Engineering Design and Material Selection Lecture 6 - CAD: Freeform Modeling PDF

Engineering Design and Material Selection Lecture 6 — CAD: Freeform Modeling Dr. Tino Stankovic Prof. Dr. Kristina Shea Prof. Kristina Shea 1  Course Schedule Week/ Topic...

Engineering Design and Material Selection Lecture 6 — CAD: Freeform Modeling Dr. Tino Stankovic Prof. Dr. Kristina Shea Prof. Kristina Shea 1  Course Schedule Week/ Topic Case study Quiz Lecturer Dates 1 Introduction and Sketching 2 Introducing Engineering Design Health Prof. Dr. Kristina Shea 3 Technical Drawing: Projections and Cuts 4 CAD: Introduction and Modeling Operations 5 CAD: Features and Parametric Modeling Future Mobility 6 CAD: Freeform Modeling Dr. Tino Stankovic 7 CAD: Assemblies and Standard Mechanical Parts X (45 min) 8 Technical Drawing: Dimensioning Health 9 Sustainability in Engineering Design 10 Materials and their Properties 11 Manufacturing Processes with Focus on Additive Manufacturing Sustainable Materials Prof. Dr. Kristina Shea 12 Material Selection 13 Review and Q+A X (75 min) Prof. Kristina Shea Engineering Design + Computing Laboratory 2  Learning Objectives ▪ Learn different models to represent curves and surfaces (Bézier, B-Spline, and NURBS) in CAD. ▪ Learn the importance of continuity conditions to connect curves and surfaces to represent freeform objects. ▪ WARNING: There will be some simple math…but I am sure you can handle it. Prof. Kristina Shea Engineering Design + Computing Laboratory 3  How can we create freeform shapes in CAD? Ventilator paddle Kyburz PLUS II body Prof. Kristina Shea Engineering Design + Computing Laboratory 4  Kyburz PLUS II Lights Prof. Kristina Shea Engineering Design + Computing Laboratory 5  Freeform Surfaces - Automotive Surfaces Class-B – Stiffening plate Class-A– Outer panel Source: Audi AG Prof. Kristina Shea Engineering Design + Computing Laboratory 6  Freeform Surfaces – Surface Reflection Analysis Surface model used to analyze Real reflections on prototype reflection curves (ICEM Surf) Source: Audi AG Prof. Kristina Shea Engineering Design + Computing Laboratory 7  Unisurf System, Renault, P.E. Bézier,1971 Image source: P. E. Bézier, “Example of an Existing System in the Motor Industry: The Unisurf System”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , Vol. 321, No. 1545, A Discussion on Computer Aids in Mechanical Engineering Design and Manufacture (Feb. 9, 1971), pp. 207-218 Prof. Kristina Shea Engineering Design + Computing Laboratory 8  Types of Geometric Models Geometric models 3D models 2D models 3D 3D surface 3D solid 2D wireframe models models drawings Bézier curves Boundary and surfaces representations (B-reps) B-spline curves Solid Primitives and surfaces NURBS curves and surfaces Prof. Kristina Shea Engineering Design + Computing Laboratory 9  + Volumes 3D Models + Faces Points Solid model Edges Surface model Wireframe model Prof. Kristina Shea Engineering Design + Computing Laboratory 10  3D Surface Model Representation through surfaces: ▪ Use 3D curves and control points to represent a series of surface patches ▪ Surfaces can be represented using different numerical forms (Bézier, B-Spline, NURBS, etc.) ▪ Surface models have no information about volume Teapot represented by 32 Bézier patches Prof. Kristina Shea Engineering Design + Computing Laboratory 11  Representations: Analytic vs. Parametric Geometric Analytic (non-parametric) Parametric representation representation representation  x( u,v) p(u,v) =  y(u,v)  𝑢 𝑣 𝑓(𝑥, 𝑦, 𝑧) = 0 𝑝(𝑢, 𝑣) 𝑧 𝑦 z( u,v)  𝑥 𝑏 𝑧 Elliptic cylinder 𝑎 2 2 𝑥 = 𝑎 ∗ cos(𝑢) 𝑥 𝑦 + =1 𝑦 = 𝑏 ∗ sin(𝑢) 𝑎2 𝑏 2 𝑥 𝑦 z= 𝑣 𝑧 𝑥 = 𝑟 ∗ sin(𝑢)∗ cos(𝑣) 𝑦 = 𝑟 ∗ sin(𝑢)∗ sin(𝑣) Sphere 𝑥2 + 𝑦2 + 𝑧2 = 𝑟2 2r 𝑧 = 𝑟 ∗ cos(𝑢) 𝑥 𝑦 centered at origin (0,0,0) Prof. Kristina Shea Engineering Design + Computing Laboratory 12  Basics: Control Points, Interpolation and Approximation Characteristic polygon y Control point x y P4 y P5 P2 P2 P4 P3 P1 P1 P3 x x Interpolation Approximation Prof. Kristina Shea Engineering Design + Computing Laboratory 13  Surface Models Parametric representation 1,0 1,1 u v 0,0 𝐩(𝑢, 𝑣) 0,1 𝑧 Characteristic polyhedron Control points 𝑦 𝑥 ▪ Surface models are divided into surface patches ▪ A surface can be defined by interpolation or approximation of control points Prof. Kristina Shea Engineering Design + Computing Laboratory 14  Continuity between Curves and Surface Patches ▪ Curves and surface patches are combined to model a composite curve or surface by enforcing constraints on continuity. ▪ Point (0th order, C0) – two curves A and B are joined end to end and share a common end point. ▪ Tangent (1st order, C1) – two curves A and B are joined end to end and share a common end point as well as the same slope at this point. ▪ Curvature (2nd order, C2) – two curves A and B are joined end to end and share a common end point as well as the same slope and curvature at this point. 0th order 1st order 2nd order Prof. Kristina Shea Engineering Design + Computing Laboratory 15  Bézier Curves and Surfaces ▪ Defined by a network of control points, either in approximated or interpolated form ▪ Degree of the curve is determined by the number of control points ▪ Pass exactly through the two end points (curve) or four corner points (surface) ▪ Tangent to the straight-line network at the end points (curve) or corner points (surface) P1 P5 P4 P3 P7 P0 P6 P2 Two Bézier curves joined with continuity of image source: tangents Two Bézier patches Foley et al. (1994) Prof. Kristina Shea Engineering Design + Computing Laboratory 16  𝐩 𝑡 = 1 − 𝑡 2 𝐏0 + 2𝑡 1 − 𝑡 𝐏1 + 𝑡 2 𝐏2 Bézier Curves 𝑛 Definition and Basis Function “Think of expanding 1−𝑡 +𝑡 into a sum” ▪ Approximate real continuous function 𝑓𝑖 𝑡 1 Basis functions 𝐵𝑖,𝑛 on a closed interval using a set of 0.9 (Bernstein polynomials) polynomial functions. 0.8 for 𝑛 = 0, 1, 2 2 1−𝑡 𝑡2 ▪ A point 𝐩 on a Bézier curve: 0.7 𝑛 0.6 2𝑡 1 − 𝑡 𝐩 𝑡 = ෍ 𝐵𝑖,𝑛 𝑡 𝐏𝑖 0.5 𝑖=0 0.4 0.3 𝑡 ∈ 𝑡1 , 𝑡2 Polynomial functions, or 𝑛 + 1 control 0.2 Basis functions points 0.1 Degree of a Bézier curve is 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑝𝑜𝑖𝑛𝑡𝑠 − 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Adapted from: Stroud, I., & Nagy, H. (2011). Solid modelling and CAD systems: how to survive a CAD system. Springer Science & Business Media. 𝑡 = 0, 1 Prof. Kristina Shea Engineering Design + Computing Laboratory 17  𝐩 𝑡 = 1 − 𝑡 3 𝐏0 + 3𝑡 1 − 𝑡 2 𝐏1 + 3𝑡 2 1 − 𝑡 𝐏2 + 𝑡 3 𝐏3 𝐩 𝑡 = 𝐵03 𝐏0 + 𝐵13 𝐏1 + 𝐵23 𝐏2 + 𝐵33 𝐏3 Bézier Curves 𝐩 𝑡 = σ𝑛𝑖=0 𝐵𝑖,𝑛 𝑡 𝐏𝑖 Cubic Bézier Curve Example 1 Control points 0.9 𝐵03 𝐵33 Point 𝐩(𝑡) 0.8 0.7 0.6 𝑡 ∈ 0, 1 0.5 𝐵13 0.4 𝐵23 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Prof. Kristina Shea Engineering Design + Computing Laboratory 18  Question: Bézier curves Consider a Bézier curve defined by 100 control points. Which statements are correct? A. The curve has a degree of 100. B. If one point is moved, the complete curve needs to be recomputed. C. The calculation effort to calculate such a curve is much higher than for a curve defined by 10 control points. Prof. Kristina Shea Engineering Design + Computing Laboratory 19  Bézier Curves Summary ▪ Degree (or the power of the polynomials in Basis function) of a Bézier curve equals: 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑝𝑜𝑖𝑛𝑡𝑠 − 1 ▪ Consequently, the degree of the curve or surface is determined by the number of control points ▪ Further, the calculation effort increases as the number of control points increases. ▪ While the intent of the mathematical model was an “intuitive feel”, moving a single control point affects the whole curve, which can make it difficult for a user to manipulate the control points to achieve the desired curve or surface. Prof. Kristina Shea Engineering Design + Computing Laboratory 20  Splines Image source: Grandine, T. A. (2005). The extensive use Image source: : Farin, G. E. (2002). Curves and surfaces for CAGD: a practical guide. of splines at Boeing. SIAM News, 38(4), 3-6. Morgan Kaufmann Prof. Kristina Shea Engineering Design + Computing Laboratory 21  B-spline Curves Image source: Shah, J. J., & Mäntylä, M. (1995). Parametri and feature-based CAD/CAM: concepts, techniques, and ▪ Generalized form of Bézier curves and surfaces applications. John Wiley & Sons. ▪ A curve consisting of polynomial segments of degree 𝑘 − 1 and a network of 𝑛 + 1 control points 𝐏𝑖 ▪ Goal: Obtain better local control 𝑛 𝐩 𝑡 = ෍𝑁 𝑡 𝑖,𝑘 𝐏𝑖 𝑖=0 Basis functions 𝑛 + 1 control points 𝑡 ∈ 𝐮, where 𝐮 is a knot vector 𝐮 = (𝑢0 , 𝑢1 , … , 𝑢𝑛 , … , 𝑢𝑛+𝑘 ), with 𝑢0 ≤ 𝑡 ≤ 𝑢𝑛+𝑘 Adapted from: Hirz, M., Dietrich, W., Gfrerrer, A., & Lang, J. (2013). Integrated computer-aided design in automotive development. Springer-Verl. Berl.-Heidelb. DOI, 10, 978-3. Prof. Kristina Shea Engineering Design + Computing Laboratory 22  B-spline Curves Basis functions 𝑁 𝑡 𝑖,𝑘 example for 𝑛 = 4 (5 control points), 𝑘 = 3, and control point 𝑖 = 1 ▪ The knot vector 𝐮 = (𝑢0 , 𝑢1 , … , 𝑢𝑛=4 , … , 𝑢𝑛+𝑘=7 ): 𝑁𝑖=1,𝑘=1 (𝑡) 1 1 Controls 𝑡 ∈ ൣ𝑢1 , 𝑢2 ) 0.9 𝐵03 𝐵33 0.9 Recall the cubic Bézier example: 𝑛 0.8 𝑁𝑖=1,𝑘=2 (𝑡) 0.8 𝐩 𝑡 = ෍ 𝐵𝑖,𝑛 𝑡 𝐏𝑖 0.7 Controls 𝑡 ∈ 𝑢1 , 𝑢3 0.7 𝑖=0 0.6 0.6 𝑡 = 0, 1 0.5 𝑁𝑖=1,𝑘=3 (𝑡) 0.5 Controls 𝑡 ∈ 𝑢1 , 𝑢4 𝐵13 0.4 0.4 𝐵23 0.3 𝑛 0.3 0.2 𝐩 𝑡 = ෍𝑁 𝑡 𝑖,𝑘 𝐏𝑖 0.2 𝑖=0 0.1 0.1 Degree 𝑘 = 3 − 1 = 2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 𝑢0 0 1 𝑢 1 𝑢2 2 𝑢3 3 4 𝑢 4 5𝑢 5 6 𝑢 6 7 𝑢 7 Prof. Kristina Shea Engineering Design + Computing Laboratory 23  B-spline Curves 𝑛+𝑘 =4+3=7 Control points Examples 5 control points and for 𝑘 = 3 𝐮 = (0, 1, 2, 3, 4, 5, 6, 7) Knots 𝑘−1=2 𝑖=2 𝑖=3 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [2, 5] Each curve segment is a parabolic function (𝑘 − 1 = 2) controlled by exactly 𝑘 = 3 control points in this case. Prof. Kristina Shea Engineering Design + Computing Laboratory 24  B-spline Curves 𝑛+𝑘 =4+3=7 Control points Examples 5 control points and for 𝑘 = 3 𝐮 = (0, 1, 2, 3, 4, 5, 6, 7) Knots 𝑘−1=2 𝑖=2 𝑖=3 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [2, 5] Each curve segment is a parabolic function (𝑘 − 1 = 2) controlled by exactly 𝑘 = 3 control points in this case. Prof. Kristina Shea Engineering Design + Computing Laboratory 25  B-spline Curves 𝑛+𝑘 =4+2=6 Control points Examples 5 control points and for 𝑘 = 2 𝐮 = (0, 1, 2, 3, 4, 5, 6) Knots 𝑘−1=1 𝑖=2 𝑖=3 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [1, 5] Each curve segment is a line (𝑘 − 1 = 1) controlled by exactly 𝑘 = 2 control points. Control polygon coincides with curve segments. Prof. Kristina Shea Engineering Design + Computing Laboratory 26  B-spline Curves 𝑛+𝑘 =4+2=6 Control points Examples 5 control points and for 𝑘 = 2 𝐮 = (0, 1, 2, 3, 4, 5, 6) Knots 𝑘−1=1 𝑖=2 𝑖=3 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [1, 5] Each curve segment is a line (𝑘 − 1 = 1) controlled by exactly 𝑘 = 2 control points. Control polygon coincides with curve segments. Prof. Kristina Shea Engineering Design + Computing Laboratory 27  B-spline Curves End-point Interpolation for 5 control points and for 𝑘 = 3 1 1 (2.5, 0.9) (2.5, 0.9) 0.9 0.9 (1.0, 0.8) (1.0, 0.8) 0.8 (3.5, 0.8) 0.8 (3.5, 0.8) 0.7 0.7 0.6 0.6 0.5 𝑛+𝑘 =4+3=7 0.5 0.4 𝐮 = (0, 1, 2, 3, 4, 5, 6, 7) 0.4 𝐮 = (0, 0, 0, 1, 2, 3, 3, 3) 0.3 0.3 𝑘-times knot 0.2 0.2 multiplicity at each end 0.1 0.1 (0.0, 0.0) 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [2, 5] (5.0, 0.0) (0.0, 0.0) 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [0, 3] (5.0, 0.0) 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Prof. Kristina Shea Engineering Design + Computing Laboratory 28  B-spline Curves Generalization of Bézier curves using B-splines 1 1 (2.5, 0.9) (2.5, 0.9) 0.9 0.9 (1.0, 0.8) (1.0, 0.8) 0.8 (3.5, 0.8) 0.8 (3.5, 0.8) 0.7 0.7 0.6 𝑘−1=4 0.6 0.5 0.5 0.4 0.4 B-Spline with end-point interpolation, 5 Bézier curve with 5 control points and 0.3 0.3 control points and 𝑘 = 5, from which it degree 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑝𝑜𝑖𝑛𝑡𝑠 − 1 = 0.2 follows that the degree is 𝑘 − 1 = 4 0.2 5−1=4 0.1 0.1 (0.0, 0.0) 𝐮 = (0, 0, 0, 0, 0, 1, 1, 1, 1, 1) (5.0, 0.0) (0.0, 0.0) (5.0, 0.0) 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 𝑡 ∈ 𝑢𝑘−1 , 𝑢𝑛+1 = [0, 1] 𝑡 = 0, 1 Prof. Kristina Shea Engineering Design + Computing Laboratory 29  “Splines” in NX 𝑘 − 1 = 1 (linear) 𝑘−1=2 𝑘 − 1 = 3 (cubic) (quadratic) Two splines joined with a tangent constraint 𝑘 − 1 = 4 (4th degree) Prof. Kristina Shea Engineering Design + Computing Laboratory 30  B-spline Curves and Surfaces II ▪ Control points only influence a local area, defined by the curve degree, 𝑘 − 1 ▪ Approximated and interpolated forms Moving a control point location, P3, affects Interpolated B-spline only part of the curve. surface patch Prof. Kristina Shea Engineering Design + Computing Laboratory image source: Shah and Mäntylä (1995) 31  B-spline Curves Summary ▪ A B-spline is a composite curve composed of separate curves, i.e. 𝐩1 (𝑡), … , 𝐩4 (𝑡) ▪ The curves are joined at the points at which the parameter values are knot values. ▪ If the knot intervals are of equal lengths, then the knot vector is called uniform. ▪ The degree of each curve segment equals 𝑘 − 1. ▪ Increasing the number of control points does not affect the degree of each curve segment. Prof. Kristina Shea Engineering Design + Computing Laboratory 32  B-spline Curves Summary ▪ The continuity C at the knot equals C𝑘−2. ▪ Each separate curve, e.g. 𝐩1 (𝑡), is affected by 𝑘 control points. ▪ Each control point, 𝐏𝑖 , affects a maximum of 𝑘 separate curves. ▪ If the number control points is set equal to 𝑘, the curve represents the special case of the Bézier curve. Prof. Kristina Shea Engineering Design + Computing Laboratory 33  Question: Which control point(s) 𝐏𝑖 affect the segment 𝐩4 (𝑡) if 𝑘 = 3? A. P0 B. P1 𝐩4 (𝑡) 𝑘=3 𝐩3 (𝑡) C. P2 𝐩2 (𝑡) D. P3 𝐩1 (𝑡) E. P4 F. P5 Prof. Kristina Shea Engineering Design + Computing Laboratory 34  Rational Curves 𝑛 1 (0.0, 1.0) (1.0, 1.0) ▪ Recall: 𝐩 𝑡 = ෍ 𝐵𝑖,𝑛 𝑡 𝐏𝑖 0.9 𝑖=0 0.8 ▪ For 𝑛 = 2 we rewrite 𝐩 𝑡 using weights 𝑤𝑖 as: 0.7 2 2 1 − 𝑡 2 𝐏0 𝑤0 + 2𝑡 1 − 𝑡 𝐏1 𝑤1 + 𝑡 2 𝐏2 𝑤2 0.6 ( , ) 𝐩 𝑡 = 2 2 1 − 𝑡 2 𝑤0 + 2𝑡 1 − 𝑡 𝑤1 + 𝑡 2 𝑤2 0.5 0.4 Bézier in the rational form 0.3 ▪ By setting 𝑤0 = 𝑤2 = 1 ,we solve 0.2 T 2 2 2 0.1 Quadratic Bézier Curve, 𝑛 = 2 ▪ 𝐩(𝑡 = 0.5) = , 2 2 for 𝑤1 and obtain: 𝑤1 = 2 Arc 𝑅 = 1 (1.0, 0.0) 0 Adapted from: Stroud, I., & Nagy, H. (2011). Solid modelling and CAD systems: how to survive a 0 0.2 0.4 0.6 0.8 1 CAD system. Springer Science & Business Media. Prof. Kristina Shea Engineering Design + Computing Laboratory 36  Rational Curves 1 1 (0.0, 1.0) (1.0, 1.0) (0.0, 1.0) (1.0, 1.0) 0.9 0.9 0.8 0.8 0.7 0.7 2 2 0.6 ( , ) 0.6 2 2 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 Quadratic Rational Bézier Curve 0.1 Quadratic Bézier Curve, 𝑛 − 1 = 2 0.1 Quadratic Bézier Curve Arc 𝑅 = 1 (1.0, 0.0) Arc 𝑅 = 1 (1.0, 0.0) 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Prof. Kristina Shea Engineering Design + Computing Laboratory 37  NURBS Curves and Surfaces ▪ NURBS = Non-Uniform Rational B-Spline ▪ Increased generality of B-spline to describe almost any freeform curve and surface ▪ Allow “knot values” to be non-uniformly spaced ▪ Control points can now be weighted Curves and control points Control Point P4 P2 3 P5 1 Knot 1,5 Knot 2 Weights σ𝑛𝑖=0 𝑤𝑖 𝑁 𝑡 𝑖,𝑘 𝐏𝑖 Surface P6 𝐩 𝑢 = 𝑛 P1 Knot & P3 Knot & Control Point σ𝑖=0 𝑤𝑖 𝑁 𝑡 𝑖,𝑘 Control Point Prof. Kristina Shea Engineering Design + Computing Laboratory 38 example created with Rhinoceros®  Lofting – Extrusion Based on Several Profiles ▪ Solid is created by interpolating a series of sections. ▪ Common applications in wing, hull, and forged objects design. ▪ Can be difficult to control. Goal: Combustion engine model Connecting rod Prof. Kristina Shea Engineering Design + Computing Laboratory 39  Lofting Example - Connecting Rod Modeling I 1. 2. 3. 4. 5. 6. Prof. Kristina Shea Engineering Design + Computing Laboratory 40  Lofting Example - Connecting Rod Modeling II 7. 8. 9. 10. 11. 12. Prof. Kristina Shea Engineering Design + Computing Laboratory 41  Creating Splines and Surfaces in NX (1) ▪ Create sections curves ▪ spline and line ▪ set different continuity conditions ▪ First Section: “G1” First (G1) ▪ Middle Section: “G0” ▪ Last Section: “G2” Middle (G0) Last (G2) Prof. Kristina Shea Engineering Design + Computing Laboratory 42  Creating Splines and Surfaces in NX (2) ▪ Reflections of extrude operation based on the splines: First (G1) ▪ First Section: “G1” ▪ Middle Section: “G0” ▪ Last Section: “G2” Middle (G0) Last (G2) Prof. Kristina Shea Engineering Design + Computing Laboratory 43  Creating Splines and Surfaces in NX (3) ▪ Surface (Studio Surface) based on the three original First (G1) section curves ▪ Reflections showing that Middle (G0) continuity in section curves is not “G2” (curvature continuity) Last (G2) Prof. Kristina Shea Engineering Design + Computing Laboratory 44  Creating Splines and Surfaces in NX (4) ▪ Surface (Studio Surface) based on the three section First (G2) curves with “G2” continuity ▪ Reflections showing that Middle (G2) curvature continuity in the sections curves is now constrained (“G2”) Last (G2) Prof. Kristina Shea Engineering Design + Computing Laboratory 45  Question: Which category of curves offers the most possibilities to change the shape of the curve? A. Bézier curves B. B-Splines C. NURBS Prof. Kristina Shea Engineering Design + Computing Laboratory 46  Freeform Modeling: Wrap-Up Freeform surfaces are widely used for bodywork, wings, and forged parts. Freeform surfaces cannot be created often with extrude, revolve or sweep operations. Curves and surfaces can be parametrized with Bézier, B- Splines or NURBS using control points. Different continuity conditions define how different curves and surfaces are connected. Reflections in CAD can be used to visualize surfaces and their continuity conditions. Prof. Kristina Shea Engineering Design + Computing Laboratory 47  Exercise 6 – Freeform modeling in CAD Feature-based design (continuation) Free-form modeling (Splines, surfaces, and continuity) Prof. Kristina Shea Engineering Design + Computing Laboratory 48  Exercise 6 – Freeform modeling in CAD Design your own car top! Prof. Kristina Shea Engineering Design + Computing Laboratory 49