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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | (OCoLC)1081000105 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Alvin Penner |

ISBN: | 9783030125516 3030125513 9783030125523 3030125521 |

OCLC Number: | 1088555544 |

Description: | 1 online resource (xii, 79 pages) : illustrations (some color). |

Contents: | Intro; Preface; Contents; 1 Introduction; 2 Least Squares Orthogonal Distance Fitting; 2.1 Definition of Error Function; 2.2 Character of Solution; 2.3 Optimization of F(a, u); References; 3 General Properties of Splines; References; 4 ODF Using a Cubic Bézier; 4.1 Fitting a Function with a Double Inflection Point; 4.2 Initializing a Cubic Bézier; 4.3 Optimizing the Fit; 4.4 Continuity of the rms Error; 4.5 Character of Coalescing Solutions; References; 5 Topology of Merges/Crossovers; 5.1 Center of Mass Fit; 5.2 Example of Two Types of Merge/Crossover; 5.3 Response to Change in g(t) 5.4 Distinguishing Between Type 1 and Type 2 EventsReferences; 6 ODF Using a 5-Point B-Spline; 6.1 Initializing a 5-Point B-Spline; 6.2 Basis Functions of a 5-Point B-Spline; 6.3 Decomposition into Two Bézier Segments; 6.4 ODF Results for a 5-Point B-Spline; 7 ODF Using a 6-Point B-Spline; 7.1 Initializing a 6-Point B-Spline; 7.2 Basis Functions of a 6-Point B-Spline; 7.3 Decomposition into Three Bézier Segments; 7.4 ODF Results for a 6-Point B-Spline; 8 ODF Using a Quartic Bézier; 8.1 Initializing a Quartic Bézier; 8.2 ODF Results for a Quartic Bézier; 8.3 Enumeration of Solutions |

Series Title: | SpringerBriefs in computer science |

Responsibility: | Alvin Penner. |

### Abstract:

This 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.

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