INTRODUCTION

Originating as the study of geometry, Lofting can be traced to the Egyptians of 1700 B.C. It was the early Greeks, however, who used conic curves in the design of ship hulls to increase speed and maneuverability.

Draftsmen and engineers relate what designers have specified in terms of three view drawings and tabulated data. From these drawings and data, the Loftsman creates a smooth three-dimensional surface. In creating definitions, the modern Loftsman uses the tools of the past, straight edge, splines, curves and compasses. He also has added all of the modern techniques; mathematical analysis, digital computers, automated drafting machines, and CRT scopes, are used to aid his definitions.

With all of the advanced equipment and techniques, one still finds the old Loftsman sighting down a line saying, “It’s a good surface because it’s pleasing to the eye.”

In the last decade many engineering firms have introduced numerous numerical control cutting machines into their facilities. These machines are used primarily to cut highly complicated parts, and to close accuracy, at lower cost and greater speeds.

The aerospace industry was one of the first users of automated machines. Years of research and production in controlling these machines have been logged by individual companies. They researched the field of bounded three dimensional surfaces in space to produce data for these machines. Because of this, various method and special theories developed in surface definitions, each varying throughout the industry. Divisions within one corporation can have diametric techniques of surface definitions. Still, the intent is the same, to produce accurate mathematical surfaces for the numerically milling and analysis of parts.

This paper will explore the history of surface definitions, the various methods employed, their applications, and restrictions. The techniques used by the McDonnell Company, with present day applications, are discussed. Projected applications in this field of surface definitions also are treated.

LOFTING

Lofting is the science of generating real two and three dimensional surfaces in space. These surfaces may have originated from conceptual sketches, structural requirements, aerodynamics, thermodynamics, hydrodynamics, minimum envelope designs or some combination of these, and other input parameters. Lofting encompasses both graphical and mathematical techniques; and, the graphics, in the final stages, being tedious and error ridden, has been surpassed by the use of mathematics. The graphical solution is basically the same throughout the industry, but various approaches have been taken for the mathematical. Two major branches of lofting have evolved; the conic lofting technique and a new innovation, the parametric surface method.

Lofting originated as the study of geometry. Walter Vaughn in his book “Aircraft Descriptive Geometry” states:

“The oldest written record on the subject of geometry is an ancient Egyptian manuscript. It was written in the year 1700 B.C. by Ahmes, an Egyptian priest. The wisdom of the Egyptians spread over the civilized world. As early as 600 B.C. many students and scholars went to Egypt to study. Among these students were the Greeks, many of whom studied the science of geometry and returned to their own country to teach it.”1

Conic lofting is the science of developing surfaces based on the sectional properties of the cone. The cone was first analyzed around 500 B.C. by the Greeks. “Euclid is given credit for the study of geometry, but it was Apollonius of Perga who studied the cone and its sections.”2




Apollonius working only with clay cones found that by cutting the cone with a knife, certain figures would emerge. Parallel sections produced larger or smaller figures of the same family.


Appolonius wrote eight books on conic sections, of which the subject were; Ellipses, Parabolas, Hyperbolas; Circles, Propositions involving the Foci, the Intersection of Conics, The Lines of Least and Greatest Distance From a Point on the Diameter to the Periphery, and The Construction of Conics and the Normals to Them.

No other great works on conic sections appeared until 1676 when Blaise Pascal, then sixteen years old, wrote his “Essay on Conics.” 3 Although Pascal’s works were of great importance, they were not printed in his lifetime.

Two hundred years later C.J. Brianchon, still a student at the Ecole Polytechnique in Paris, published a statement of the “Allied Theorem,” that lines connecting opposite vertices of a circumscribed hexagon meet in a point.4

The early shipbuilders found that by using Appolonius’ conic curves in designing their ships hulls, speed and maneuverability greatly increased. The shipbuilders could not afford to build an unfavorably designed ship, so scaled models were built and tested against older designs. Although the testing was crude, faster and larger sailing vessels were built while the art of developing these designs was a closely guarded secret. (See figure 2.)

As the shipping industry spread to other countries, so did lofting. “To know precisely the complete definition of a ship’s surface, a full scale set of master lines defining this surface must be available. To accomplish this, a huge platform or floor was built, at considerable height above the ground. It was located adjacent to the ships “ways” where the hull was being constructed. Because of its height and spaciousness, this floor became known as the loft, or mold loft. The men who worked there were termed loftsmen.”5



Graphical Construction of Conics




(Figure 2.0)

By connecting the numbered points, we may draw four sides of the hexagon of PASCAL’s Theorem, and find on point (P) on PASCAL’s line. The problem is to locate a sixth point such that it will be on the conic.

1.