The Central Idea
Computer-aided design represents intended geometry in a form that a computer can construct, modify, measure, verify, display, and exchange.
A useful summary is
\[ \boxed{ \text{CAD} = \text{geometry} + \text{constraints} + \text{parameters} + \text{construction rules} }. \]The goal is not merely to draw a shape but to encode how that shape is defined.
From Drawing to Model
A traditional drawing records a visible representation of an object.
A CAD model may additionally store:
- exact coordinates;
- dimensions;
- angles;
- radii;
- parallel and perpendicular relationships;
- symmetry;
- feature dependencies;
- material or assembly information.
CAD Is Primarily Vector Geometry
CAD systems generally represent geometry using coordinates, curves, surfaces, and solids rather than fixed pixels.
This allows a line to remain mathematically straight and a circle to remain mathematically circular regardless of display zoom.
The model is rasterized only when it is shown on a screen or exported as an image.
Sketches
Many CAD workflows begin with a two-dimensional sketch.
A sketch may contain:
- points;
- line segments;
- circles;
- arcs;
- ellipses;
- splines;
- construction geometry.
Sketch Geometry Is Usually Constrained
A segment drawn by hand with a mouse may initially have approximate coordinates.
Constraints can then specify the intended geometry exactly.
Examples include:
\[ \boxed{ \text{horizontal, vertical, parallel, perpendicular, tangent, concentric} }. \]Dimensional Constraints
A dimensional constraint assigns a numerical quantity such as:
- length;
- distance;
- angle;
- diameter;
- radius;
- offset.
For example,
\[ \|B-A\|=50 \]constrains segment \(AB\) to have length \(50\) in the chosen units.
Geometric Constraints
A geometric constraint specifies a relationship rather than a measurement.
For direction vectors \(\mathbf{u}\) and \(\mathbf{v}\):
parallelism can be expressed as
\[ \mathbf{u}\times\mathbf{v}=0, \]while perpendicularity can be expressed as
\[ \boxed{ \mathbf{u}\cdot\mathbf{v}=0 }. \]Constraint Solving
A constrained sketch forms a system of equations and geometric conditions.
The CAD system solves that system to determine coordinates that satisfy the requested relationships.
Thus sketching can become a problem in algebraic and numerical constraint solving.
Degrees of Freedom
A free point in the plane has two positional degrees of freedom:
\[ x \qquad\text{and}\qquad y. \]Constraints remove degrees of freedom.
A fully constrained sketch has enough independent conditions to determine the intended geometry.
Underconstrained Geometry
A sketch is underconstrained when some geometric freedom remains.
For example, a circle may have a fixed radius but an unfixed center.
It can still move while satisfying the known constraint.
Fully Constrained Geometry
A sketch is fully constrained when the intended geometry is determined up to the CAD system's accepted conventions.
A fully constrained model is generally more predictable when dimensions are later changed.
Overconstrained Geometry
A sketch can contain too many constraints.
If the conditions conflict, there may be no solution.
For example, requiring the same segment to have both
\[ \|B-A\|=40 \]and
\[ \|B-A\|=50 \]is inconsistent.
Redundant Constraints
Constraints may also be redundant without being contradictory.
If two independent conditions already force a third relationship, explicitly adding that third condition may not add new geometric information.
Constraint solvers must distinguish redundancy from inconsistency.
Parametric Design
In parametric CAD, dimensions and features depend on named parameters.
Suppose
\[ W=100 \]and
\[ H=\frac{W}{2}. \]Changing \(W\) automatically updates \(H\).
Parameters Create Relationships
Instead of specifying isolated values, a designer can encode design logic:
\[ \boxed{ d_2=2d_1+5 }. \]The model then updates according to the relationship whenever \(d_1\) changes.
Feature-Based Modeling
Many solid CAD systems organize construction as a sequence of features.
Examples include:
- extrusions;
- revolutions;
- holes;
- fillets;
- chamfers;
- patterns;
- shell operations.
Feature History
A model may store an ordered construction history:
\[ F_1\to F_2\to F_3\to\cdots\to F_n. \]Later features can depend on faces, edges, dimensions, or geometry created by earlier features.
Regeneration
When an early parameter changes, the CAD system may recompute later features in dependency order.
This process is often called regeneration.
Parametric modeling therefore combines geometry with a dependency graph.
Extrusion
An extrusion extends a planar profile along a direction.
If profile point \(P\) is moved by distance \(t\) along direction \(\mathbf{d}\), then
\[ \boxed{ P(t)=P+t\mathbf{d} }. \]Extruding a closed planar region produces a three-dimensional solid.
Revolution
A revolution rotates a profile around an axis.
In two dimensions, rotation by angle \(\theta\) about the origin uses
\[ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}. \]Three-dimensional revolved solids extend this same transformation idea.
Sweep
A sweep moves a profile along a path.
The path may be:
- a line;
- an arc;
- a spline;
- a spatial curve.
The profile's orientation may remain fixed or change along the path.
Loft
A loft creates a surface or solid that transitions through a sequence of cross-sectional profiles.
The geometry must interpolate or approximate the specified sections while maintaining suitable continuity.
Curves in CAD
CAD systems rely heavily on parametric curves.
A curve can be represented as
\[ \boxed{ C(t)= \bigl(x(t),y(t),z(t)\bigr) }. \]Different curve families provide different control over shape and smoothness.
Bézier Curves
A Bézier curve is controlled by a finite set of control points.
For a quadratic Bézier curve with control points \(P_0,P_1,P_2\),
\[ \boxed{ B(t)= (1-t)^2P_0+ 2(1-t)tP_1+ t^2P_2 } \]for
\[ 0\le t\le1. \]Splines
Splines use piecewise polynomial curves joined with controlled continuity.
They allow smooth complex shapes without requiring one extremely high-degree polynomial.
CAD systems commonly use B-splines and related forms.
NURBS
NURBS stands for non-uniform rational B-splines.
NURBS are important because they can represent:
- free-form curves;
- free-form surfaces;
- circles;
- ellipses;
- many engineering shapes.
Surface Representation
A parametric surface uses two parameters:
\[ \boxed{ S(u,v)= \bigl(x(u,v),y(u,v),z(u,v)\bigr) }. \]Holding one parameter constant traces a curve across the surface.
Surface Patches
Complex surfaces are often assembled from multiple patches.
Adjacent patches may be required to meet with:
- positional continuity;
- tangent continuity;
- curvature continuity.
Solid Modeling
A solid model represents a filled three-dimensional region rather than only its visible surface.
This supports questions such as:
- What is the volume?
- What is inside or outside?
- Do two parts intersect?
- What remains after material is removed?
Boundary Representation
A common solid representation is boundary representation, or B-rep.
The solid is described through connected:
- vertices;
- edges;
- faces;
- surface geometry;
- topological relationships.
Geometry and Topology in B-rep
Geometry tells us where surfaces, curves, and points are located.
Topology tells us how they are connected.
Two faces may lie on similar geometric surfaces yet represent different parts of a solid because their topological roles differ.
Constructive Solid Geometry
Another solid-modeling approach builds complicated solids from simpler ones using Boolean operations.
Common primitives include:
- boxes;
- cylinders;
- spheres;
- cones;
- prisms.
Boolean Union
The union of solids \(A\) and \(B\) is
\[ \boxed{ A\cup B }. \]It contains every point belonging to either solid.
Boolean Intersection
The common volume is
\[ \boxed{ A\cap B }. \]This operation retains only the region occupied by both solids.
Boolean Difference
Subtracting \(B\) from \(A\) produces
\[ \boxed{ A\setminus B }. \]This is a natural mathematical model of machining away material or cutting a hole.
Boolean Operations Require Intersection Geometry
To combine solids correctly, a CAD kernel may need to:
- intersect surfaces;
- split curves and faces;
- classify pieces as inside or outside;
- reconstruct valid topology.
This directly uses the intersection and point-in-region ideas studied earlier.
Fillets
A fillet replaces a sharp edge with a rounded transition.
In a simple planar case, a circular arc is inserted tangent to two adjacent edges.
Three-dimensional filleting generalizes the same smooth-transition idea along solid edges.
Chamfers
A chamfer replaces a sharp corner or edge with a flat beveled face.
The new geometry may be controlled by:
- one distance;
- two distances;
- a distance and angle.
Patterns
Repeated features can be generated from transformations.
A linear pattern may use
\[ P_k=P_0+k\mathbf{d}. \]A circular pattern may use repeated rotations by
\[ \Delta\theta= \frac{2\pi}{n}. \]Local Coordinate Systems
CAD models often use local coordinate systems attached to:
- parts;
- assemblies;
- sketch planes;
- features;
- machines or tools.
Transformations connect these local frames to a global frame.
Homogeneous Transformations
In three-dimensional CAD and graphics, rotations and translations are often combined using homogeneous coordinates.
A transformation can be written as
\[ \boxed{ \begin{bmatrix} \mathbf{x}'\\ 1 \end{bmatrix} = \begin{bmatrix} R & \mathbf{t}\\ \mathbf{0}^{T} & 1 \end{bmatrix} \begin{bmatrix} \mathbf{x}\\ 1 \end{bmatrix} }. \]Assembly Modeling
A CAD assembly combines multiple parts.
Relationships between parts may constrain:
- coincident faces;
- concentric holes;
- parallel planes;
- fixed distances;
- rotational or translational motion.
Assembly Constraints
If two cylindrical axes should coincide, their direction vectors must be parallel and their axis lines must share the same geometric locus.
Assembly constraints therefore create another geometric constraint system.
Interference Detection
CAD assemblies often need to detect whether two parts occupy the same physical space.
This is a collision-detection problem:
\[ \boxed{ A\cap B\ne\varnothing }. \]Bounding volumes and narrow-phase intersection tests may be used internally.
Clearance
Two parts may not intersect yet still be too close.
Clearance analysis asks for the minimum distance
\[ \boxed{ d_{\min}(A,B) }. \]This can be important in mechanical motion, manufacturing, and assembly.
Mass Properties
A valid solid model can support calculations of:
- volume;
- surface area;
- centroid;
- mass;
- moments of inertia.
These quantities depend on the modeled geometry and assigned material properties.
Volume from Cross Sections
Conceptually, a solid volume can be viewed as accumulated cross-sectional area:
\[ \boxed{ V= \int_a^b A(x)\,dx }. \]CAD kernels may use exact formulas, surface integrals, tessellations, or specialized geometric algorithms depending on the model.
Tessellation for Display
Exact CAD surfaces are usually converted into triangles for interactive display.
This process is called
\[ \boxed{\text{tessellation}}. \]The display mesh approximates the exact CAD geometry.
Display Mesh versus Exact Model
A visible triangular edge on the screen may belong only to the display tessellation.
The underlying CAD face may still be one smooth exact surface.
It is important not to confuse the rendering approximation with the design geometry.
Tessellation Tolerance
A display tessellation can be controlled by an allowed deviation
\[ \varepsilon. \]Smaller tolerance generally produces more triangles and a closer visual approximation to curved geometry.
Manufacturing Tolerance
A mathematical CAD model may specify an ideal dimension such as
\[ 25.000\text{ mm}. \]A real manufactured part cannot be produced with infinite precision.
A tolerance specifies an acceptable interval, for example
\[ \boxed{ 25.000\pm0.025\text{ mm} }. \]Geometric Tolerancing
Manufacturing specifications may control more than size.
They can constrain:
- flatness;
- parallelism;
- perpendicularity;
- position;
- circularity;
- profile.
Model Tolerance versus Manufacturing Tolerance
These are different concepts.
A CAD kernel may use a numerical tolerance to decide whether two computed points are effectively coincident.
A manufacturing tolerance specifies how much physical deviation from the ideal design is acceptable.
Geometric Kernels
The core geometry engine inside a CAD system is often called a geometric modeling kernel.
It may provide algorithms for:
- curve and surface evaluation;
- intersections;
- Boolean operations;
- offsets;
- fillets;
- topology construction;
- tessellation.
Robustness Is Essential in CAD
CAD operations often combine geometry computed through many previous steps.
Small numerical errors can accumulate and create questions such as:
- Do two curves truly meet?
- Are two vertices the same?
- Is a face inside or outside a solid?
- Does a Boolean operation produce a closed result?
Near-Coincident Geometry
Suppose two points satisfy
\[ \|P-Q\| \]that is very small but not exactly zero in floating-point arithmetic.
A CAD kernel may need to decide whether to treat them as the same geometric vertex.
This decision must be consistent with the model scale.
Invalid Solid Models
A solid model can become invalid if it contains:
- gaps between faces;
- overlapping faces;
- self-intersections;
- nonmanifold edges;
- inconsistent orientation;
- zero-area or degenerate faces.
Manifold Solid Boundaries
In a typical closed manifold B-rep solid, each ordinary boundary edge is incident to exactly two faces.
This supports a consistent local distinction between inside and outside.
Face Orientation
Solid boundaries often use consistently oriented faces.
Surface normals point outward according to the chosen convention.
Inconsistent orientation can make inside/outside classification ambiguous.
Interoperability
CAD data is frequently exchanged between different software systems.
Exchange must preserve as much as possible of:
- geometry;
- topology;
- units;
- assemblies;
- attributes;
- manufacturing information.
Exact Geometry versus Exchange Approximation
A model may be exported as:
- exact curves and surfaces;
- a polygonal mesh;
- a drawing;
- a raster image.
Each representation preserves different information.
Mesh Export
Exporting a CAD model as a triangular mesh converts exact surfaces into an approximation.
This may be appropriate for:
- 3D printing;
- visualization;
- simulation preprocessing;
- graphics pipelines.
Units Matter
A coordinate value such as
\[ 25 \]is incomplete without a unit convention.
CAD systems must track whether values represent:
- millimeters;
- inches;
- meters;
- degrees;
- radians;
- other engineering units.
Unit Conversion
If
\[ 1\text{ in}=25.4\text{ mm}, \]then a length of \(2\) inches becomes
\[ 2(25.4)=50.8\text{ mm}. \]Incorrect unit interpretation can scale an entire model incorrectly.
Direct Modeling
Not all CAD systems require a long feature history.
In direct modeling, faces and geometric features may be pushed, pulled, moved, or edited more directly.
The underlying model still requires geometric and topological consistency.
Parametric versus Direct Modeling
Parametric modeling emphasizes design intent through dependencies and dimensions.
Direct modeling emphasizes immediate geometric modification.
Modern systems may combine both approaches.
Design Intent
A strong CAD model encodes not only current dimensions but also how the model should behave when changed.
For example, a hole can be constrained to remain centered in a plate rather than positioned by fixed coordinates.
This captures the designer's intended relationship.
Example: Centered Hole
Suppose a rectangular plate has width \(W\) and height \(H\).
A centered hole has center
\[ \boxed{ C= \left( \frac{W}{2}, \frac{H}{2} \right) }. \]Changing either plate dimension automatically moves the hole center.
Example: Parametric Bolt Circle
Suppose \(n\) holes are distributed evenly on a circle of radius \(R\).
Hole \(k\) may be located at
\[ \boxed{ P_k= \left( R\cos\frac{2\pi k}{n}, R\sin\frac{2\pi k}{n} \right) }. \]Changing \(n\) or \(R\) regenerates the pattern.
Example: Extruded Plate
A rectangle
\[ 0\le x\le W, \qquad 0\le y\le H \]extruded through thickness \(T\) produces
\[ \boxed{ 0\le x\le W,\quad 0\le y\le H,\quad 0\le z\le T }. \]The simple feature creates a rectangular solid.
Common Error: Treating a CAD Model as Only a Picture
A CAD model can contain exact geometric and parametric structure that is not visible in a static image.
Screenshots preserve appearance but generally discard most modeling information.
Common Error: Overconstraining a Sketch
Adding more constraints does not always improve a model.
Conflicting or redundant constraints can make a sketch difficult or impossible to solve.
Common Error: Leaving Important Geometry Underconstrained
Unconstrained degrees of freedom can allow geometry to move unexpectedly during later edits.
Important design relationships should be expressed deliberately.
Common Error: Depending on Fragile Feature References
A later feature may reference an edge or face created by an earlier feature.
If regeneration changes that topology, the reference may no longer identify the intended object.
Robust design intent should avoid unnecessary dependence on unstable references.
Common Error: Confusing Display Tessellation with Exact Geometry
A curved CAD face may look faceted because of coarse display tessellation.
That does not necessarily mean the underlying model is polygonal.
Common Error: Ignoring Units during Import or Export
A model created in millimeters can appear \(25.4\) times too large or too small if another system interprets the same values as inches.
Common Error: Applying Boolean Operations to Invalid Geometry
Self-intersections, gaps, or degenerate faces can make Boolean classification ambiguous.
Geometry should be validated before complex solid operations.
Common Error: Assuming Numerical Tolerance Means Manufacturing Tolerance
Kernel tolerance governs computational decisions.
Manufacturing tolerance governs acceptable physical variation.
They serve different purposes.
A Reliable Procedure for a Parametric Sketch
- Choose an appropriate sketch plane and coordinate system.
- Create the basic geometric primitives.
- Add geometric constraints such as parallelism or tangency.
- Add dimensional constraints.
- Identify remaining degrees of freedom.
- Remove contradictory or unnecessary constraints.
- Verify that the sketch changes predictably when dimensions are edited.
A Reliable Procedure for Feature-Based Solid Modeling
- Create a stable base sketch or primitive.
- Generate the first solid feature.
- Add later features in logical dependency order.
- Use parameters to encode design intent.
- Regenerate after important changes.
- Check for failed references or invalid topology.
- Verify dimensions, clearances, and mass properties as needed.
A Reliable Procedure for Boolean Modeling
- Confirm that both input solids are valid.
- Detect and compute surface intersections.
- Split affected faces and edges.
- Classify resulting pieces as inside or outside the other solid.
- Select pieces according to union, intersection, or difference.
- Reconstruct consistent topology.
- Verify that the result is closed and manifold when a solid is expected.
A Reliable Procedure for CAD Model Verification
- Check units and coordinate systems.
- Check sketch constraint status.
- Check feature regeneration.
- Check for self-intersections or gaps.
- Check face orientation and manifold connectivity.
- Check interference and clearance in assemblies.
- Check important dimensions and tolerances.
- Distinguish exact model geometry from display tessellation.
Practice 1: Constraint Type
A CAD sketch requires two lines to meet at \(90^\circ\). Is this primarily a geometric or dimensional constraint?
Practice 2: Degrees of Freedom
How many positional degrees of freedom does an unconstrained point have in the plane?
Practice 3: Underconstrained Circle
A circle has a fixed radius, but its center is free to move anywhere in the sketch plane. Is it fully constrained?
Practice 4: Extrusion
What kind of three-dimensional object is produced by extruding a closed rectangular profile through a positive distance?
Practice 5: Boolean Difference
Which set operation models removing solid \(B\) from solid \(A\)?
Practice 6: B-rep
Name three types of topological entities commonly stored in a boundary representation.
Practice 7: Parametric Relationship
If
\[ H=\frac{W}{2} \]and \(W\) changes from \(80\) to \(120\), what is the new value of \(H\)?
Practice 8: Tessellation
Why is an exact curved CAD surface commonly converted into triangles for display?
Practice 9: Interference
What mathematical condition indicates that two solid parts interfere?
Practice 10: Tolerance
Why should numerical kernel tolerance not be confused with manufacturing tolerance?
Practice Answers
Practice 1
Perpendicularity is primarily
\[ \boxed{\text{a geometric constraint}}. \]Practice 2
An unconstrained planar point has
\[ \boxed{2} \]positional degrees of freedom: \(x\) and \(y\).
Practice 3
No. The radius is fixed, but the center still has positional freedom.
\[ \boxed{\text{The circle is underconstrained}.} \]Practice 4
A rectangular prism is produced.
\[ \boxed{\text{rectangular solid}} \]Practice 5
\[ \boxed{ A\setminus B }. \]Practice 6
Valid answers include:
\[ \boxed{ \text{vertices, edges, and faces} }. \]Practice 7
\[ H= \frac{120}{2} = \boxed{60}. \]Practice 8
Graphics hardware renders triangles efficiently, so the exact surface is approximated with a display mesh.
Practice 9
Two parts interfere when
\[ \boxed{ A\cap B\ne\varnothing }. \]Practice 10
Numerical kernel tolerance controls computational geometric decisions, while manufacturing tolerance specifies acceptable physical deviation from the ideal design.
What to Remember
Computer-aided design uses computational geometry to encode precise design intent.
- CAD models store geometry rather than only appearance;
- sketches combine geometric primitives with dimensional and geometric constraints;
- degrees of freedom help explain under-, fully, and overconstrained sketches;
- parametric models encode relationships between design quantities;
- feature history creates geometric dependencies;
- extrusions, revolutions, sweeps, and lofts construct three-dimensional form;
- curves and surfaces are commonly represented parametrically;
- B-rep combines geometry with topology;
- Boolean operations construct solids through union, intersection, and difference;
- assemblies use coordinate transformations and geometric constraints;
- tessellation approximates exact CAD geometry for display and some downstream uses;
- robust numerical tolerances are essential to reliable geometric modeling.
The larger computational principle is that design becomes far more powerful when geometry is represented not only by coordinates, but also by the relationships and rules that generated it.
Where We Go Next
Computer-aided design focuses on precise constructive geometry and design intent. The next lesson broadens the view to the geometric mathematics used to create, transform, display, and manipulate visual models.
Computer Graphics and Geometric Modeling develops the computational relationship between geometric models, transformations, projections, surfaces, meshes, visibility, and rendered images.