ME 555: Computer-Aided Design
Estimated study time: 9 minutes
Table of contents
Sources and References
- Mortenson, Geometric Modeling, 3rd ed., Industrial Press.
- Piegl and Tiller, The NURBS Book, 2nd ed., Springer.
- Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th ed., Morgan Kaufmann.
- Shah and Mäntylä, Parametric and Feature-Based CAD/CAM, Wiley.
- Lee, Principles of CAD/CAM/CAE Systems, Addison-Wesley.
- Hoffmann, Geometric and Solid Modeling, Morgan Kaufmann.
Chapter 1: Why Geometric Modelling
Every computer-aided tool in design, analysis, and manufacturing operates on a model of geometry. The quality and type of that model determine what questions can be asked of it. A wireframe describes only edges; it cannot answer “is this point inside the part?” A surface model describes faces but not enclosed volume. A solid model completely describes a three-dimensional region and supports queries, measurements, and operations that wireframes and surfaces cannot.
1.1 Historic Development
Early numerical control systems worked from two-dimensional drawings translated to coded sequences. Sketchpad in the early 1960s pioneered interactive graphical design. The 1970s produced wireframe and polyhedral models and then Constructive Solid Geometry (CSG). The 1980s saw boundary representation (B-rep) mature and the introduction of parametric, feature-based design. Non-uniform rational B-splines (NURBS) unified the representation of curves and surfaces across industries. Contemporary systems combine B-rep geometry, a feature-based modelling history, and assembly mating constraints.
1.2 Engineering Roles of CAD
CAD outputs feed manufacturing (CAM toolpaths), analysis (FEA meshes, CFD domains), documentation (dimensioned drawings), visualization (rendering, VR), and downstream lifecycle activities. Design changes propagate through the model via the parametric history; a well-structured model amplifies design productivity while a brittle model punishes every revision.
Chapter 2: Curves
2.1 Parametric Representation
A parametric curve \( \mathbf{P}(t) \) maps a parameter interval to three-dimensional space. Parametric forms accommodate multi-valued shapes (loops, overhangs) that explicit functions cannot represent. Polynomial bases in \( t \) are the building block because they are closed under the operations CAD needs (differentiation, integration, evaluation).
2.2 Bézier Curves
A Bézier curve of degree \( n \) is
\[ \mathbf{P}(t) = \sum_{i=0}^{n} B_{i,n}(t)\,\mathbf{P}_i, \qquad t \in [0,1], \]with Bernstein basis \( B_{i,n}(t) = \binom{n}{i} t^i (1-t)^{n-i} \). The control polygon \( \{\mathbf{P}_i\} \) provides an intuitive sculpting handle; the curve lies within the convex hull of the control points and passes through the endpoints. De Casteljau’s algorithm evaluates and subdivides a Bézier curve using only linear interpolations and is numerically robust.
2.3 B-Splines and NURBS
A single Bézier segment of high degree is unwieldy; raising degree magnifies changes far from their source. B-splines decompose curves into low-degree pieces with controlled continuity.
\[ \mathbf{P}(t) = \sum_{i=0}^{n} N_{i,p}(t)\,\mathbf{P}_i, \]with \( N_{i,p} \) recursively defined from a knot vector. NURBS generalize this to weighted control points, enabling the exact representation of conic sections (circles, ellipses, hyperbolas) that polynomials cannot capture:
\[ \mathbf{P}(t) = \frac{\sum_i w_i N_{i,p}(t) \mathbf{P}_i}{\sum_i w_i N_{i,p}(t)}. \]NURBS are the industry-standard representation for sculpted geometry in aerospace, automotive, and industrial design.
Chapter 3: Surfaces
3.1 Tensor-Product Surfaces
A tensor-product surface is
\[ \mathbf{S}(u,v) = \sum_{i}\sum_{j} N_{i,p}(u) N_{j,q}(v)\,\mathbf{P}_{i,j}. \]Bézier and NURBS surfaces follow the same pattern with the corresponding bases. They are ideal for rectangular topology and for the large majority of car-body, aircraft-skin, and consumer-product surfacing.
3.2 Non-Tensor Patches
Triangular Bézier patches, Gregory patches, and T-splines handle topologies that tensor products cannot: holes, star-points, and local refinement. Subdivision surfaces start from a coarse polygonal mesh and apply refinement rules (Catmull–Clark, Loop) that converge to smooth limit surfaces; they have become dominant in animation and are increasingly important in industrial design.
3.3 Continuity
Two patches joined along a boundary can be \( C^0 \) (same position), \( G^1 \) (tangent-plane continuous), \( C^1 \) (same first derivative), and so on. Aesthetic surfacing typically demands at least \( G^2 \) to avoid visible reflection discontinuities on shiny bodies; strict \( C^k \) matters for downstream analysis.
Chapter 4: Solid Modelling
4.1 Constructive Solid Geometry
CSG represents a solid as a Boolean expression over primitive solids (blocks, cylinders, spheres, cones, tori) and regularized set operations. The model is a tree; the leaves are primitives and the internal nodes are operations. CSG is compact and unambiguous but must be evaluated (usually into a B-rep) for most queries.
4.2 Boundary Representation
B-rep describes a solid by its bounding surfaces (faces), their edges, and their vertices, together with the topology that glues them. A standard winged-edge or half-edge data structure encodes adjacencies. Euler’s formula for a closed, connected, genus-\( g \) solid without voids is
\[ V - E + F = 2(1 - g). \]Valid B-reps satisfy Euler–Poincaré relations; constructive operations on B-reps are defined as Euler operators that maintain validity by construction.
4.3 Hybrid and Decomposition Models
Most commercial systems are hybrid: user operations may be expressed as CSG-like Booleans, but internal storage is B-rep. Decomposition models — voxels, octrees, cell complexes — are useful for specific applications such as volumetric analysis, medical imaging, and additive manufacturing slicing.
Chapter 5: Parametric and Feature-Based Design
5.1 Constraints and Dimensions
A parametric sketch is a two-dimensional drawing constrained by geometric relations (parallel, perpendicular, tangent, coincident) and dimensional relations (length, radius, angle). The sketch solver computes positions satisfying the constraints; over-constraint and under-constraint are flagged. Well-constrained sketches support robust downstream edits.
5.2 Features and History
A feature is a parametrized operation with semantic intent: extrude, revolve, sweep, loft, shell, rib, hole, fillet, chamfer, pattern. The model is the ordered history of these features applied to the workpiece. Each feature’s parameters are linked to dimensions, part parameters, or equations, producing a design that can be updated by changing high-level parameters.
Design intent — the logical structure that the designer wants preserved under changes — is captured by the choice of features, sketches, and relationships. A change-resilient model has: minimal essential constraints, no extraneous reference geometry, features that reflect manufacturing steps, and parameters that reflect design drivers.
5.3 Assemblies and Mates
An assembly is a collection of parts positioned by mating constraints (coincident, concentric, distance, angle). Motion degrees of freedom remaining after mates implement mechanisms: revolute, prismatic, cylindrical. Top-down assembly modelling defines part geometry in the assembly context so that interfaces between mating parts remain consistent under change.
5.4 CAD in Practice
The laboratory component of CAD education is essential: sculpting a mouse body, assembling a mechanism, documenting a plate with GD&T, and taking a part from CAD to CAM toolpath. Through these exercises the abstractions above become concrete, and the designer develops habits—naming conventions, parameter discipline, feature ordering—that separate robust models from fragile ones.
