Interpolating Cubic SplinesA spline is a thin flexible strip composed of a material such as bamboo or steel that can be bent to pass through or near given points in the plane, or in 3-space in a smooth manner. Mechanical engineers and drafting specialists find such (physical) splines useful in designing and in drawing plans for a wide variety of objects, such as for hulls of boats or for the bodies of automobiles where smooth curves need to be specified. These days, physi cal splines are largely replaced by computer software that can compute the desired curves (with appropriate encouragment). The same mathematical ideas used for computing "spline" curves can be extended to allow us to compute "spline" surfaces. The application ofthese mathematical ideas is rather widespread. Spline functions are central to computer graphics disciplines. Spline curves and surfaces are used in computer graphics renderings for both real and imagi nary objects. Computer-aided-design (CAD) systems depend on algorithms for computing spline functions, and splines are used in numerical analysis and statistics. Thus the construction of movies and computer games trav els side-by-side with the art of automobile design, sail construction, and architecture; and statisticians and applied mathematicians use splines as everyday computational tools, often divorced from graphic images. |
Contents
| 1 | |
| 3 | |
Smoothing Splines | 10 |
51 | 18 |
Curves | 31 |
ix | 37 |
ASpline Curves With Range Dimension d | 75 |
31 | 89 |
Double Tangent Cubic Splines | 94 |
Global Cubic Space Curve Splines | 101 |
Rational Cubic Splines | 156 |
Tensor Product Surface Splines | 193 |
Boundary Curve Based Surface Splines | 210 |
Physical Splines | 217 |
References | 233 |
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Common terms and phrases
3-space a x b angle arc length parameterized B-spline basis bicubic compute coordinate system coordinate vectors corresponding cross product cubic polynomial segment cubic polynomial space cubic spline segment curvature curve segment curve x data points defined denotes derivative diagonal direction double tangent Exercise geometrically continuous global cubic spline graph Hermite cubic polynomial Hermite cubic spline inflection point inner product interpolates the points interpolation function join order length parameterized curve linear m₁ matrix mi+1 minimal monotonic normal vector Note osculating circle osculating plane p₁ parameter limit values parameter values parameterized space curve patch physical spline Pi+1 planar plane equation point x(t points P1 polynomial space curve quadratic range dimension scalar Show slope Solution specified spline curve spline function spline interpolation subspace tangent line tangent vectors unit vectors vector space Xi+1 xy-plane y₁ Yi+1