A basis for the representation, manufacturing tool path generation and scanning measurement of smooth freeform surfaces

Freeform surfaces find wide application, particularly in optics, from unique single-surface science programmes to mobile phone lenses manufactured in billions. This thesis presents research into the mathematical and algorithmic basis for the generation and measurement of smooth freeform surfaces. Two globally significant cases are reported: 1) research in this thesis created prototype segments for the world’s largest telescope; 2) research in this thesis made surfaces underpinning the redefinition of one of the seven SI base units – the kelvin - and also what will be the newly (and permanently) defined value for the Boltzmann constant. Theresearchdemonstratestwounderlyingphilosophiesofprecisionengineering, the critical roles of determinism and of precision measurement in precise manufacturing. The thesis presents methods, and reports their implementation, for the manufacture of freeform surfaces through a comprehensive strategy for tool path generation using minimum axis-count ultra-precision machine tools. In the context of freeform surface machining, the advantages of deterministic motion performance of three-axis machines are brought to bear through a novel treatment of the mathematics of variable contact point geometry. This is applied to ultra-precision diamond turning and ultra-precision large optics grinding with the Cranfield Box machine. New techniques in freeform surface representation, tool path generation, freeform tool shape representation and error compensation are presented. A comprehensive technique for very high spatial resolution CMM areal scanning of freeform surfaces is presented, with a new treatment of contact error removal, achieving interferometer-equivalent surface representation, with 1,000,000+ points and sub-200 nm rms noise without the use of any low-pass filtering