What is CAGD about?

Why is CAGD necessary?

The Basics of CAGD

Real World Examples

History of CAGD

What is CAGD about? --

Computer-aided geometric design (CAGD) is a new field that initially developed to bring the advantages of computers to industries such as automotive, aerospace, and shipbuilding. CAGD expanded rapidly and now pervades many areas, from pharmaceutical design to animation. We are surrounded by products that were first visualized on a computer. These products were modified and refined entirely within the computer, so that when the product entered production, the tools and dies were produced directly from the geometry stored in the computer. This process is known as virtual prototyping. Computer visualizations of new products reduce the design cycle by easing the process of design modification and tool production. CAGD is based on the creation of curves and surfaces and is accurately described as curve and surface modeling. Using CAGD tools with elaborate user interfaces, designers create and refine their ideas to produce complex results. They combine large numbers of curve and surface segments to realize their ideas. However, the individual segments they use are relatively simple, and it is at this level that the study of CAGD is concentrated.

The Need for CAGD --

When creating products and artwork, designers face tasks such as this: Given two points in a plane and two directions associated with the points, find the curve that passes through the points that is tangent to the given directions. This is a simple pencil-and-paper task for anyone who is familiar with parametric forms, the types of curves used in CAGD. CAGD tools are meant to be a intuitive a simple to use. Sometimes the mathematics underlying the tools becomes quite sophisticated, yet the result is meant to be easily understood and geometrically intuitive. The technical person often benefits from these intuitive and visually related tools when considering deeper mathematical problems. The geometry of CAGD is very amenable to visual demonstration.

The Basics of CAGD: From Points to Teapots --

There is a natural progression of the geometry behind CAGD. With small incremental steps, it is possible to describe complex objects in terms of simple primitives, such as points and lines.

Control Points: The Start of CAGD
Take four points in a plane and connect them to form a polygon. The points are known as control points, and the polygon as the control polygon. The control points and polygon determine the approximate shape of the curve to be formed.
A CAGD Favorite: The Bézier Curve
A special curve known as the Bézier curve may be generated by the control points. Note that while the curve will pass through the first and last control points, it may only come close to other points.
Three-Dimensional Control Polygons for Surfaces (Control Meshes)
Bézier curves behave just as well in three dimensions as in two.
Three-Dimensional Wireframe Model
When the Bézier curves are created and connected in three dimensions, a wireframe model of the object is produced.
Three-Dimensional Shaded Object
If the surface produced by the three-dimensional Bézier patches is illuminated and shaded, an object with a realistic appearance results. The Utah teapot, on display at the Boston Museum of Computer History, is a classic in CAGD and computer graphics.

Real World Applications --

Automobile, aircraft, aerospace, and ship designers use CAGD techniques in the design of various types of vehicles. Wire-frame drawings are used to model indivdual components and plan surface contours for automobiles, airplanes, spacecraft, and ships.

Individual surface sections and vehicle components can be designed separately and fitted together to display the total object. Simulations of the operation of a vehicle are often run to test the vehicle performance as in the figure. Realistic renderings allow the designer to see how the finished product will appear.

History of CAGD --

Computer-aided geometric design has mathematical roots that stretch back to Euclid and Descartes. Its practical application began with automated machinery to compute, draft, and manufacture objects with free-form surfaces. Production pressures in the aircraft industry during World War II stimulated many new devices to enhance and accelerate design and manufacturing. For example, in 1944, Liming designed fuselage spars with a "superelliptic" method that could be implemented with an electromechanical calculator.

Shipbuilders also became interested in CAGD early on for many reasons. One example, that may sound trivial but was a serious impediment to ship design, was that the only place large enough to draw full-scale plans for a ship was in the loft of the shipbuilders' dry dock. The huge drawings would warp and shrink in the moist air, causing very real manufacturing problems. Computers provided the greatest stimulus because of their power to enable new ideas.

In 1963, Ferguson developed one of the first surface patch systems by which individual curvilinear patches are joined smoothly to create the surface "quilt". He also introduced the notion of parametrically defined surfaces, which has become the standard because it provides freedom from an arbitrarily fixed coordinate system. Vertical tangent vectors can be defined by differentiation, for instance, which is not possible in explicit Cartesian form.

In the mid 1960s, automotive companies became involved in CAGD as a way to drive milling machines. Car bodies were designed by artists using clay models. Painstaking measurements produced data that could drive numerically controlled milling machines to produce stamp molds.
The initial use of CAGD was to represent the data as a smooth surface for numerical control. It soon became apparent that the surfaces could be used for the design.

In 1971, Pierre Bézier reformulated Ferguson's ideas so that a draftsman without any extensive mathematical training could design a surface. Bézier's system, UNISURF, was used by Renault and became a milestone in the development of CAGD. It epitomized the difference between surface fitting and surface design. The purpose of design was to provide the draftsman, who had strong intuition about shape but limited mathematical training, with computer tools that empowered him or her to use the sophisticated mathematics of surface representation.

In the meantime, the mathematical underpinnings of CAGD continued to advance. DeCasteljau examined triangular patches and developed evaluation techniques. Coons unified much of the previous work into a general scheme that became the basis of the early modeler PDGS made by Ford. At General Motors in 1974, Gordon and Riesenfeld exploited the properties of B-spline curves and surfaces for design.

Driven primarily by the automotive, shipbuilding, and aerospace industries, both the mathematics of CAGD and the designer interface tools continued to improve through the 1970s. The first CAGD conference was organized by Barnhill and Riesenfeld in 1974, where the term "CAGD" was first used.

In the 1980s, the power and versatility of computer aided designing seemed suddenly to be discovered by anyone who had a free-form geometric surface application. Industrial designers were smitten with the power of computer design, and many commercial modelers became the basis of several substantial applications, including CATIA, EUCLID, STRIM, ANVIL, and GEOMOD. Geoscience used CAGD methods to represent seismic horizons; computer graphics designers modeled their objects with surfaces, as did molecule designers for pharmaceuticals. Architects discovered CAGD, word processing and drafting programs based their interface protocols on free-form curves (PostScript), and even moviemakers discovered the power of animating with such surfaces, beginning with TRON, continuing through Jurassic Park, and beyond.