Image files {vector file} {geometric file} can use mathematical formulas that software {object-oriented graphics} {vector graphics} {computer graphics} {graphics} {draw program} manipulates. Vector-graphics images transform easily, look good on higher-resolution displays, and use less memory than bitmaps.
fonts
Scalable fonts {True Type font} {vector font} {scalable font} {object-oriented font} {outline font} use vector graphics.
parameters
Main parameter is geometric-object type, such as rectangle. x, y, and z coordinates locate figure. Width and height are dimensions. Lines have color and size. Fill has color, opacity, orientation, and pattern.
spline
Splines represent lines and can represent boundary lines that divide surfaces into polygons.
surfaces
Closed surfaces can be polygons, such as triangles. Surfaces can be parallel lines or grids.
volumes
Splines and generalized ellipses or ellipsoids can represent volumes.
formulas
Vector graphics uses formulas for points, lines, splines, polygons, circles, ellipses, Bézier curves, Bézier splines {bezigon}, text in Bézier curve fonts, Catmull-Rom splines, NURBS, and rectangle bitmaps. For example, circle information includes radius, center point, line style, line color, fill style, and fill color. Formulas can combine.
formulas: transformations
Most formulas allow transformations, such as rotation, translation, reflection, scaling, stretching, and skewing. Programs can perform logical and set operations. Programs can also add surface-texture bitmaps, add shading, and smooth irregularities.
bit map
Images can be pixel arrays {bit map} {raster file} that software {raster graphics} manipulates. Programs {paint program} can make bit maps smaller or larger, but enlarged images have blank or repeated areas. Vector graphics convert to raster graphics for dot-matrix printers, laser printers, and display monitors, but not plotters.
Computer-graphics algorithms {de Casteljau's algorithm} {Farin-Boehm algorithm} can evaluate parametric curves {Bézier curve} or surfaces {Bézier surface}. Linear, quadratic, and cubic Bézier-curve {Bézier polygon} splines can draw any shape.
history
Paul de Casteljau invented de Casteljau's algorithm [1959]. Pierre Bézier of France designed automobiles [1962].
equations
Lines can use two points P0 and P1 {control points} and have equation f(t) = (1 - t) * P0 + t * P1, where 0 <= t <= 1. Quadratics can start at P0, go toward P1, and end at P3: f(t) = (1 - t)^2 * P0 + 2 * t * (1 - t) * P1 + t^2 * P2, where 0 <= t <= 1. Cubics can start at P0, go toward first P1 then P2, and end at P3. f(t) = (1 - t)^3 * P0 + 3 * t * (1 - t)^2 * P1 + 3 * t^2 * (1 - t) * P2 + t^3 * P3, where 0 <= t <= 1.
interface
Drawing programs display control points, which control lines.
Real functions can have domain-value sets {level set} {fiber, set} that make function constant. Real-equation roots are level sets. For two-dimensional domains f(x,y), level sets are curves {implicit curve} {isocontour}. For three-dimensional domains f(x,y,z), level sets are surfaces {implicit surface} {isosurface}.
function
Functions are positive inside surface, zero on surface, and negative outside surface. At points, function gradient is perpendicular to level-set surface.
shape
Closed planar curves can be isocontours. Closed surfaces can be isosurfaces. Shape boundaries are level sets. Shapes have real-function positive values. As object changes, level-set function translates by non-parametric methods {level set method} {Eulerian approach} (Stanley Osher and James Sethian) [1988].
Computers can test all rays entering screen to see where they land on object-indexed kd-tree scene {ray tracing} {ray processing unit} (RPU). Ray tracing can check for shadows, reflection, refraction, and color.
Rasterization software renders three-dimensional scenes for two-dimensional screens {graphical processing unit} (GPU). GPU first makes scene into independent plane polygons, typically triangles. It must find which ones are closer {z-buffer}. Because light reflects from objects onto other objects {global illumination}, GPU must later add indirect light.
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Date Modified: 2022.0225