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This document discusses various concepts in computer-aided design and manufacturing, particularly focusing on curves and surfaces representation through parametric equations and continuity conditions. It covers different types of splines, including interpolation and approximation splines, as well as projection methods like parallel and perspective projections. The importance of continuity in curve fitting and the mathematical representation of curves is emphasized throughout.
CAD_transformation 3-dimensional presentation file
- 1. 1 National Institute of Technology Jamshedpur Department of Mechanical Engineering Department of Mechanical Engineering National Institute of Technology Jamshedpur Deepak Kumar, Department of Mechanical Engineering, NIT Jamshedpur deepak.me@nitjsr.ac.in Computer Aided Design/Computer Aided Manufacturing
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- 8. 8 3D Scaling National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 11. 11 The position vector is assumed to be a row vector in right-handed system National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 17. 17 Polynomial Fit National Institute of Technology Jamshedpur
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- 21. 21 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
- 22. 22 Continuity When two curves are joined, we typically want some degree of continuity across the boundary (the knot) – C0, “C-zero”, point-wise continuous, curves share the same point where they join Let C1(u) and C2(u) , be two parametric Curves. 0 1 u C1(1) = C2(0) National Institute of Technology Jamshedpur
- 23. 23 – C1, “C-one”, continuous derivatives, curves share the same parametric derivatives where they join C´1(1)= C´2(0) National Institute of Technology Jamshedpur
- 24. 24 – C2, “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
- 25. 25 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
- 26. 26 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.
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- 30. 30 PROJECTIONS 1. Parallel Projections a) Orthographic Projections b) Axonometric Projections 2. Perspective Transformations and Projections National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 37. 37 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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- 62. 62 PLANE CURVES 1. Analytical Curves 2. Synthetic Curves National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 69. 69 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 (uor θ) (2) Nonparametric equation x, y, z coordinates are related by a function National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 70. 70 Curve Equations National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 71. 71 National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 72. 72 Parametric Equations –Advantages over nonparametric forms National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 92. 92 y x Curve Fitting Data point approximated by straight Line National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 93. 93 Is a straight line suitable for each of these cases ? National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 94. 94 Polynomial Fit National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 95. 95 National Institute of Technology Jamshedpur Department of Mechanical Engineering
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- 98. 98 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
- 99. 99 Continuity When two curves are joined, we typically want some degree of continuity across the boundary (the knot) – C0, “C-zero”, point-wise continuous, curves share the same point where they join Let C1(u) and C2(u) , be two parametric Curves. 0 1 u C1(1) = C2(0) National Institute of Technology Jamshedpur Department of Mechanical Engineering
- 100. 100 – C1, “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
- 101. 101 – C2, “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
- 102. 102 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
- 103. 103 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