CAD Software Fundamentals: Mathematics, Transformations, and Curves

A set of vocabulary flashcards covering core CAD software concepts, including coordinate systems, transformations, and curve/surface representations.

Cartesian coordinate system

A 3D coordinate framework with orthogonal axes x, y, z and an origin at (0,0,0).

Point

A precise location in 3D space defined by coordinates (x, y, z).

Vector form

Representing a point or set of points as a column vector [x, y, z]^T to simplify computations.

3D solid

A volume bounded by surfaces; conceptually defined by a collection of points, lines, and planes.

Translation

Rigid-body motion moving every point by a fixed offset (Δx, Δy, Δz); p' = p + Δ.

Delta (Δx, Δy, Δz)

A change in coordinate values used to describe translation.

Rotation

Turning an object about an axis by an angle, described by rotation matrices.

Rotation matrix

A matrix that rotates coordinates around an axis; e.g., Rx(θ), Ry(θ), Rz(θ).

Scaling

Resizing an object by multiplying coordinates with scale factors; represented by a diagonal scaling matrix.

Homogeneous coordinates

Extending 3D coordinates to four dimensions by appending 1, enabling translation to be included in matrix multiplication.

Composite transformation

A single matrix that combines translation, rotation, and scaling, allowing multiple transforms to be applied in one step.

Order of transformations

The sequence in which translation, rotation, and scaling are applied; different orders yield different results.

Implicit polynomial

A curve defined by F(x, y) = 0, such as a circle (e.g., x^2 + y^2 − r^2 = 0).

Explicit polynomial

A curve defined by y = f(x) (or z = f(x, y)); the explicit form can be transformed to implicit form if needed.

Parametric polynomial

A curve described by x(t), y(t), z(t) with a parameter t, often expressed using a basis vector [t^3, t^2, t, 1].

Tangent

A vector indicating the direction of a curve at a given parameter value, obtained by differentiating with respect to the parameter.

Ferguson curve

A cubic curve defined by two end points P0, P1 and tangents T0, T1, allowing control over curvature through endpoints and tangents.

Bezier curve

A curve defined by four control points, offering greater control over shape through additional control points.

NURBS (Non-Uniform Rational B-Splines)

A versatile CAD curve representation that generalizes Bezier/B-spline curves using weights and knot vectors.

Bezier surface

A two-dimensional extension of Bezier curves defined by a grid of control points, creating complex surfaces.

NURBS surface

A two-dimensional generalization of NURBS curves used to model smooth, complex surfaces.

Control points

Points that influence the shape of Bezier/NURBS curves and surfaces.

Knots

The knot vector in NURBS that parameterizes curves/surfaces and controls parameter spacing.