Curves, Surfaces and Colour Models
Smooth curves and surfaces are essential to modern graphics. Fonts, character animation, industrial design, and natural scenery all require representations that are both compact and capable of producing smooth shapes. Equally essential is a principled approach to colour, because human perception of colour is nuanced and varies with the display technology in use.
Parametric Curves
A parametric curve is defined by a function P(t) = (x(t), y(t), z(t)) for t in some interval. Parametric representations allow curves that are multi-valued in the Cartesian sense, such as circles and figure eights. The simplest smooth curves are polynomials of low degree, with cubic polynomials offering a good balance between flexibility and computational cost.
Bézier Curves
A Bézier curve of degree n is defined by n + 1 control points and a set of Bernstein basis polynomials. The curve passes through the first and last control points and is tangent to the line segments at the endpoints. The curve lies within the convex hull of its control points, a property that simplifies clipping and intersection. Bézier curves are widely used in computer typography, illustration software, and motion paths.
de Casteljau's Algorithm
Bézier curves can be efficiently evaluated using de Casteljau's algorithm, which recursively interpolates between pairs of control points. The algorithm is numerically stable and produces the exact point on the curve for any parameter value. It also supports subdivision: any Bézier curve can be split at parameter t into two Bézier curves meeting at P(t), which is useful for adaptive rendering.
B-splines and NURBS
While Bézier curves are defined globally—moving one control point affects the entire curve—B-splines provide local control through a piecewise polynomial defined over a knot vector. Non-Uniform Rational B-splines (NURBS) extend B-splines with weights, enabling exact representation of conic sections such as circles and ellipses. NURBS are the industry standard in CAD, animation, and engineering.
Surfaces
Parametric surfaces extend these ideas to two parameters. A Bézier surface patch is defined by a grid of control points and a product of Bernstein basis polynomials in u and v. B-spline surfaces and NURBS surfaces similarly extend their curve counterparts. Large smooth objects are constructed by stitching together many surface patches with continuity conditions ensuring smooth transitions.
Subdivision Surfaces
Subdivision surfaces begin with a coarse polygonal cage and repeatedly refine it to produce smooth limit surfaces. Popular subdivision schemes include Catmull–Clark, Loop, and Doo–Sabin. Their appeal lies in combining the topological flexibility of polygonal meshes with the smoothness of parametric surfaces; they are the dominant representation in modern animation studios.
Colour Models
A colour model is a mathematical system for representing colours. The RGB model is additive: red, green, and blue light combine to produce colours on a display. The CMY or CMYK model is subtractive and is used in printing. The HSV and HSL models parameterize colours by hue, saturation, and brightness, which often align better with user intuition. The YCbCr model separates luminance from chrominance and underlies most video compression standards.
Colour Gamut and Perception
The colour gamut of a device is the range of colours it can display. Human perception is described by the CIE XYZ colour space, with the visible gamut forming a horseshoe shape on the chromaticity diagram. Device gamuts are always subsets of this shape. Displays and printers require colour management, including calibration and ICC profiles, to ensure consistent appearance across media.
Dithering and Halftoning
When a display or printer cannot reproduce a colour directly, dithering or halftoning creates the illusion of intermediate colours by arranging available colours in patterns. Ordered dithering uses a fixed matrix; error-diffusion algorithms such as Floyd–Steinberg distribute quantization error to neighbouring pixels. These techniques are especially important in printing and low-bit-depth displays.
Summary
Curves and surfaces provide the mathematical language for smooth shapes, while colour models provide the framework for representing and manipulating colour. Together with transformations and clipping, these tools form the foundation of realistic and stylistically varied computer graphics.