Geometry and Interpolation of Curves and Surfaces
This text takes a practical, step-by-step approach to algebraic curves and surface interpolation motivated by the understanding of the many applications in engineering analysis, approximation, and curve plotting problems. Because of its usefulness for computing, the algebraic approach is the main theme, but a brief discussion of the synthetic approach is also presented as a way of gaining additional insight before proceeding with the algebraic manipulation. The authors start with simple interpolation, including splines, and extend this in an intuitive fashion to the production of conic sections. They then introduce projective coordinates as tools for dealing with higher order curves and singular points. They present many applications and concrete examples, including parabolic interpolation, geometric approximation, and the numerical solution of trajectory problems. In the final chapter, they apply the basic theory to the construction of finite element basis functions and surface interpolants over nonregular shapes. Professionals, students, and researchers in applied mathematics, solid modeling, graphics, robotics, and engineering design and analysis will find this a useful reference.
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acnode affine plane affine transformation algebraic curve approximation asymptotes Axiom axis base points basis functions boundary segments central projections Chapter chord circle coefficients collinear complex projective plane consider construction coordinate system corresponding crunode cubic spline cubic surface cubies curve of order cusp defined determine dimension discussed distinct points domain double point ellipse Example finite interpolation property finite linear interpolation four points geometry given curve given points hence Hermite interpolation homogeneous coordinates homogeneous polynomial hyperbola ideal line inflection point intersection line meets line pair linear interpolation problem maps matrix meets the curve method multiple points nodes nonsingular parabola parallel parametrization pass pencil of conies piecewise point of multiplicity polynomial of degree projective plane projective transformation quartic rational curve result satisfy Section Show shown in Figure simple points slope solution Steiner surface tangent plane Theorem three points triangle of reference variables vector space zero