Controlling the interpolation of NURBS curves and surfaces

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements. A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$^1$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$^2$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods. Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry. For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini. Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $N$ patches to form an $N$-sided region is considered. It is shown that regions with odd $N$ can be joined with G$^1$ continuity, but those with even $N$ or requiring G$^2$ continuity can only be obtained for specific geometries

Outlook and appraisal [June 1988]

The growth of the Scottish economy is now moving into line with the rest of the UK. Industrial production is buoyant. The growth of demand is becoming more broadly based between domestic and external sectors. Yet, construction output still remains relatively depressed, and in a significant number of other industries the ratio of Scottish to UK output remains below that of 1980. As yet there is little indication of the favourable changes in unemployment and vacancies during 1987 being reflected in the most recent employment data. But short-term prospects are more favourable than for several years, while growth forecasts for the medium term compare favourably with those for the UK as a whole

The Scottish economy [June 1988]

With the delay in the publication of this quarter's Commentary it should be borne in mind that responses to both the Scottish Chamber's Business Survey (SCBS) and CBI surveys now refer to developments of two months ago. Accordingly, responses do not take account of, or could reasonably be assumed to have anticipated, the recent increases in nominal interest rates and the depreciation of sterling

The world economy [June 1988]

Growth in the world economy appears to be continuing at a brisk rate. Recent forecasts suggest that output should expand by about 3% this year compared with 3.IX in 1987. The outlook therefore appears to be more favourable than when we last reported; a reflection of the limited impact to date of the October stock market crash. Some progress has been made in adjusting the key trade imbalances but progress remains slow

The British economy [June 1988]

The rate of growth of output in the British economy appears to be slowing down in line with earlier expectations for 1988. The rate of growth of domestic demand remains strong. Manufacturing investment is forecast to rise to record levels during 1988 and there is little evidence that the growth in consumer demand is slackening. The slow-down in the rate of growth is therefore largely a reflection of decline in the external demand for exports and buoyant import growth. The progressive upward movement of sterling in recent months and the evidence of increasing inflationary pressure, has placed the Government in a significant policy dilemma. Recent events are a clear illustration that interest rate policy cannot be used both to target the exchange rate and to regulate the expansion of domestic money and credit

The British economy [March 1988]

Growth in the British economy continues at a high rate. To date there is little evidence that the collapse of share prices in October has had much effect on consumer demand or on business confidence. The expected decline in the UK growth rate during 1988 is likely to prevent any occurence of the symptoms of overheating which were evident before the events of last October. Further significant deterioration in the balance of payments is to be expected this year. The size of the deterioration and the extent of downward pressure on the exchange rate will in part depend on the degree to which the economy can contain incipient cost inflationary pressures

The world economy [March 1988]

The outlook for the world economy is, on balance, probably more favourable than when we last reported in November: a further significant deterioration in equity prices has been avoided and business confidence appears to be holding up well. World economic growth will be lower in 1988 than would have been the case but there appears a good chance that recession can be avoided. The key to future exchange rate stability and the maintenance of business confidence in the world economy rests on the progress being made in the US, Japan and West Germany in introducing domestic macro-economic policies to remove the continuing financial imbalances. West Germany's weakening resolve in this connection is a major cause for concern

The British economy [November 1987]

The British economy continues to grow strongly at a rate in excess of other major industrial countries. However, future prospects have been considerably clouded following the dramatic collapse of share prices in the world's major financial centres. The outlook for 1988 and beyond is likely to be heavily influenced by developments elsewhere in the world economy

The Scottish economy [March 1988]

Since October 1984 there have been two regular surveys of Scottish businesses, one carried out by the Chambers of Commerce and the other by the CBI. These have the great advantage that they provide indications of the trends in a number of economic variables several months in advance of the publication of the official statistics . Moreover, they provide a basis for making assessments about the short-term prospects for the economy

The world economy [November 1987]

Recent events have been dominated by a major collapse of share prices in the world's stock markets followed by instability in the foreign exchange markets and a significant downward movement in the exchange value of the dollar. By the end of the first week of November, the stock markets in London, Frankfurt, New York and Tokyo had fallen by 32 30 25% and 15 respectively, since the middle of October. Uncertainty about the attempts to cut the US Budget deficit, the course of the dollar and the future of the Louvre accord on exchange rate stabilisation , continued to have a depressing effect on share prices. While the depressing effects on trade and growth in the world economy of a once-and-for-all fall in share prices of this magnitude should be limited and not of recessionary proportions, continued uncertainty could seriously damage business confidence making a major recession more likel