Theory And Application Of Low Degree Variable Degree Splines

Variable degree spline is a generalization of traditional spline function. After givinga sequence of knots, variable degree spline can be defined on them as piecewise polyno-mials. The degree of polynomials can be di?erent on di?erent knot intervals.In chapter one, we reviewed three di?erent approaches defining variable degreesplines. We analyzed their advantages and limitations respectively.In chapter two, we gave a more general definition of variable spline function and itsspecial case: low degree variable degree spline. We put forward and proved the dimensionformula of low degree variable degree spline. Then, we constructed a set of functions inlow degree variable degree spline space and proved that they are basis functions of thisspace. We also proved that this group of basis functions process three properties: non-negativity, unity partition and local support. Additionally, we showed that point evaluationof variable degree spline functions can be implemented by de Casteljau algorithm.In chapter three, we illustrated an application of low degree variable degree spline inrepresenting the contour of TrueType fonts. We presented the transformation algorithmand analyzed the storage space saving brought by this new representation method in detail.After that, we chose three types of most widely used TrueType fonts. Statistic showsthat representing these fonts'outlines by variable degree splines significantly saves spacestorage.