In this chapter we begin the study of best approximations. In this case we study the best (min) polynomial approximation in the uniform (max) norm. The existence a best approximating polynomial is first presented. The more subtle issue of uniqueness is then discussed. To show uniqueness the celebrated de la Vallee Poussin, and Chebyshev equi-oscillation theorems are presented. A first error estimate is then presented. The problems of interpolation, discussed in the previous chapter, and best approximation are then related via the Lebesgue constant. Chebyshev polynomials are then introduced, and their most relevant properties presented. Interpolation at Chebyshev nodes, and the mitigation of the Runge phenomenon are then discussed. Finally; Bernstein polynomials; moduli of continuity and smoothness; are detailed in order to study Weierstrass approximation theorem.

This chapter describes the essential aspects of geometry definition of flow channels, blades and vanes in radial compressors. The impeller blades and flow channel make up a complex three-dimensional shape, and a general method for defining such geometries using Bezier surfaces is described. The meridional channel is defined as a series of Bezier patches whose geometry is parametrised to allow changes to be made quickly and efficiently during the design. Radial compressor impeller blades are typically defined as a distribution of camber line angle and thickness on the hub and casing meridional sections. A three-dimensional shape is generated by joining the hub and casing sections by straight lines to make up a ruled surface. This simplifies simplify manufacture by flank milling. In transonic flows, a more complex definition is used in which the geometry is defined along a series of separate meridional planes. A description of the definition of asymmetric components, such as the volute, is also provided.