- Email: [email protected]
In memory of Steven Anson Coons
Foreword
In Memory of Steven Anson Coons b y J, H a t v a n y , P. B6zier and B. H e r z o g
Steven A. Coons in Munich (1970).
North-Holland Publishing Company Computers in Industry 3 (1982) I-8 0166-3615/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
2
Foreword
1. Introductory Remarks, by J. Hatvanv Computer and Automation Institute, Hungarian Academy of Sciences, H-1502 BUDAPEST, XI, Kende Utca 13-17, Hungary
Mausoleums, eulogies and reminiscences appeal only to the sentiments of those whose personal emotions they resuscitate. The memorial which the world of science raises to its departed, is the intellectual continuation of their work through successive generations and in all corners of the Globe of international learning. When Professor Bertram Herzog, an earstwhile pupil of Steve Coons' and his host at Boulder University during the last months of Steve's life, called on me in Budapest in 1980, we discussed an appropriate way to commemorate Steve. We soon asked Professor Pierre B6zier of Paris, who had modestly and lovingly dedicated the compendium of his life's work "to my Teacher and Master: Steve Coons", to join us in our endeavour. Together, the three of us invited a broad spectrum of researchers from m a n y countries who happened to be familiar to us through their published work in the areas of surface representation and computational geometry, to contribute to this Memorial Issue. We did not ask for papers about Steve Coons, or about his work. We asked for papers about the authors' own work, in the conviction that they would together demonstrate how Steve's scientific achievement lives on and inspires an ever growing circle of mathematicians, computer scientists and engineers in universities, research institutes and industries throughout the world. The first six papers of this Issue are primarily of a theoretical nature. Cohen and Riesenfeld present a general matrix representation method. They have written to say that "the late Professor Coons might have enjoyed this paper because 1) he was an enthusiastic promoter of splines, 2) he deeply
"~mTputers in Industr)'
admired the B6zier technique, and 3) matrix formulations were his favorite representations whenever they could be applied. He reasoned with matrices more comfortably than other people we knew: they "talked" to him about the characteristics and behavior of the schemes they represented. As the field developed, the implementational facility found in matrix formulations gave more motivation and acceptance to this approach to dealing with curves and surfaces." The papers of Bars~iv and of Semenkov and Vasilet, are concerned with their important new results in the B-spline technique of surface representation. Barnhill proposes novel solutions to some problems concerned with the uses of Coons' patches, while Sablonnibre reports on an original treatment of Lagrange interpolation problems by quadratic splines. The contribution of Knapp demonstrates an extension of Coons' techniques to the case of nonuniform basis B-splines. The next batch of three papers deals with various aspects of man-machine interaction in the surface design and representation environment -- a field which since "Project M A C " days was always of concern to Steve Coons. Forrest deals with a pragmatic approach to man-machine interaction in the C A D context, Bonitz tackles this problem from another aspect: the development of methods permitting the stylist easier control over car-body shape in a computer-based system. Posdamer presents new results of a multi-disciplinary approach to the automatic measurement of the surface geometry of three-dimensional objects. The final group of seven papers describes industrial systems which have been developed and are being used in a number of countries. Okino, Kakazu, Kubo, Hashimoto and Shimora describe the celebrated TIPS-1 system, Strasser writes about a fast interactive shape design process, Macurek reports on the C K D A P T approach to the manufacture of turbine blades, Arm# writes of a highperformance 3D graphics systems. V~irady discusses practical considerations for C A D / C A M applied to sculptured surfaces, Renner presents a new engineering approach to curve representation, while Janse shows a C A D system for "free-form'" surface design. Steve Coons was equally at home in each of these areas: theory, methods and applications. For him they always existed together in the totality of a design-tool, design and implementation complex.
Computers in Indust(v
Foreword
3
Originally Professors B6zier, Herzog and I, as responsible for assembling this Issue, planned to blend our individual editorial contributions into a single presentation. The contributions have, however, turned out to be so different in style and character, that it was deemed preferable for them to retain their individual flavours. Pierre Bbzier has offered an appreciation of Steven Coons' SCIE N T I F I C A C H I E V E M E N T . Bert Herzog, in his eulogy delivered at the memorial service (of which that presented here is a slightly abbreviated version) speaks of him as a PERSON, a SCIENTIST, a TEACHER and a FRIEND. (My own thoughts on Steve have already been published in this Journal. *) We hope we have taken an appropriate step towards commemorating a great scientist and a warm-hearted man. * C O M P U T E R IN I N D U S T R Y 1 (1979) 63-64
2. Steven Anson Coons: The Scientist, by P. Bbzier 12 Avenue Gourgaud, 75017 PARIS, France
There is no doubt that Steven Anson Coons, our friend Steve, will be remembered as one of the men who have been responsible for dramatic progress in industries such as aircraft, ship and car manufacturing. The mathematical theory he invented and the systems derived from it are now used throughout the world. In order to fully understand the value of his contribution, it seems advisable to recall the process which was widely used before Steve's invention, since one easily forgets the difficulties previously encountered, after they have been overcome. Man invented the machine tool milleniums ago;
Steven Anson Coons (left) in Budapest with Jozsef Hatvany (right).
Egyptian bas-reliefs and paintings display lathes and drills; the need for accuracy, as we understand it today, began about the middle of the eighteenth century; to minimize the wear of the parts which had to slide or rotate, it was necessary that the surfaces remain in full contact, so as to keep the specific load to a minimum. Hence, the motion of the tool relative to the part to be machined was either a linear translation or a rotation. Consequently, the surfaces generated by the combination of two such motions were defined by segments of lines and circles, and were limited to planes, cylinders, cones, spheres and toruses. It is worth mentioning that the speed of the combined motions had not to be related by an accurate ratio. For obtaining leadscrews or gear flanks, the relation between the two speeds had to be kept in strict proportion; this was obtained by successive improvements till the advent of optical or electrical systems. But mechanical industry must sometimes manufacture parts the shape of which can not be exclusively described with lines and circles. The reason is twofold: on the one hand, the efficiency of some mechanisms is related with aero-, hydro- or thermodynamics, and their shape must comply with physical laws; in the same manner,
4
Foreword
some cams must control motions that are free of shocks. On the other hand, stylists generally conceive curves and surfaces the curvatures of which vary smoothly from end to end. Before the advent of numerical control, the only method available in such cases was to copy-turn, grind or mill, using a template or a three dimensional master. Nevertheless, one has to admit that copymilling has a limited accuracy, due to the deflection of the feeler, and the milled parts have to be hand-finished by highly skilled operators. For inspection, one generally had to rely on templates, or on direct contact (i.e. die-spotting) with a standard part. For either type of parts, the first step was the building of a model; it was the result of iterative adjustments performed according to the measures obtained during tests, and it was the case for turbine foils, propellers blades, airwings or boat hulls; or it was the expression of a stylist's intention or will. A typical example is that of car bodies, furniture, household appliances etc.. When aesthetics played a dominant role, the model was hand made of clay or plaster of Paris. Then, one had to pick the coordinates of points located on the surface of the model. If the shape was rather complicated, or if accuracy was sought for, the points were many, sometimes in the thousands. Owing to the nature of the measuring machines, those points were located on cross-sections parallel with the machine referential. On a drawing board, the designer traced curves running through, or by, those points, making use of splines or French curves. In the meantime, he or she endeavoured to correct any defect, kink or bump, that became apparent. After that stage, templates made of steel or aluminium sheets, and duplicating the drawn curves were used to adjust cross-sections on a master. Interpolation between those sections was performed by highly skilled patternmakers or diesetters. Whilst performing that operation, they often had to correct the cross-sections so as to obtain a very smooth shape, and eliminate defects that the drawings could not disclose. From the moment the master was finished and accepted, it became the only standard, and the drawings or templates were considered only approximative. Of course, the importance of a master was great; it was made of very expensive mahogany, of
Uomputers in lndust O,
stable resin or of metal. Wood or resin parts were kept in air and moisture-conditioned rooms so as to avoid warping. Evidently, so precious a part could not be used on a copy-milling machine; plaster of Paris or resin replicas served for machining and inspection. Due to the deflection of the feeler, there is a slight discrepancy between part and replica, and it is the task of a setter to correct the former. So, from model to drawing, to template, to master, to copy, to part, small differences did occur, the sum of which was not always negligible. Moreover, the process was slow and retouches were expensive as well as time consuming. As soon as data processing and numerical control had made significant progress regarding cost, power and reliability, it became obvious that it was the best way to improve accuracy and reduce leadtime. Aircraft, ship and car building industries were among those most interested in promoting it. For a car. it is admitted that a few millimetres difference in the overall length is negligible, but, for assembling parts together by welding, crimping or gluing, the range must be kept within a portion of a millimeter. Another problem arose fi'om the fact that a stylist is interested in building the shape of an object starting not from cross-sections but from character lines which are free-form, i.e. twisted lines and curves; the same often happens with technical surfaces: for instance, the leading and trailing edges of a turbine foil or of an aircraft wing are seldom plane curves. Such a requirement naturally implies that curves and surfaces be parametric rather than Cartesian. It is relatively easy, on a sculptured surface, to define two families of piecewise curves which enclose patches; each segment is limited by the nodes located at the corners of the patches. It is much more difficult to give a patch a mathematical definition such that it correctly blends with the surfaces surrounding it up to order two, which is tangency, or three, osculation. It can safely be stated that Steve Coons gave this rather difficult problem its first general solu-
tion. Representing the overall surface of an object with a great number of point coordinates is feasible so long as the computer is powerful enough, but it is much better to express it with a limited quantity of coefficients of a set of polynomial
Computers in Industry
functions. It is advisable to reduce the quantity of numerical data so as to minimize the possiblity of mistakes. In the meantime, it is possible to take into account (in real time) the allowance left after a roughing cut, to compensate for springback, and to compute the offset - whichever be the tool dimension. Moreover, the use of parametric functions is compulsory for machining patterns, dies and stamping tools, when the part is in a tilted position. The first solution, as it was described in Coons' MIT report*, could be applied to a large variety of problems, but it had a weak point: the properties of the boundary curves can not give any information regarding mixed derivatives at the corners of the patch, which control the torsion or twist of the surface. In principle, one could decide that their value is nil. But, though the blending of the patches be correct, one could detect at the corners of the patch, a small portion that looks flat and this is, aesthetically speaking, a defect. Steve Coons found the remedy at once, namely: slightly moving four points located inside the patch, from which the values of the twist vectors are deduced. From one patch to the next, the derivatives were selected in order to erase the flat portions. The consequence was that the system no longer was purely algorithmic, and that it somewhat relied on the taste and experience of the operator. Tangential blending is good enough to solve most of the aesthetic problems, but when it comes to fluid dynamics, any discontinuity of curvature causes shocks and consequently a loss of energy and efficiency. Steven Coons easily solved the problem, raising the order of his interpolating functions up to five. Then among the thirty-six coefficients of a biquintic polynomial, only twenty are provided by the boundary lines; unless cancelling the sixteen others, the operator has to perform an interactive task, moving as many points to give the surface an overall definition. In some very particular cases, specialists insist on having a continuity up to the third derivative, so as to avoid what is called "jerk". The Coons' formulae adapt perfectly to this requirement; extending it to any order has no limitation but the power of the computer (... and the operator's endurance). The need to have to rely partly on the operator is no intrinsic weakness in Coons' formulae. It comes directly from the fact that a line can only
Foreword
5
yield derivatives related to itself. Any algorithm intended to go further would be liable by its own nature to fail at a given moment, as can any algorithm. Inventing the method that bears his name, Steven Coons perfectly solved the problem he was faced with; and in doing so, he created a system Which is used throughout the worm and has brought to him an international reputation. Leaving aside the scientific aspect of Steve Coons' contribution, we must ask ourselves what is the lesson we received from him - since it is our duty to try and complement the results already available. At first sight, it is strange to consider that a problem which had (for a fair amount of time) attracted the attention of industry, was not solved by a professional mathematician, but by an engineer. No doubt Steven Coons had a gift for mathematics, and in particular for matrices and tensors. Looking at the figures of a tensor, he was able to imagine broadly the shape and properties of the surface so defined. During a meeting held at Salt Lake City in the year 1974 (the subject of which was "Computer Aided Geometric Design"), he exchanged information with a colleague - only scribbling figures in a table of numbers. Moreover, to make vector computation faster, he invented a special set of symbols - but, one had to work hard to develop the ability to use them. Steven Coons was not only able to solve a difficult problem raised by the aircraft and car industries; but well aware of the possibilities of data processing and of his theory in particular, he thought that the data broadly defining a car body could be chosen at the very beginning of the process, in the styling department, then refined and completed by designers, and used all along from methods to stamping tools design, pattern shop, tool shop and inspection. Very few people at that time thought that such an opinion was realistic; today, there is proof that Steve was right - years and years afterwards. There can be no doubt that Steven Coons was a brilliant man, with outstanding intelligence and broad knowledge. His professional experience he had acquired while working as an engineer. He knew that before inventing a solution, one first has to carefully consider the data of the problem and, most of all, to take into account the behaviour and
6
Foreword
C'omputers
reaction of the people of all ranks who will have to work with it. In addition to all the aforementioned qualities. Steve was a generous man, dispensing freely what he knew, and h a p p y to feel that he had trans-
3. Steven Anson Coons: The Man, by B. Herzof~ Herzog Associates, Inc. P.O. Box 3169, Boulder, CO 80307, USA
Steven Anson Coons was born in Palatine, New York, on March 7, 1912. Depending upon Steve's mood, his childhood was either happy or not quite so. In any case, eventually he enrolled at M I T only to leave about a year later to end his formal student days under adverse conditions - as he would be quick to tell you. This was always most surprising to those of us who first met him under professional circumstances. He was often and mistakenly introduced as "Dr. Coons". Steve would then quietly chuckle with close friends about his "flunking" out of M I T and remaining degreeless. Or, as it is stated in a Syracuse University publication, " H e has never earned a university degree, proving that a degree is not necessary - if one is a genius." The first professional association we know about is his work as an engineer with Vought-Sikorsky Aircraft Corporation from 1940 to 1948. During this period he demonstrated his keen insight into matters of graphical presentation of complex objects and shaped surfaces such as airplane fuselages, wings and other components. His work obtained the notice of the well-known teacher of engineering drawing, John Rule of MIT, and now retired in New Mexico. Professor Rule persuaded Steve Coons to return to MIT, the institution that had been unable to deal with, and nurture his genius during his abbreviated student days. He remained on the faculty of M I T until 1969.
in lndust O,
mitted part of his knowledge to his students, colleagues and friends. Steven Anson Coons was a man of humor and gentleness, and a wonderful friend.
Before and during this period he was a photograpic illustrator and color consultant to Ginn and Company, Publishers. With his colleague, Professor Rule, he published several well-known engineering graphics texts. Most of us are better acquainted with his important contribution in computer graphics. About 1962 it became apparent that computers would be able to produce graphical images and, more important, to receive instructions from humans in human ways, i.e., one could sketch pictures which a computer could understand. Steve Coons was an important member of the thesis committee of Ivan Sutherland. The thesis is entitled Sketchpad - a scheme which clearly demonstrated this concept. Steve Coons advised on this hallmark achievement. More important, however, was Steve's ability to perceive, and perceive very sharply, the significance of this achievement. He spent much of his time delivering lectures nationally and internationally to illustrate the potential of computer graphics. His keen sensitivity could rapidly appraise the background of an audience. Then he would quickly relate the kernel of the notion of computer graphics to that audience in terms they easily understood. One could say he "had a way with words". Indeed, he had. He could achieve, with a few slowly ,paced words and dramatic pauses, more than most of us are unable to achieve with diagrams, laboriously thought out prose and explanations. His rich vocabulary, one of the manifestations of his brilliant and sensitive mind, often challenged the beholder to attend more closely to what Steve said and to ponder the rich implications of the subtle meaning in his choice of words. He was much concerned with the use of words .... and their abuse. Many of us recall his teasing objection to those time sharing computers which greeted one with "good morning" or "good day". Steve's reaction: "I wish that computer would say good morning only if it were true and it really meant it!"
Computers in Industry
Similarly he would remark, "The mathematics is telling us something, but we aren't listening". This comment would be despairingly directed at himself or, with gentle encouragement, at his colleagues and students when a subtle point revealed in an equation escaped his or our notice. Such occasions arose in the telling or reading of his important contribution known as Coons' Surfaces. Steve's extensive experience in aircraft design, his teaching of engineering drawing, and his intuitive and insightful way of using mathematics and mathematical notation, seized the opportunity to combine "man and machine", i.e., people and computers, to design surfaces. Not just artistically pleasing surfaces, but surfaces for manufactured products such as cars, ships, planes, shoe lasts, bottles, etc., etc. Steve again brought his clear perception to how an every-day problem might be better solved with new tools. New tools which permit the exploitation of new methods to solve old problems. His name is embedded forever in the technical literature for computer generated surfaces. We are indebted to Steve for this contribution of Coons' surfaces. He, however, felt much more satisfied and proud of his students who, inspired
Steven A. Coons in Stockholm (1970).
Foreword
7
by his Project MAC monograph, produced so many good extensions of his work. He left MIT in 1969 to become Full Professor at Syracuse University. There he continued much of the initial work on surfaces begun at MIT and during a year at Harvard. The work flourished under his tutelage and his students produced, and continue to produce, important results. Steve was a highly sought after lecturer in short courses. He regularly gave a series of lectures on computational geometry and surfaces during summers at the University of Michigan. He spent a sabbatical year at the University of Utah. At MIT, at Harvard, at Michigan, at Utah and elsewhere he inspired students, each in their own creative way, to extend and amplify Steve's important contribution. This is the reward most cherished by a teacher. Steve Coons was the consummate teacher and he took great satisfaction and pride in these accomplishments. His students, who quickly in his mind became colleagues, honored and repaid his inspiration by their accomplishments. It would be wrong to say that he was only a teacher of students. Many of us on the faculties of universities or engineers in industry were similarly inspired and were thus more effective in our work.
8
Foreword
As his health became less robust he retired from Syracuse University and spent a little over a year at the Automation Institute of the Hungarian Academy of Science in Budapest. There, without the strain of classroom teaching, he wrote more papers and inspired research. They miss him sorely since he left to return to the United States in 1977. Much of what I have recalled here deals with Steve Coons' technical contributions as engineer, teacher and researcher. Less well-known is his
(;omputers m lndusto,
sensitive use of the camera. He also played the violin with great sensitivity and feeling. Lately he preferred to discuss elements of music rather than perform. In these discussions he always had some revealing insight that made professional musicians glow with appreciation for his perception. And, he was an equally ardent admirer of poetry. We were all deeply saddened by his passing. Steve lives on in what he taught and what he was to all of us.
In Memory of Steven Anson Coons b y J, H a t v a n y , P. B6zier and B. H e r z o g
Steven A. Coons in Munich (1970).
North-Holland Publishing Company Computers in Industry 3 (1982) I-8 0166-3615/82/0000-0000/$02.75
© 1982 N o r t h - H o l l a n d
2
Foreword
1. Introductory Remarks, by J. Hatvanv Computer and Automation Institute, Hungarian Academy of Sciences, H-1502 BUDAPEST, XI, Kende Utca 13-17, Hungary
Mausoleums, eulogies and reminiscences appeal only to the sentiments of those whose personal emotions they resuscitate. The memorial which the world of science raises to its departed, is the intellectual continuation of their work through successive generations and in all corners of the Globe of international learning. When Professor Bertram Herzog, an earstwhile pupil of Steve Coons' and his host at Boulder University during the last months of Steve's life, called on me in Budapest in 1980, we discussed an appropriate way to commemorate Steve. We soon asked Professor Pierre B6zier of Paris, who had modestly and lovingly dedicated the compendium of his life's work "to my Teacher and Master: Steve Coons", to join us in our endeavour. Together, the three of us invited a broad spectrum of researchers from m a n y countries who happened to be familiar to us through their published work in the areas of surface representation and computational geometry, to contribute to this Memorial Issue. We did not ask for papers about Steve Coons, or about his work. We asked for papers about the authors' own work, in the conviction that they would together demonstrate how Steve's scientific achievement lives on and inspires an ever growing circle of mathematicians, computer scientists and engineers in universities, research institutes and industries throughout the world. The first six papers of this Issue are primarily of a theoretical nature. Cohen and Riesenfeld present a general matrix representation method. They have written to say that "the late Professor Coons might have enjoyed this paper because 1) he was an enthusiastic promoter of splines, 2) he deeply
"~mTputers in Industr)'
admired the B6zier technique, and 3) matrix formulations were his favorite representations whenever they could be applied. He reasoned with matrices more comfortably than other people we knew: they "talked" to him about the characteristics and behavior of the schemes they represented. As the field developed, the implementational facility found in matrix formulations gave more motivation and acceptance to this approach to dealing with curves and surfaces." The papers of Bars~iv and of Semenkov and Vasilet, are concerned with their important new results in the B-spline technique of surface representation. Barnhill proposes novel solutions to some problems concerned with the uses of Coons' patches, while Sablonnibre reports on an original treatment of Lagrange interpolation problems by quadratic splines. The contribution of Knapp demonstrates an extension of Coons' techniques to the case of nonuniform basis B-splines. The next batch of three papers deals with various aspects of man-machine interaction in the surface design and representation environment -- a field which since "Project M A C " days was always of concern to Steve Coons. Forrest deals with a pragmatic approach to man-machine interaction in the C A D context, Bonitz tackles this problem from another aspect: the development of methods permitting the stylist easier control over car-body shape in a computer-based system. Posdamer presents new results of a multi-disciplinary approach to the automatic measurement of the surface geometry of three-dimensional objects. The final group of seven papers describes industrial systems which have been developed and are being used in a number of countries. Okino, Kakazu, Kubo, Hashimoto and Shimora describe the celebrated TIPS-1 system, Strasser writes about a fast interactive shape design process, Macurek reports on the C K D A P T approach to the manufacture of turbine blades, Arm# writes of a highperformance 3D graphics systems. V~irady discusses practical considerations for C A D / C A M applied to sculptured surfaces, Renner presents a new engineering approach to curve representation, while Janse shows a C A D system for "free-form'" surface design. Steve Coons was equally at home in each of these areas: theory, methods and applications. For him they always existed together in the totality of a design-tool, design and implementation complex.
Computers in Indust(v
Foreword
3
Originally Professors B6zier, Herzog and I, as responsible for assembling this Issue, planned to blend our individual editorial contributions into a single presentation. The contributions have, however, turned out to be so different in style and character, that it was deemed preferable for them to retain their individual flavours. Pierre Bbzier has offered an appreciation of Steven Coons' SCIE N T I F I C A C H I E V E M E N T . Bert Herzog, in his eulogy delivered at the memorial service (of which that presented here is a slightly abbreviated version) speaks of him as a PERSON, a SCIENTIST, a TEACHER and a FRIEND. (My own thoughts on Steve have already been published in this Journal. *) We hope we have taken an appropriate step towards commemorating a great scientist and a warm-hearted man. * C O M P U T E R IN I N D U S T R Y 1 (1979) 63-64
2. Steven Anson Coons: The Scientist, by P. Bbzier 12 Avenue Gourgaud, 75017 PARIS, France
There is no doubt that Steven Anson Coons, our friend Steve, will be remembered as one of the men who have been responsible for dramatic progress in industries such as aircraft, ship and car manufacturing. The mathematical theory he invented and the systems derived from it are now used throughout the world. In order to fully understand the value of his contribution, it seems advisable to recall the process which was widely used before Steve's invention, since one easily forgets the difficulties previously encountered, after they have been overcome. Man invented the machine tool milleniums ago;
Steven Anson Coons (left) in Budapest with Jozsef Hatvany (right).
Egyptian bas-reliefs and paintings display lathes and drills; the need for accuracy, as we understand it today, began about the middle of the eighteenth century; to minimize the wear of the parts which had to slide or rotate, it was necessary that the surfaces remain in full contact, so as to keep the specific load to a minimum. Hence, the motion of the tool relative to the part to be machined was either a linear translation or a rotation. Consequently, the surfaces generated by the combination of two such motions were defined by segments of lines and circles, and were limited to planes, cylinders, cones, spheres and toruses. It is worth mentioning that the speed of the combined motions had not to be related by an accurate ratio. For obtaining leadscrews or gear flanks, the relation between the two speeds had to be kept in strict proportion; this was obtained by successive improvements till the advent of optical or electrical systems. But mechanical industry must sometimes manufacture parts the shape of which can not be exclusively described with lines and circles. The reason is twofold: on the one hand, the efficiency of some mechanisms is related with aero-, hydro- or thermodynamics, and their shape must comply with physical laws; in the same manner,
4
Foreword
some cams must control motions that are free of shocks. On the other hand, stylists generally conceive curves and surfaces the curvatures of which vary smoothly from end to end. Before the advent of numerical control, the only method available in such cases was to copy-turn, grind or mill, using a template or a three dimensional master. Nevertheless, one has to admit that copymilling has a limited accuracy, due to the deflection of the feeler, and the milled parts have to be hand-finished by highly skilled operators. For inspection, one generally had to rely on templates, or on direct contact (i.e. die-spotting) with a standard part. For either type of parts, the first step was the building of a model; it was the result of iterative adjustments performed according to the measures obtained during tests, and it was the case for turbine foils, propellers blades, airwings or boat hulls; or it was the expression of a stylist's intention or will. A typical example is that of car bodies, furniture, household appliances etc.. When aesthetics played a dominant role, the model was hand made of clay or plaster of Paris. Then, one had to pick the coordinates of points located on the surface of the model. If the shape was rather complicated, or if accuracy was sought for, the points were many, sometimes in the thousands. Owing to the nature of the measuring machines, those points were located on cross-sections parallel with the machine referential. On a drawing board, the designer traced curves running through, or by, those points, making use of splines or French curves. In the meantime, he or she endeavoured to correct any defect, kink or bump, that became apparent. After that stage, templates made of steel or aluminium sheets, and duplicating the drawn curves were used to adjust cross-sections on a master. Interpolation between those sections was performed by highly skilled patternmakers or diesetters. Whilst performing that operation, they often had to correct the cross-sections so as to obtain a very smooth shape, and eliminate defects that the drawings could not disclose. From the moment the master was finished and accepted, it became the only standard, and the drawings or templates were considered only approximative. Of course, the importance of a master was great; it was made of very expensive mahogany, of
Uomputers in lndust O,
stable resin or of metal. Wood or resin parts were kept in air and moisture-conditioned rooms so as to avoid warping. Evidently, so precious a part could not be used on a copy-milling machine; plaster of Paris or resin replicas served for machining and inspection. Due to the deflection of the feeler, there is a slight discrepancy between part and replica, and it is the task of a setter to correct the former. So, from model to drawing, to template, to master, to copy, to part, small differences did occur, the sum of which was not always negligible. Moreover, the process was slow and retouches were expensive as well as time consuming. As soon as data processing and numerical control had made significant progress regarding cost, power and reliability, it became obvious that it was the best way to improve accuracy and reduce leadtime. Aircraft, ship and car building industries were among those most interested in promoting it. For a car. it is admitted that a few millimetres difference in the overall length is negligible, but, for assembling parts together by welding, crimping or gluing, the range must be kept within a portion of a millimeter. Another problem arose fi'om the fact that a stylist is interested in building the shape of an object starting not from cross-sections but from character lines which are free-form, i.e. twisted lines and curves; the same often happens with technical surfaces: for instance, the leading and trailing edges of a turbine foil or of an aircraft wing are seldom plane curves. Such a requirement naturally implies that curves and surfaces be parametric rather than Cartesian. It is relatively easy, on a sculptured surface, to define two families of piecewise curves which enclose patches; each segment is limited by the nodes located at the corners of the patches. It is much more difficult to give a patch a mathematical definition such that it correctly blends with the surfaces surrounding it up to order two, which is tangency, or three, osculation. It can safely be stated that Steve Coons gave this rather difficult problem its first general solu-
tion. Representing the overall surface of an object with a great number of point coordinates is feasible so long as the computer is powerful enough, but it is much better to express it with a limited quantity of coefficients of a set of polynomial
Computers in Industry
functions. It is advisable to reduce the quantity of numerical data so as to minimize the possiblity of mistakes. In the meantime, it is possible to take into account (in real time) the allowance left after a roughing cut, to compensate for springback, and to compute the offset - whichever be the tool dimension. Moreover, the use of parametric functions is compulsory for machining patterns, dies and stamping tools, when the part is in a tilted position. The first solution, as it was described in Coons' MIT report*, could be applied to a large variety of problems, but it had a weak point: the properties of the boundary curves can not give any information regarding mixed derivatives at the corners of the patch, which control the torsion or twist of the surface. In principle, one could decide that their value is nil. But, though the blending of the patches be correct, one could detect at the corners of the patch, a small portion that looks flat and this is, aesthetically speaking, a defect. Steve Coons found the remedy at once, namely: slightly moving four points located inside the patch, from which the values of the twist vectors are deduced. From one patch to the next, the derivatives were selected in order to erase the flat portions. The consequence was that the system no longer was purely algorithmic, and that it somewhat relied on the taste and experience of the operator. Tangential blending is good enough to solve most of the aesthetic problems, but when it comes to fluid dynamics, any discontinuity of curvature causes shocks and consequently a loss of energy and efficiency. Steven Coons easily solved the problem, raising the order of his interpolating functions up to five. Then among the thirty-six coefficients of a biquintic polynomial, only twenty are provided by the boundary lines; unless cancelling the sixteen others, the operator has to perform an interactive task, moving as many points to give the surface an overall definition. In some very particular cases, specialists insist on having a continuity up to the third derivative, so as to avoid what is called "jerk". The Coons' formulae adapt perfectly to this requirement; extending it to any order has no limitation but the power of the computer (... and the operator's endurance). The need to have to rely partly on the operator is no intrinsic weakness in Coons' formulae. It comes directly from the fact that a line can only
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yield derivatives related to itself. Any algorithm intended to go further would be liable by its own nature to fail at a given moment, as can any algorithm. Inventing the method that bears his name, Steven Coons perfectly solved the problem he was faced with; and in doing so, he created a system Which is used throughout the worm and has brought to him an international reputation. Leaving aside the scientific aspect of Steve Coons' contribution, we must ask ourselves what is the lesson we received from him - since it is our duty to try and complement the results already available. At first sight, it is strange to consider that a problem which had (for a fair amount of time) attracted the attention of industry, was not solved by a professional mathematician, but by an engineer. No doubt Steven Coons had a gift for mathematics, and in particular for matrices and tensors. Looking at the figures of a tensor, he was able to imagine broadly the shape and properties of the surface so defined. During a meeting held at Salt Lake City in the year 1974 (the subject of which was "Computer Aided Geometric Design"), he exchanged information with a colleague - only scribbling figures in a table of numbers. Moreover, to make vector computation faster, he invented a special set of symbols - but, one had to work hard to develop the ability to use them. Steven Coons was not only able to solve a difficult problem raised by the aircraft and car industries; but well aware of the possibilities of data processing and of his theory in particular, he thought that the data broadly defining a car body could be chosen at the very beginning of the process, in the styling department, then refined and completed by designers, and used all along from methods to stamping tools design, pattern shop, tool shop and inspection. Very few people at that time thought that such an opinion was realistic; today, there is proof that Steve was right - years and years afterwards. There can be no doubt that Steven Coons was a brilliant man, with outstanding intelligence and broad knowledge. His professional experience he had acquired while working as an engineer. He knew that before inventing a solution, one first has to carefully consider the data of the problem and, most of all, to take into account the behaviour and
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C'omputers
reaction of the people of all ranks who will have to work with it. In addition to all the aforementioned qualities. Steve was a generous man, dispensing freely what he knew, and h a p p y to feel that he had trans-
3. Steven Anson Coons: The Man, by B. Herzof~ Herzog Associates, Inc. P.O. Box 3169, Boulder, CO 80307, USA
Steven Anson Coons was born in Palatine, New York, on March 7, 1912. Depending upon Steve's mood, his childhood was either happy or not quite so. In any case, eventually he enrolled at M I T only to leave about a year later to end his formal student days under adverse conditions - as he would be quick to tell you. This was always most surprising to those of us who first met him under professional circumstances. He was often and mistakenly introduced as "Dr. Coons". Steve would then quietly chuckle with close friends about his "flunking" out of M I T and remaining degreeless. Or, as it is stated in a Syracuse University publication, " H e has never earned a university degree, proving that a degree is not necessary - if one is a genius." The first professional association we know about is his work as an engineer with Vought-Sikorsky Aircraft Corporation from 1940 to 1948. During this period he demonstrated his keen insight into matters of graphical presentation of complex objects and shaped surfaces such as airplane fuselages, wings and other components. His work obtained the notice of the well-known teacher of engineering drawing, John Rule of MIT, and now retired in New Mexico. Professor Rule persuaded Steve Coons to return to MIT, the institution that had been unable to deal with, and nurture his genius during his abbreviated student days. He remained on the faculty of M I T until 1969.
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mitted part of his knowledge to his students, colleagues and friends. Steven Anson Coons was a man of humor and gentleness, and a wonderful friend.
Before and during this period he was a photograpic illustrator and color consultant to Ginn and Company, Publishers. With his colleague, Professor Rule, he published several well-known engineering graphics texts. Most of us are better acquainted with his important contribution in computer graphics. About 1962 it became apparent that computers would be able to produce graphical images and, more important, to receive instructions from humans in human ways, i.e., one could sketch pictures which a computer could understand. Steve Coons was an important member of the thesis committee of Ivan Sutherland. The thesis is entitled Sketchpad - a scheme which clearly demonstrated this concept. Steve Coons advised on this hallmark achievement. More important, however, was Steve's ability to perceive, and perceive very sharply, the significance of this achievement. He spent much of his time delivering lectures nationally and internationally to illustrate the potential of computer graphics. His keen sensitivity could rapidly appraise the background of an audience. Then he would quickly relate the kernel of the notion of computer graphics to that audience in terms they easily understood. One could say he "had a way with words". Indeed, he had. He could achieve, with a few slowly ,paced words and dramatic pauses, more than most of us are unable to achieve with diagrams, laboriously thought out prose and explanations. His rich vocabulary, one of the manifestations of his brilliant and sensitive mind, often challenged the beholder to attend more closely to what Steve said and to ponder the rich implications of the subtle meaning in his choice of words. He was much concerned with the use of words .... and their abuse. Many of us recall his teasing objection to those time sharing computers which greeted one with "good morning" or "good day". Steve's reaction: "I wish that computer would say good morning only if it were true and it really meant it!"
Computers in Industry
Similarly he would remark, "The mathematics is telling us something, but we aren't listening". This comment would be despairingly directed at himself or, with gentle encouragement, at his colleagues and students when a subtle point revealed in an equation escaped his or our notice. Such occasions arose in the telling or reading of his important contribution known as Coons' Surfaces. Steve's extensive experience in aircraft design, his teaching of engineering drawing, and his intuitive and insightful way of using mathematics and mathematical notation, seized the opportunity to combine "man and machine", i.e., people and computers, to design surfaces. Not just artistically pleasing surfaces, but surfaces for manufactured products such as cars, ships, planes, shoe lasts, bottles, etc., etc. Steve again brought his clear perception to how an every-day problem might be better solved with new tools. New tools which permit the exploitation of new methods to solve old problems. His name is embedded forever in the technical literature for computer generated surfaces. We are indebted to Steve for this contribution of Coons' surfaces. He, however, felt much more satisfied and proud of his students who, inspired
Steven A. Coons in Stockholm (1970).
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by his Project MAC monograph, produced so many good extensions of his work. He left MIT in 1969 to become Full Professor at Syracuse University. There he continued much of the initial work on surfaces begun at MIT and during a year at Harvard. The work flourished under his tutelage and his students produced, and continue to produce, important results. Steve was a highly sought after lecturer in short courses. He regularly gave a series of lectures on computational geometry and surfaces during summers at the University of Michigan. He spent a sabbatical year at the University of Utah. At MIT, at Harvard, at Michigan, at Utah and elsewhere he inspired students, each in their own creative way, to extend and amplify Steve's important contribution. This is the reward most cherished by a teacher. Steve Coons was the consummate teacher and he took great satisfaction and pride in these accomplishments. His students, who quickly in his mind became colleagues, honored and repaid his inspiration by their accomplishments. It would be wrong to say that he was only a teacher of students. Many of us on the faculties of universities or engineers in industry were similarly inspired and were thus more effective in our work.
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As his health became less robust he retired from Syracuse University and spent a little over a year at the Automation Institute of the Hungarian Academy of Science in Budapest. There, without the strain of classroom teaching, he wrote more papers and inspired research. They miss him sorely since he left to return to the United States in 1977. Much of what I have recalled here deals with Steve Coons' technical contributions as engineer, teacher and researcher. Less well-known is his
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sensitive use of the camera. He also played the violin with great sensitivity and feeling. Lately he preferred to discuss elements of music rather than perform. In these discussions he always had some revealing insight that made professional musicians glow with appreciation for his perception. And, he was an equally ardent admirer of poetry. We were all deeply saddened by his passing. Steve lives on in what he taught and what he was to all of us.