Finding an Envelope is an Optimization Problem
The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding op...
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| Published in | Journal of optimization theory and applications Vol. 186; no. 2; pp. 453 - 458 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.08.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-3239 1573-2878 |
| DOI | 10.1007/s10957-020-01707-9 |
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| Abstract | The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding optimization problem is established. Extensions of the envelope problem to multiple parameters and families of surfaces are considered, as well as envelope problems that cannot be solved by the standard approach, for example, problems involving multiple parameters subject to constraints on the parameters. |
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| AbstractList | The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding optimization problem is established. Extensions of the envelope problem to multiple parameters and families of surfaces are considered, as well as envelope problems that cannot be solved by the standard approach, for example, problems involving multiple parameters subject to constraints on the parameters. |
| Author | Hull, David G. |
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| CitedBy_id | crossref_primary_10_3390_math13030382 |
| Cites_doi | 10.1109/TSMC.1987.6499313 10.1007/978-3-662-05382-9 10.1109/TAC.1976.1101338 |
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| Copyright | Springer Science+Business Media, LLC, part of Springer Nature 2020 Springer Science+Business Media, LLC, part of Springer Nature 2020. |
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| DOI | 10.1007/s10957-020-01707-9 |
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| Keywords | Parameter optimization Envelopes Family of curves |
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| References | LiDHaimesYYThe envelope approach for multiobjective optimization problemsIEEE Trans. Syst. Man Cybern.1987171026103892365410.1109/TSMC.1987.6499313 BronshteinINHandbook of Mathematics2004Upper Saddle RiverSpringer23723810.1007/978-3-662-05382-9 TaylorAEAdvanced Calculus1955New YorkGinn and Company3934040067.02503 Google Search.: “envelope of a family of curves”. Found are many websites from sources such as Wikipedia, Math24, Math Curve, Mathematics Stack Exchange, and more, containing examples and figures of envelopes (2020) CourantRDifferential and Integral Calculus1936New YorkInterscience Publishers169183 GreenbergMPFoundations of Applied Mathematics1978Upper Saddle RiverPrentice-Hall3994000381.00001 LinJGMultiple-objective problems: pareto-optimal solutions by method of proper equality constraintsIEEE Trans. Automat. Contr.19762164165040768810.1109/TAC.1976.1101338 D Li (1707_CR7) 1987; 17 R Courant (1707_CR1) 1936 AE Taylor (1707_CR2) 1955 1707_CR5 IN Bronshtein (1707_CR4) 2004 MP Greenberg (1707_CR3) 1978 JG Lin (1707_CR6) 1976; 21 |
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| SubjectTerms | Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Curves Engineering Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Parameters Theory of Computation |
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| Title | Finding an Envelope is an Optimization Problem |
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