Part 4-Types and mathematical representations of Curves .pptx

lecture 4

1. Part #4 Types and Mathematical Representations of Curves 1 Oct., 2022

2. Part #4-Outline Lecture 1: Introduction Types of Curves Representation of Curves Parametric Representation of Curves Interpolation and Approximation of Curve Properties for Curve Design Lecture 2 & 3: Synthesis (Free-Form) Curves Hermite Curve Bezier Curve B-Spline Curves Rational B-splines Curves (including Non-uniform rational B-splines – NURBS) Lecture 4: Techniques for Surface Modeling Surface Patch Coons and Bicubic Patches Bezier and B-Spline Surfaces.

3. Objectives: Develop the various mathematical representations of the curves and surfaces used in geometric construction Understand the parametric representation of curves and surfaces and their relationship with computer graphics.

4. Types, Representation & Properties of Curves Lecture 1: Introduction Types of Curves Representation of Curves Parametric Representation of Curves Interpolation and Approximation of Curve Properties for Curve Design

5. Introduction This class presents the available types of entities of the modeling technique and their related mathematical representations to enable good understanding of how and when to use these entities in engineering applications. Users usually have to decide on the type of modeling technique based on the ease of using the technique during the construction phase and on the expected utilization of the resulting database later in the design and manufacturing processes. The most mathematically straight forward geometric entities are curves. A curve is an integral part of any design and an engineer needs to draw one or other type of curves or curved surfaces applicable to many engineering components used in automotive, aerospace and hydrospace industries. Different types of shape constraints (e.g., continuity and/or curvature ) are imposed to accomplish specific shapes of the curve or curved surfaces. When a curve is two-dimensional , it lies entirely in a plane known as planar curve . However, three-dimensional curve lies in space called space curve .

6. Introduction (a) Evenly Spaced data points (b) Non-evenly spaced data points

7. Analytical Curve & Synthetic Curve are a two types of the curves used in geometric modeling. Analytical Curve: This types of curve can be represented by simple analytical (mathematical) equations for which input is standard analytical mathematical equation like point, line, arc, circle, ellipse, parabola & hyperbola . Conic curves or conics are the curves formed by the intersection of a plane with a right circular cone (parabola, hyperbola and sphere). These basic entities has a fixed form & cannot be modified to achieve a shape that violates the mathematical equations. They can be combined together with various end conditions to generate the overall curve design. Types of curves Analytical Curve: A parabola is the curve created when a plane intersects a right circular cone parallel to the side (elements) of the cone. An ellipse is the curve created when a plane cuts all the elements ( sides ) of the cone but its not perpendicular to the axis.

8. A hyperbola is the curve created when a plane parallel to the axis and perpendicular to the base intersects a right circular cone. This type of curve representation has the following advantages: Types of curves Analytical Curve:

9. Types of curves Synthetic Curve : Synthetic Curve: Unfortunately, it is not possible to represent all types of curves required in engineering applications analytically ; therefore, the method based on the data points ( synthetic curves ) is very useful in designing the objects with curved shapes. Products such as car bodies, ship hulls, airplane fuselage and wings, propeller blades, shoe insoles, and bottles are a few examples that require free-form , or synthetic curves and surfaces . It is computed by using geometric input parameters like point, tangent. These parameters are processed to generate curves such as Bezier , Hermite cubic , B-spline .

10. Types of curves Synthetic Curve:

11. Mathematically, curves can be described by two techniques: Non-Parametric Equations can be explicit or implicit , and Parametric Equations Non-Parametric Equations: Representation of curves

12. Non-Parametric Equations: Representation of curves

13. Representation of curves Non-Parametric Equations:

14. Non-Parametric Equations: Representation of curves

15. Non-Parametric Equations: Representation of curves Nonparametric equations must be solved simultaneously to determine points on the curves , inconvenient process  . If the slope of a curve at a point is vertical or near vertical , its value becomes infinity or very large . However; the shapes of most engineering objects are intrinsically independent of any coordinate system, the equations are dependent on the choice of coordinate system . If the curve is to be displayed as a series of point or straight-line segments , the computations involved could be extensive . Although non-parametric representation of curve equations are used in some cases, they are not in general suitable for CAD. There are four problems with describing curves using nonparametric equations:

16. Parametric Curves: Representation of curves

17. Parametric Curves: Representation of curves

18. Parametric Curves: Representation of curves

19. Parametric Curves: Representation of curves

20. Parametric Curves: Representation of curves

21. Parametric Representation of Analytic Curves

22. Parametric Representation of Analytic Curves Lines:

23. Parametric Representation of Analytic Curves

24. Parametric Representation of Analytic Curves Circles:

25. Parametric Representation of Analytic Curves Circles:

26. Parametric Representation of Analytic Curves Circles:

27. Parametric Representation of Analytic Curves

28. Parametric Representation of Analytic Curves

29. The analytical form of planar curves is not suitable for designing the complex three-dimensional curves and surfaces used for designing the complex shaped objects. The need for synthetic curves in design arises on two occasions; when a curve is represented by a collection of measured data points and when an existing curve must change to meet new design requirements. The designer prefers the synthetic curve, which passes through the set of data points, because designer has full control on its shape as per the new design requirements. A spline curve is defined by giving a set of coordinate positions , call control points , which indicate the general shape of the curve. These control points are then fitted with piecewise continuous parametric polynomial functions . Mathematically, curve fitting ( interpolation ) and curve fairing ( approximation ) techniques are used for generating the curves & surfaces in CAD : Interpolation Techniques: If the problem of curve design is a problem of data fitting, the classic interpolation solutions are used. When polynomial sections are fitted so that the curve passes through all the points , or through each control point , the resulting curve is said to interpolate the set of control points. Approximation Techniques: On the other hand, if the problem is dealing with free-form design with smooth shapes, approximation methods are used. when the polynomials are fitted to the general control point path without necessarily passing through any control point , the resulting curve is said to approximate the set of control points. Interpolation and Approximation of Curve:

30. Interpolation and Approximation of Curve:

31. Interpolation and Approximation of Curve:

32. Interpolation and Approximation of Curve:

33. Interpolation and Approximation of Curve:

34. Interpolation and Approximation of Curve:

35. Interpolation and Approximation of data points (a) curve fitting (b) curve fairing Interpolation and Approximation of Curve:

36. Interpolation and Approximation of Curve:

37. Properties for Curve Design

38. Properties for Curve Design

39. Properties for Curve Design

40. Properties for Curve Design

41. Properties for Curve Design

42. Properties for Curve Design (a) Zero Order Continuity (b) First Order continuity (c) Second Order Continuity

43. Properties for Curve Design

44. Lecture 2 & 3: Synthesis (Free-Form) Curves Hermite Curve Bezier Curve B-Spline Curves Rational B-splines Curves (including Non-uniform rational B-splines – NURBS) Synthesis (Free-Form) Curves

45. Synthesis (Free-Form) Curves

46. Synthesis (Free-Form) Curves

47. Synthesis (Free-Form) Curves

48. Synthesis (Free-Form) Curves

49. Synthesis (Free-Form) Curves

50. Synthesis (Free-Form) Curves: Spline Curves

51. Synthesis (Free-Form) Curves: Spline Curves

52. Synthesis (Free-Form) Curves: Spline Curves

53. Synthesis (Free-Form) Curves: Spline Curves

54. Synthesis (Free-Form) Curves: Spline Curves

55. Synthesis (Free-Form) Curves: Spline Curves

56. Classifications of Spline Curves

57. Classifications of Spline Curves

58. Classifications of Spline Curves

59. Hermite Curves

60. Hermite Curves

61. Hermite Curves

62. Hermite Curves

63. Hermite Curves

64. Hermite Curves

65. Hermite Curves

66. Hermite Curves

67. Hermite Curves

68. Hermite Curves

69. Hermite Curves

70. Hermite Curves

71. Hermite Curves

72. Hermite Curves

73. Hermite Curves Shape Control

74. Hermite Curves

75. Hermite Curves

76. Hermite Curves

77. Hermite Curves

78. Hermite Curves Limitations

79. Hermite Curves Limitations

80. Hermite Curves Example:

81. Hermite Curves Example:

82. Hermite Curves

83. Hermite Curves

84. Hermite Curves

85. Hermite Curves

86. Hermite Curves

87. Bezier Curves In fact, two outside points control the shape of Bézier curves

88. Bezier Curves

89. Bezier Curves

90. Bezier Curves

91. Bezier Curves

92. Bezier Curves Effect of position of control points on the shape of cubic Bézier curves

93. Bezier Curves Blending Functions Formulation

94. Bezier Curves Blending Functions Formulation

95. Bezier Curves Blending Functions Formulation

96. Bezier Curves Blending Functions Formulation

97. Bezier Curves Blending Functions Formulation

98. Bezier Curves Blending Functions Formulation

99. Bezier Curves Blending Functions Formulation

100. Bezier Curves Blending Functions Formulation

101. Bezier Curves Blending Functions Formulation

102. Bezier Curves Blending Functions Formulation (a) Bézier/Bernstein blending functions for three control points (b) Bézier/Bernstein blending functions for four control points (c) Bézier/Bernstein blending functions for five control points

103. Bezier Curves The properties of Bézier curves depend upon the properties of Bernstein polynomial . Properties of Bézier Curves

104. Bezier Curves Properties of Bézier Curves The properties of Bézier curves are:

105. Bezier Curves Properties of Bézier Curves

106. Bezier Curves Properties of Bézier Curves

107. Bezier Curves Properties of Bézier Curves

108. Bezier Curves Properties of Bézier Curves This represents that the sequence of control points defining the curve can be changes without modify of the curve shape.

109. Bezier Curves Properties of Bézier Curves

110. Bezier Curves Properties of Bézier Curves

111. Bezier Curves Composite Bézier Curves

112. Bezier Curves Composite Bézier Curves

113. Bezier Curves Composite Bézier Curves

114. Bezier Curves Example:

115. Bezier Curves Example:

116. Bezier Curves Example:

117. Bezier Curves Drawbacks of Composite Bézier Curves:

118. Bezier Curves Drawbacks of Composite Bézier Curves:

119. Bezier Curves Drawbacks of Composite Bézier Curves:

120. Essential Requirements for Synthetic Curves

121. B-Spline Curves

122. B-Spline Curves

123. B-Spline Curves

124. B-Spline Curves

125. B-Spline Curves

126. B-Spline Curves

127. B-Spline Curves

128. B-Spline Curves Types of B-spline Curves:

129. Techniques for Surface Modeling Lecture 4: Techniques for Surface Modeling Surface Patch Coons and Bicubic Patches Bezier and B-Spline Surfaces.