The Bézier Curve A Bézier curve is a line that may have turns or form loops

A Bézier curve is a line that may have turns, spirals or form loops. Figure 1 is made using Bézier curves Straight lines and arcs are the only geometric objects that can be reproduced reliably on paper. More intricate lines, such as loops, s-curves and spirals, require complex calculations. Early designers utilized hand- drawn estimations, a ruler and a pair of compasses or even the French curve. (The French curve, which appeared in 1885, is a curved piece of flat material used as a guide when drawing curves.) However, all of
	Figure 1
these proved to be difficult when it comes to the specificity needed in engineering and design. The mathematical formula for describing Bézier curves was first used in 1972 to shape automobile surfaces. Pierre Bézier and Paul de Casteljau discovered the curve at about the same time. Both were engineers in France and worked Renault and Citroen, respectively. de Casteljau's work was slightly earlier than Bézier's work but as it was never published, the curve was named for Bézier. However, the algorithm, which forms the basis for the Bézier curve, is now credited to de Casteljau.

In computer graphics, a spline is a smooth curve that passes through two or more points. Splines are generated with mathematical formulas. Two of the most common types of splines are b-spline curves and Bézier curves. A Bézier curve requires at least three points. The two endpoints of the soon-to-be curve are called the anchor points. The points which will define the shape of the curve are called handles (also, tangent points or nodes) and allow the designer to modify the actual shape of a curve. In most drawing applications two control points attach to each handle. By moving these handles or control points, you can change the shape from a straight line to a curve. Figure 2 uses Paint Shop Pro and illustrates the actions necessary to create an arc. A line was drawn from 1a to 1b. Points 1a and 1b become anchor points. Handles were maneuvered to points 2a and 2b making the arc. Figure 3 is a very primitive representation of the anchor points and handles used to make Figure 2. The handles can be pulled from anywhere along the rectangle between 1a and 1b. Each handle or node is selected individually and is pulled up and over to 2a (or 2b), then to 2b. When released, the arc forms. Figure 4 shows how to create an s-curve. A line is drawn between 1a and 1b. The handle on the left is pulled up to point 2 to while the handle on the right is pulled down to point 3. When released, an s-curve appears. Figure 5 shows the actions necessary to create a loop. Point 1a is clicked, then 1b is clicked followed by a click on point 3, thus, forming a loop.

The Bézier curve remains the cornerstone in the field of study called geometric modeling or computer-aided design (CAD). Those in this field use the Bézier's mathematical calculations to design curves that are useful in modeling and design because the curve is flexible and allows for greater precision.

There are two ways that applications use geometrical formulas to represent images. 1) Vector graphics; 2) Bitmaps. Bitmapped or raster images consist of a pattern of dots that when viewed as a whole appear as an image. Vector-oriented images, including the Bézier curve, are more flexible than bitmaps. A designer may change the shape a vector-oriented image by controlling handles or nodes. In addition, images stored as vectors look better on higher resolution monitors and printers, while bitmapped images will appear the same regardless of resolution. Vector graphics often require less memory than bitmapped images do.

Draw programs allow the designer to create and manipulate vector graphics, while paint programs only allow the designer to manipulate bit-mapped graphics. For the most part, draw programs support Bézier curves. Most advanced graphics systems, including CAD systems and animation software, use vector graphics. Most output devices (including dot matrix printers, laser printers, and display monitors) are raster devices. Plotters are the exception. In addition, many printers, PostScript, for example, use vector graphics. PostScript printers use a raster image processor (RIP) that performs the translation within the printer. Fonts represented as vectors are called vector, scalable, object-oriented or outline fonts.

All objects, even vector objects, must be translated into bitmaps before the final output andafter all manipulations are complete. Thus, it is possible to output vector-oriented images on any device no matter the resolution or size.