
23. 23. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves Surfaces APPLICATION Manufacture r = √ x 2+ y 2 i i Metrology Inspection r SUMMARY yi xi x A circle is the path of a point that moves such that it is at a constant distance from a given co-planar point.
24. 24. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves Surfaces x + y 2=r 2 2 APPLICATION Manufacture Metrology Inspection SUMMARY x Implicit equation of a circle.
25. 25. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves Surfaces y=±√ r −x 2 2 APPLICATION Manufacture Metrology Inspection SUMMARY x Explicit equation of a circle.
26. 26. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves Surfaces xi APPLICATION cos (θi )= r Manufacture yi Metrology sin(θ i )= Inspection r r yi SUMMARY θi xi x Derivation of parametric representation.
27. 27. Curves andPlane Curves SurfacesMathematical representation of a circle D. ElKott PREAMBLE MATH TALE MANUFACTURE The Circle CURVES & SURFACES y Curves x=r cos (θ) Surfaces y=r sin (θ) 0⩽θ⩽2 π APPLICATION Manufacture Metrology Inspection SUMMARY θ x Parametric representation.
28. 28. Curves andSpace Curves SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE MANUFACTURE Space Curve CURVES & SURFACES Curves Space curve Surfaces APPLICATION Manufacture ( x(ui ) , y (ui ) , z(ui )) Metrology Inspection SUMMARY x =x (u) y= y (u) z =z (u) u 1⩽u⩽u 2 z y x A space curve is the path of a 3-dimensional point in motion.
29. 29. Curves andSpace Curves SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE MANUFACTURE Space Curve CURVES & SURFACES Curves Space curve Surfaces APPLICATION Manufacture ( x(ui ) , y (ui ) , z(ui )) Metrology Inspection SUMMARY x =x (u) y= y (u) u z =z (u) u1 ui u2 u 1⩽u⩽u 2 z y x Mapping from the parametric space to the Cartesian space.
30. 30. Curves andSpace Curves SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE MANUFACTURE Space Curve CURVES & SURFACES Curves Space curve Surfaces ⃗ bi APPLICATION ⃗i t Manufacture Metrology Inspection p i =C (ui ) SUMMARY C (u) ni ⃗ z y x Tangent, principal normal, and binormal vectors at point pi .
31. 31. Curves andSpace Curves SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE MANUFACTURE Space Curve CURVES & SURFACES Rectifying plane Curves Space curve Surfaces ⃗ bi APPLICATION ⃗i t Normal plane Manufacture Metrology Inspection p i =C (ui ) SUMMARY Osculating C (u) ni ⃗ plane z y x Moving trihedron (Frenet Frame) of space curve.
32. 32. Curves andSurfaces SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE Parametric Representation of Surfaces MANUFACTURE CURVES & SURFACES Surface Curves ( x(ui , v j ), y (ui , v j ), z (ui , v j )) Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY x =x (u , v ) y= y (u , v) z z =z (u , v) u 1⩽u⩽u 2 v 1⩽v ⩽v 2 y x A surface is a subset of R3 that looks like a piece of R2 in the vicinity of any given point.
33. 33. Curves andSurfaces SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE Parametric Representation of Surfaces MANUFACTURE CURVES & SURFACES Surface Curves ( x(ui , v j ), y (ui , v j ), z (ui , v j )) Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Isoparametric lines v x =x (u , v ) y= y (u , v) z v2 z =z (u , v) u 1⩽u⩽u 2 vj v 1⩽v ⩽v 2 y v1 u x u1 ui u2 Mapping from the parametric space to the Cartesian space.
34. 34. Curves andSurfaces SurfacesRepresentation, and important properties. D. ElKott PREAMBLE MATH TALE Parametric Representation of Surfaces MANUFACTURE CURVES & SURFACES Surface Curves Surfaces n ⃗i, j p i , j = S (ui , v j ) Su i,j APPLICATION Manufacture Metrology Inspection Sv i,j SUMMARY Tangent z plane S (u , v) y x Tangent plane, and normal vector at point pi,j .
35. 35. Curves and Surfaces D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture MetrologyII. Application of Curve and Surface Methods Inspection SUMMARY
36. 36. Curves andBackground SurfacesTheory, and experience. D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection “Experience without theory is blind, but theory without SUMMARY experience is mere intellectual play”. Immanuel Kant (1724 1804)
37. 37. Curves andManufacture SurfacesThe manufacturing process, revisited D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Reject Reject Accept Accept
38. 38. Curves andManufacture SurfacesThe manufacturing process, revisited D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process - Inspection SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Reject Reject Accept Accept
39. 39. Curves andManufacture SurfacesThe manufacturing process, revisited D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & The Manufacturing Process - Inspection Feedback SURFACES Curves Design Process Design Process Surfaces Design Intent Conceptual design. APPLICATION Conceptual design. ● ● CAD Model CAD Model Material Requirement ●FEA. FEA. Manufacture ● (with tolerances) (with tolerances) ●Assembly design. ●Assembly design. Metrology ●Design for manufacture. ●Design for manufacture. Design Constraints .... Inspection .... SUMMARY Generate Generate Manufacture Manufacture Physical Object Physical Object Machining Machining (with manufacturing errors) Part Part (with manufacturing errors) Instructions Instructions (CNC) (CNC) (CAM) (CAM) Feedback Inspection Planning Inspection Planning Measure Measure Analyse Analyse ●Where to measure. ●Where to measure. Part Part Measurement Measurement ●Sample size. Sample size. (CMM) Results ● .... (CMM) Results .... Feedback Reject Reject Accept Accept
40. 40. Curves andMetrology SurfacesThe science of measurement D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Metrology Inspection SUMMARY Metrology is derived from the Greek words: Metron - Measure, and Logos - calculation, study, reason.
41. 41. Curves andMetrology SurfacesThe science of measurement D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Metrology Inspection SUMMARY Metrology is derived from the Greek words: Metron - Measure, and Logos - calculation, study, reason. Metrology is the science of measurement.
42. 42. Curves andMetrology SurfacesThe science of measurement D. ElKott PREAMBLE MATH TALE MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Coordinate Metrology Inspection Coordinate metrology is concerned with measuring the spatial SUMMARY coordinates, mostly of physical objects.
43. 43. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE The Coordinate Measuring Machine MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY The CMM is a Cartesian robot which is utilised to measure the coordinates of surface points.
44. 44. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE The Coordinate Measuring Machine Sensor MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY The CMM is a Cartesian robot which is utilised to measure the coordinates of surface points.
45. 45. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Coordinate Measurement MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY
46. 46. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Coordinate Measurement MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY
47. 47. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Micro-Measuring Technology MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Micro-measuring using hybrid sensors.
48. 48. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Micro-Measuring Technology MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Tactile Sensor Workpiece Optical Sensor Micro-measuring using hybrid sensors.
49. 49. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Micro-Measuring Technology MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Micro-measuring using hybrid sensors.
50. 50. Curves andCoordinate Metrology SurfacesMeasuring physical objects D. ElKott PREAMBLE MATH TALE Micro-Measuring Technology MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION Manufacture Metrology Inspection SUMMARY Micro-components
51. 51. Curves andCoordinate Metrology SurfacesChallenge D. ElKott PREAMBLE MATH TALE Measurement Inspection MANUFACTURE CURVES & SURFACES Curves Surfaces Upper tolerance limit APPLICATION CAD geometry Manufacture Metrology Inspection SUMMARY Lower tolerance limit Physical model Test conformity of the geometry of physical model to the design intent.
52. 52. Curves andCoordinate Metrology SurfacesChallenge D. ElKott PREAMBLE MATH TALE The Inspection Process, Revisited MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION CAD Model Accept Manufacture Accept Metrology Inspection Inspection Process SUMMARY Inspection Process Physical Object Reject Reject
53. 53. Curves andCoordinate Metrology SurfacesChallenge D. ElKott PREAMBLE MATH TALE The Inspection Process - Sources of Error MANUFACTURE CURVES & SURFACES Curves Surfaces APPLICATION CAD Model Accept Manufacture Accept Metrology Inspection Inspection Process SUMMARY Inspection Process Physical Object Reject Reject Algorithmic Software Hardware Human Errors Errors Errors Errors The inspection process is inﬂuenced by several sources of error.