436-105 Engineering Communications GL9:1 GL9: CAD techniques Curves Surfaces Solids Techniques
Parametric curves GL9:2 x = a 1 + b 1 u + c 1 u 2 + d 1 u 3 + y = a 2 + b 2 u + c 2 u 2 + d 2 u 3 + z = a 3 + b 3 u + c 3 u 2 + d 3 u 3 + Again, u is a parameter. Vector notation is more compact. E.g., a parametric cubic curve: R(u) = A + Bu + Cu 2 + Du 3 It is difficult to control the shape of the curve by altering the coefficients.
GL9:3 Bézier curves Input a series of points to form a track Curve passes through first and last track points Gradient at first and last points set by first and last track lines P 1 P 2 P 3 P 0
3.5 Effect of moving P 2 GL9:4 3 2.5 2 z 1.5 1 0.5 0 0 1 2 3 4 5 6 7 x
3.5 Effect of moving P 2 GL9:5 3 2.5 2 z 1.5 1 0.5 0 0 1 2 3 4 5 6 7 x
3.5 Effect of moving P 2 GL9:6 3 2.5 2 z 1.5 1 0.5 0 0 1 2 3 4 5 6 7 x
3.5 Effect of moving P 2 GL9:7 3 2.5 2 z 1.5 1 0.5 Disadvantage: moving one control point affects shape of whole curve 0 0 1 2 3 4 5 6 7 x
GL9:8 Splines composite curve passing through a series of defined points continuity of slope parametric cubic curve segments
GL9:9 B-splines Combine controllability of the Bézier curve, with stability over long spans of composite curve a blended, piecewise polynomial curve passing near a set of control points weighting functions are piecewise moving a control point affects the curve only near that point, not the whole curve order of polynomial independent of number of control points
GL9:10 B-splines original B-spline curve moved vertex adjusted curve affected only near vertex and its neighbours
GL9:11 NURBs Non-Uniform Rational B-splines control points can be weighted differently range of possibilities, from curve fitting all control points, to curve only loosely controlled by control points allow exact representation of conic curves (circle, ellipse, parabola, hyperbola) give great control over line shape use set of control points and knots spacing of knots determines range of influence of weighting functions
GL9:12 Surface modelling Cross-sectional surfaces surface specified in terms of its cross-sections Swept surface constant cross-section shape swept along a trajectory, with possible scale change Lofted surface different cross-section shapes blended (e.g., ships, aircraft)
Example swept surfaces GL9:13
Example lofted surface GL9:14
GL9:15 Surface is lofted Example cross-sections: yacht hull
GL9:16 Sculptured surfaces Just as long curves constructed piecewise with splines, extended surfaces can be formed from a number of patches e.g., Bicubic surface patch D 1 (v) P 11 C 1 (u) Need to ensure continuity of patches P 10 C 0 (u) u P 00 v D 0 (v) P 01 edge curve discontinuity
GL9:17 Solid modelling Represent solids: completely and unambiguously topology consistent Two schemes: Boundary representation (B-rep) Constructive Solid Geometry (CSG)
B-rep models GL9:18 Topological consistency all faces, edges, vertices accounted for Data structure describes the boundaries of the solid topology described in a network which represents connectivity of vertices, edges and faces geometry (points, curves and surfaces) described separately geometry linked to topology face surface edge curve vertex point
Winged-edge data structure GL9:19 Bi-directional pointers from every edge to its: two end vertices two adjacent faces, and four other edges that share a face and vertex with it Left edge anti-clockwise E Next vertex V E Right edge clockwise Left face F E Current edge F Right face Left edge clockwise E V Previous vertex E Right edge anti-c.w.
Constructive Solid Geometry (CSG) GL9:20 Solid represented as Boolean combination of simpler solids (primitives) a b
Example of CSG construction GL9:21 Record Boolean operations in a tree structure: U A B C or as a line of Boolean algebra: A U (B - C)
03sg02_a.flc Creation of a composite solid GL9:22
03sg03_a.flc Creation of a composite model GL9:23
GL9:24 CAD techniques Read Bertoline sections 7.6: 3-D Construction techniques 7.7: 3-D Viewing techniques 7.8: Object modification 7.9: Constraint-based modelling 7.10: Feature-based modelling Web-based instructions for Lab C1 access from 436-105 home page