03721nam a22004935i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050000900172072001600181072002300197072001500220082001400235100007600249245008800325250001800413260007500431300006400506336002600570337002600596338003600622347002400658490005100682505034200733520157401075650002102649650002302670650002102693650009402714650011302808710003402921773002002955776003602975776003603011830005103047856004603098912001403144942001203158950003803170999001903208978-3-030-12551-6DE-He21320190726112458.0cr nn 008mamaa190223s2019 gw | s |||| 0|eng d a97830301255169978-3-030-12551-67 a10.1007/978-3-030-12551-62doi 4aT385 7aUML2bicssc 7aCOM0120002bisacsh 7aUML2thema04a006.62231 aPenner, Alvin.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut10aFitting Splines to a Parametric Functionh[electronic resource] /cby Alvin Penner. a1st ed. 2019. 1aCham :bSpringer International Publishing :bImprint: Springer,c2019. aXII, 79 p. 32 illus., 21 illus. in color.bonline resource. atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aSpringerBriefs in Computer Science,x2191-57680 a1 Introduction -- 2 Least Squares Orthogonal Distance -- 3 General Properties of Splines -- 4 ODF using a cubic Bézier -- 5 Topology of Merges/Crossovers -- 6 ODF using a 5-Point B-spline -- 7 ODF using a 6-Point B-spline -- 8 ODF using a quartic Bézier -- 9 ODF using a Beta2-spline -- 10 ODF using a Beta1-spline -- 11 Conclusions. aThis Brief investigates the intersections that occur between three different areas of study that normally would not touch each other: ODF, spline theory, and topology. The Least Squares Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop mathematical models of the physical shapes of objects, due to the fact that it produces a fitted result that is invariant with respect to the size and orientation of the object. It is normally used to produce a single optimum fit to a specific object; this work focuses instead on the issue of whether the fit responds continuously as the shape of the object changes. The theory of splines develops user-friendly ways of manipulating six different splines to fit the shape of a simple family of epiTrochoid curves: two types of Bézier curve, two uniform B-splines, and two Beta-splines. This work will focus on issues that arise when mathematically optimizing the fit. There are typically multiple solutions to the ODF method, and the number of solutions can often change as the object changes shape, so two topological questions immediately arise: are there rules that can be applied concerning the relative number of local minima and saddle points, and are there different mechanisms available by which solutions can either merge and disappear, or cross over each other and interchange roles. The author proposes some simple rules which can be used to determine if a given set of solutions is internally consistent in the sense that it has the appropriate number of each type of solution. aComputer Science 0aComputer graphics. 0aComputer vision.14aComputer Graphics.0http://scigraph.springernature.com/things/product-market-codes/I2201324aImage Processing and Computer Vision.0http://scigraph.springernature.com/things/product-market-codes/I220212 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z978303012550908iPrinted edition:z9783030125523 0aSpringerBriefs in Computer Science,x2191-576840uhttps://doi.org/10.1007/978-3-030-12551-6 aZDB-2-SCS 2ddccEB aComputer Science (Springer-11645) c115953d115953