LEADER 03623nam 22004935 450 001 9910337576603321 005 20200705145427.0 010 $a3-030-12551-3 024 7 $a10.1007/978-3-030-12551-6 035 $a(CKB)4100000007702143 035 $a(MiAaPQ)EBC5719227 035 $a(DE-He213)978-3-030-12551-6 035 $a(PPN)235004928 035 $a(EXLCZ)994100000007702143 100 $a20190223d2019 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aFitting Splines to a Parametric Function /$fby Alvin Penner 205 $a1st ed. 2019. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2019. 215 $a1 online resource (86 pages) 225 1 $aSpringerBriefs in Computer Science,$x2191-5768 311 $a3-030-12550-5 327 $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. 330 $aThis Brief investigates the intersections that occur between three different areas of study that normally would not touch each other: ODF, spline theory, and topology. The Least Squares Orthogonal Distance Fitting (ODF) method has become the standard technique used to develop mathematical models of the physical shapes of objects, due to the fact that it produces a fitted result that is invariant with respect to the size and orientation of the object. It is normally used to produce a single optimum fit to a specific object; this work focuses instead on the issue of whether the fit responds continuously as the shape of the object changes. The theory of splines develops user-friendly ways of manipulating six different splines to fit the shape of a simple family of epiTrochoid curves: two types of Bézier curve, two uniform B-splines, and two Beta-splines. This work will focus on issues that arise when mathematically optimizing the fit. There are typically multiple solutions to the ODF method, and the number of solutions can often change as the object changes shape, so two topological questions immediately arise: are there rules that can be applied concerning the relative number of local minima and saddle points, and are there different mechanisms available by which solutions can either merge and disappear, or cross over each other and interchange roles. The author proposes some simple rules which can be used to determine if a given set of solutions is internally consistent in the sense that it has the appropriate number of each type of solution. 410 0$aSpringerBriefs in Computer Science,$x2191-5768 606 $aComputer graphics 606 $aOptical data processing 606 $aComputer Graphics$3https://scigraph.springernature.com/ontologies/product-market-codes/I22013 606 $aImage Processing and Computer Vision$3https://scigraph.springernature.com/ontologies/product-market-codes/I22021 615 0$aComputer graphics. 615 0$aOptical data processing. 615 14$aComputer Graphics. 615 24$aImage Processing and Computer Vision. 676 $a511.42 676 $a511.4223 700 $aPenner$b Alvin$4aut$4http://id.loc.gov/vocabulary/relators/aut$01065610 906 $aBOOK 912 $a9910337576603321 996 $aFitting Splines to a Parametric Function$92546871 997 $aUNINA