CAD_reflection about an arbitrary line and point

cad

1. Department of Mechanical Engineering National Institute of Technology Jamshedpur Deepak Kumar, Department of Mechanical Engineering, NIT Jamshedpur [email protected] Computer Aided Design/Computer Aided Manufacturing

2. National Institute of Technology Jamshedpur Department of Mechanical Engineering

3. National Institute of Technology Jamshedpur Department of Mechanical Engineering

4. 3D Scaling National Institute of Technology Jamshedpur Department of Mechanical Engineering

5. National Institute of Technology Jamshedpur Department of Mechanical Engineering

6. National Institute of Technology Jamshedpur Department of Mechanical Engineering

7. The position vector is assumed to be a row vector in right-handed system National Institute of Technology Jamshedpur Department of Mechanical Engineering

8. National Institute of Technology Jamshedpur Department of Mechanical Engineering

9. National Institute of Technology Jamshedpur Department of Mechanical Engineering

10. National Institute of Technology Jamshedpur Department of Mechanical Engineering

11. National Institute of Technology Jamshedpur Department of Mechanical Engineering

12. National Institute of Technology Jamshedpur Department of Mechanical Engineering

13. Polynomial Fit National Institute of Technology Jamshedpur

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17. Parametric Continuity Conditions To represent a curve as a series of piecewise parametric curves, these curves to fit together reasonably …Continuity! National Institute of Technology Jamshedpur

18. Continuity When two curves are joined, we typically want some degree of continuity across the boundary (the knot) C , “C-zero”, point-wise continuous, curves share the same point where they join Let C 1 (u) and C 2 (u) , be two parametric Curves. C 1 (1) = C 2 (0) National Institute of Technology Jamshedpur

19. C 1 , “C-one”, continuous derivatives, curves share the same parametric derivatives where they join C´ 1 (1)= C´ 2 (0) National Institute of Technology Jamshedpur

20. C 2 , “C-two”, continuous second derivatives, curves share the same parametric second derivatives where they join Higher orders possible C˝ 1 (1)= C˝ 2 (0) National Institute of Technology Jamshedpur

21. Interpolation Splines When polynomial sections are fitted so that the curve passes through each control point, the resulting curve is said to interpolate the set of control points. National Institute of Technology Jamshedpur

22. Approximation Splines When polynomial sections are fitted to the general control point path without necessarily passing through any control point, the resulting curve is said to approximate the set if control points.

23. National Institute of Technology Jamshedpur Department of Mechanical Engineering

24. National Institute of Technology Jamshedpur Department of Mechanical Engineering

25. National Institute of Technology Jamshedpur Department of Mechanical Engineering

26. PROJECTIONS 1. Parallel Projections a) Orthographic Projections b) Axonometric Projections 2. Perspective Transformations and Projections National Institute of Technology Jamshedpur Department of Mechanical Engineering

27. National Institute of Technology Jamshedpur Department of Mechanical Engineering

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32. National Institute of Technology Jamshedpur Department of Mechanical Engineering

33. PERSPECTIVE PROJECTIONS 1. Perspective Transformations and Projections a) Single point b) Two point c) Three Point 2. Vanishing points and trace points National Institute of Technology Jamshedpur Department of Mechanical Engineering

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38. National Institute of Technology Jamshedpur Department of Mechanical Engineering

39. National Institute of Technology Jamshedpur Department of Mechanical Engineering

40. National Institute of Technology Jamshedpur Department of Mechanical Engineering

41. National Institute of Technology Jamshedpur Department of Mechanical Engineering

42. National Institute of Technology Jamshedpur Department of Mechanical Engineering

43. National Institute of Technology Jamshedpur Department of Mechanical Engineering

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52. National Institute of Technology Jamshedpur Department of Mechanical Engineering

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57. National Institute of Technology Jamshedpur Department of Mechanical Engineering

58. PLANE CURVES 1. Analytical Curves 2. Synthetic Curves National Institute of Technology Jamshedpur Department of Mechanical Engineering

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65. Parametric Representation of Curves and Surfaces Two types of equations for curve representation (1) Parametric equation x, y, z coordinates are related by a parametric variable ( u or θ) (2) Nonparametric equation x, y, z coordinates are related by a function National Institute of Technology Jamshedpur Department of Mechanical Engineering

66. Curve Equations National Institute of Technology Jamshedpur Department of Mechanical Engineering

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68. Parametric Equations –Advantages over nonparametric forms National Institute of Technology Jamshedpur Department of Mechanical Engineering

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82. National Institute of Technology Jamshedpur Department of Mechanical Engineering

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85. National Institute of Technology Jamshedpur Department of Mechanical Engineering

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88. y x Curve Fitting Data point approximated by straight Line National Institute of Technology Jamshedpur Department of Mechanical Engineering

89. Is a straight line suitable for each of these cases ? National Institute of Technology Jamshedpur Department of Mechanical Engineering

90. Polynomial Fit National Institute of Technology Jamshedpur Department of Mechanical Engineering

91. National Institute of Technology Jamshedpur Department of Mechanical Engineering

92. National Institute of Technology Jamshedpur Department of Mechanical Engineering

93. National Institute of Technology Jamshedpur Department of Mechanical Engineering

94. Parametric Continuity Conditions To represent a curve as a series of piecewise parametric curves, these curves to fit together reasonably …Continuity! National Institute of Technology Jamshedpur Department of Mechanical Engineering

95. Continuity When two curves are joined, we typically want some degree of continuity across the boundary (the knot) C , “C-zero”, point-wise continuous, curves share the same point where they join Let C 1 (u) and C 2 (u) , be two parametric Curves. C 1 (1) = C 2 (0) National Institute of Technology Jamshedpur Department of Mechanical Engineering

96. C 1 , “C-one”, continuous derivatives, curves share the same parametric derivatives where they join C´ 1 (1)= C´ 2 (0) National Institute of Technology Jamshedpur Department of Mechanical Engineering

97. C 2 , “C-two”, continuous second derivatives, curves share the same parametric second derivatives where they join Higher orders possible C˝ 1 (1)= C˝ 2 (0) National Institute of Technology Jamshedpur Department of Mechanical Engineering

98. Interpolation Splines When polynomial sections are fitted so that the curve passes through each control point, the resulting curve is said to interpolate the set of control points. National Institute of Technology Jamshedpur Department of Mechanical Engineering

99. Approximation Splines When polynomial sections are fitted to the general control point path without necessarily passing through any control point, the resulting curve is said to approximate the set if control points. National Institute of Technology Jamshedpur Department of Mechanical Engineering