The technical office of the Vickers Aircraft Company ran a report series with sections for each of the main offices. Among these was the Mathematical Services department, and so these reports were given identification numbers beginning with VTO/MS/..
Even when the company became by merger, first the British Aircraft Corporation, and then British Aerospace, these identifiers continued to be used.
The following list identifies the members of the report series which are relevant to the development of the Numerical Master Geometry system within the department. The first half of the list was dated 4th October 1972. The second half is undated, but must date between then and 31st December 1974, when I left BAC.
I have tried to leave the original comments, made when the list was created, distinct from additional comments made when I was bringing this material to the web.
The papers themselves, in gzipped postscript form, are being added as links as time permits. Individual reports are being TeXed, the original text being maintained (typos in the TeXing will be corrected when they are found), but footnotes being added where the modern reader may find them helpful. No attempt is being made to create facsimiles of the original, and so the hand-drawn figures in the original are being recreated in postscript, often in cleaner form.
I hope to complete the exercise by December 1999.
VTO/MS/147 Parametric Surface Equations for Non-rectangular Regions
This gives a very complicated answer to a difficult question, an answer which is costly to use and in any case difficult to apply within a complete system. The technique has not been used in anger, and the only point of the report is that previously there were no techniques at all for dealing with this problem, and there are still no others.
1999 comment. It has been used in anger and with some success in at least two automobile companies, and there is now becoming visible a decent theory to tie together this and other ad hoc techniques which have been developed since. See Malraison's review.
VTO/MS/148 Two Basic Interrogations of Parametric Surfaces.
This illustrates in what is intended to be a naive way the methods of getting hold of information interatively which cannot be calculated explicitly and directly from the surface equations.
VTO/MS/149 Offset Parametric Surfaces.
The procedures for allowing for skin-thickness, cutter compensation etc. work out very elegantly for parametric surfaces, no iteration for the offsetting being required, however complex the offset procedure.
VTO/MS/150 General Interrogation for Parametric Surfaces.
This follows on from /148, presenting the same sort of procedures in a more general and abstract light.
VTO/MS/151 Conditions for continuity of Surface Normal between adjacent parametric surfaces.
Like /147 this tackles the `topology problem' where the surface of a shape cannot be partitioned neatly into a rectangular array. Again it is difficult, but the difficulties are essentially those inherent in irregular partitioning. The main criticism of these proposals is that the kernel of the theory requires the curvature tensors at all patch corners to be specified, and although suggestions are made for calculating these they are not demonstrated.
Also, continuity of slope across boundaries is dependent on the continuity of curvature at the corners, and the run of the boundaries is not directly controlled. The individual pieces of surface have to be rectangular or triangular, but any number can meet at each corner.
VTO/MS\152 Vectors and Parameters
An early document, written to get my own ideas straight. It contains a lot of half-thought-out stuff which might put a lay reader on to some false trails, but it could form the basis of a simple instructional document if rewritten carefully.
VTO/MS\153 Potential Surfaces
This examines the other general surface form, O(P) = 0 in a lot of detail and from many viewpoints. Most of the material is quite standard.
1999 comment. I don't think I had a good idea in 1972 of what was standard.
VTO/MS/154 Spline Curves
The many variants of spline curves are examined, both interpolating and approximating, explicit and parametric, equal and unequal interval, cubic, other odd order, even order and non-polynomial variations. Again this is advanced text-book material, not particularly original.
1999 comment. Some of it was original for the time, but was Arnold Roberts' originality not mine. The big omission is B-splines, which did not become visible to the CAGD community until a few years later. In fact they are described here as `finite width modes', but their significance totally escaped me.
VTO/MS/155 A 16-point formulation of a bicubic patch.
This describes theory similar to that developed independently by Bezier. It covers rational curves and surfaces better, and hints at the coupling of degrees of freedom to give composite piecewise surfaces later seen to be special cases of B-splines.
1999 comment. developed independently by Bernstein, de Casteljau, Bezier, Hosaka and Yamaguchi and probably many others.
VTO/MS/156 Spline Surfaces
An extension into two-way interpolation of the techniques of /154. The only really new idea, of membrane surfaces, has a serious flaw in its theory. The r^3 term should be r^2log(r^2). The correct version is documented in the NASTRAN system. Actually my version would be faster to run and probably gives as good a surface shape. Theoretical derivations in this area are only clues as to good avenues up which to look. The criterion which matters is whether the resulting interpolation has the necessary continuity and aesthetic properties.
VTO/MS/157 Step Length Control of Interrogations fo Parametric Surfaces.
This examines the dynamic control of pitch as the interrogation algorithm `marches' along a solution curve. Detailed know-how relating to economy and robustness of algorithms is included.
VTO/MS/158 Interrogations involving Two Parametric Surfaces.
As well as the obvious intersection problem, other situations involving two surfaces simultaneously are considered. Know-how relating to a detailed comparison of different formulations is included.
VTO/MS/159 First Approximations, Starts and Finishes.
This, with /148,/150,/157 and /158 more or less completes the theory of `marching method' interrogations as we have implemented them, by considering how the marching is started, kept going in the right direction and finally stopped at the right place. Know-how.
VTO/MS/160 Parametric Splines in Tension.
An addition to /154 discovered somewhat later. Of interest rather than use.
1999 comment. This material was completed and put into proper order by John Manning, who published it in a Computer Journal paper. vol 17 no 2 pp181-186. The same ideas were invented independently by Neilson as nu-splines, and were popularised by Barsky under the name of beta-splines.
VTO/MS/162 NMG `Datum' facilities
Of no interest except to a user of NMG.
VTO/MS/164 The Use of Circular Arcs for the Interpolation of Curves through Empirical Data Points.
This is some of my best work. It builds on the ideas of Shippey, but identifies an extra degree of freedom which is demonstrably used to advantage, and also, independently halves the number of arcs needed for a given number of data points. Surprisingly (and unexpectedly) the smaller number of arcs tends to give a smoother curve.
The close analogy with quadratic spline theory is a useful guide, and because the problem of assignment of parameter values to the data points does not arise, development of parametric spline theory could be stimulated.
VTO/MS/172 G3D Manual
A set of subroutines for simple 3D geometric calculations. Not very well documented, nor very efficient, but quite cost-effective at their level of use in B.A.C.
VTO/MS/187 New NMG Datum Facilities
VTO/MS/188 Trinomial Basis Functions for Interpolation in Triangular Regions.
This is an extension of Bezier theory to separate triangular surface elements, and provides the basic theory for further work on spline surface interpolation over regular triangulations.
VTO/MS/189 Machining of Wind Tunnel Models.
A commentary on a sequence of photographs showing stages in the N/C machining of a `pseudo' wind tunnel model. Know-how
VTO/MS/194 Minimum Theoretical Structure for a Bezier-type surface.
This explores the properties of Bezier surfaces and attempts to pin down minimal axioms from which they may be desired (derived ?). Some unconventional possibilities are suggested as examples.
VTO/MS/195 B-spline interpolation over regular triangular lattices.
The analogue of the tensor product B-spline basis functions over a rectangular grid are described for a regular plane triangular lattice. Some specific low-order results for the regular polyhedral lattices are also examined, but no general theory. Builds on the ideas of /188
VTO/MS/197 Original of MTDR paper (Proc 14th International Machine Tool Design and Research conference. Macmillan 1974. pp 233-238)
VTO/MS/199 Two Slope-Continuous Triangular Elements constructed from low-order polynomial pieces.
These are attempts to find triangular patches which can be joined in an irregular triangular partitioning. The results satisfy that requirement, but are not very fair. No analogue of the B-spline modes are sought.
1999 comment One of these partitionings turns out to be the Clough Tocher element. The other is the Powell-Sabin 6-way element published later in ACMToG.
VTO/MS/200 A method for displaying the intersection curve of two quadric surfaces.
A piece of academic geometry which demonstrates the power of tensor notation for this purpose.
1999 comment Later published in Computer Journal. vol 19 no 4 pp336-338
VTO/MS/204 The area of a plane Bezier curve.
Determines formulae for the area enclosed by a Bezier curve in terms of the control point co-ordinates. Then does the same in principle for the B-spline.
VTO/MS/205 A New Class of Spline Curves
A non-linear spline is described which has the property that the offset curve is of the same form, but with different coefficients. The curve of /164 is shown to be just one member of a whole family with this property. Difficulties in applying the higher order members of the family are discussed.
VTO/MS/206 The Convolution of two regions.
A generalization of the offset operator is described which has a number of possible applications.
VTO/MS/207 A Class of Surfaces closed under five important geometric operators.
This describes a class of surfaces analogous to the curve described in /205. Again some difficulties are described and are shown to be unavoidable.
VTO/MS/210 A Convenient Representation of Profiles formed from arcs of circles and straight lines.
This entry was not on the original list, but the document is available. It deals with the bulge-factor approach to representing piecewise circular profiles, as used in the Profiledata K-curves extensions later used in the CADCentre's GNC product, and also in the Applied Research of Cambridge draughting system.
This page was updated 4th November 1999