Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications

Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describ...
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Autor*in:	Sidra Maqsood [verfasserIn] Muhammad Abbas [verfasserIn] Kenjiro T. Miura [verfasserIn] Abdul Majeed [verfasserIn] Gang Hu [verfasserIn] Tahir Nazir [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces

Übergeordnetes Werk:	In: Advances in Difference Equations - SpringerOpen, 2006, (2021), 1, Seite 32
Übergeordnetes Werk:	year:2021 ; number:1 ; pages:32

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.1186/s13662-021-03614-3

Katalog-ID:	DOAJ015719162

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520			\|a Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
650		4	\|a GBTB basis functions
650		4	\|a Shape control of developable GBT-Bézier curve
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650		4	\|a Duality
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allfields	10.1186/s13662-021-03614-3 doi (DE-627)DOAJ015719162 (DE-599)DOAJ3f1a8938d0754e438617dd7333ae1ec4 DE-627 ger DE-627 rakwb eng QA1-939 Sidra Maqsood verfasserin aut Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering. GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces Mathematics Muhammad Abbas verfasserin aut Kenjiro T. Miura verfasserin aut Abdul Majeed verfasserin aut Gang Hu verfasserin aut Tahir Nazir verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2021), 1, Seite 32 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2021 number:1 pages:32 https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 kostenfrei https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2021 1 32
spelling	10.1186/s13662-021-03614-3 doi (DE-627)DOAJ015719162 (DE-599)DOAJ3f1a8938d0754e438617dd7333ae1ec4 DE-627 ger DE-627 rakwb eng QA1-939 Sidra Maqsood verfasserin aut Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering. GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces Mathematics Muhammad Abbas verfasserin aut Kenjiro T. Miura verfasserin aut Abdul Majeed verfasserin aut Gang Hu verfasserin aut Tahir Nazir verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2021), 1, Seite 32 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2021 number:1 pages:32 https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 kostenfrei https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2021 1 32
allfields_unstemmed	10.1186/s13662-021-03614-3 doi (DE-627)DOAJ015719162 (DE-599)DOAJ3f1a8938d0754e438617dd7333ae1ec4 DE-627 ger DE-627 rakwb eng QA1-939 Sidra Maqsood verfasserin aut Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering. GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces Mathematics Muhammad Abbas verfasserin aut Kenjiro T. Miura verfasserin aut Abdul Majeed verfasserin aut Gang Hu verfasserin aut Tahir Nazir verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2021), 1, Seite 32 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2021 number:1 pages:32 https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 kostenfrei https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2021 1 32
allfieldsGer	10.1186/s13662-021-03614-3 doi (DE-627)DOAJ015719162 (DE-599)DOAJ3f1a8938d0754e438617dd7333ae1ec4 DE-627 ger DE-627 rakwb eng QA1-939 Sidra Maqsood verfasserin aut Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering. GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces Mathematics Muhammad Abbas verfasserin aut Kenjiro T. Miura verfasserin aut Abdul Majeed verfasserin aut Gang Hu verfasserin aut Tahir Nazir verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2021), 1, Seite 32 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2021 number:1 pages:32 https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 kostenfrei https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2021 1 32
allfieldsSound	10.1186/s13662-021-03614-3 doi (DE-627)DOAJ015719162 (DE-599)DOAJ3f1a8938d0754e438617dd7333ae1ec4 DE-627 ger DE-627 rakwb eng QA1-939 Sidra Maqsood verfasserin aut Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering. GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces Mathematics Muhammad Abbas verfasserin aut Kenjiro T. Miura verfasserin aut Abdul Majeed verfasserin aut Gang Hu verfasserin aut Tahir Nazir verfasserin aut In Advances in Difference Equations SpringerOpen, 2006 (2021), 1, Seite 32 (DE-627)377755699 (DE-600)2132815-8 16871847 nnns year:2021 number:1 pages:32 https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 kostenfrei https://doi.org/10.1186/s13662-021-03614-3 kostenfrei https://doaj.org/toc/1687-1847 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ SSG-OLC-PHA GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2027 GBV_ILN_2055 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2021 1 32
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topic_title	QA1-939 Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications GBTB basis functions Shape control of developable GBT-Bézier curve Developable GBT-Bézier surfaces Duality Enveloping developable GBT-Bézier surfaces Spine curve developable GBT-Bézier surfaces
topic	misc QA1-939 misc GBTB basis functions misc Shape control of developable GBT-Bézier curve misc Developable GBT-Bézier surfaces misc Duality misc Enveloping developable GBT-Bézier surfaces misc Spine curve developable GBT-Bézier surfaces misc Mathematics
topic_unstemmed	misc QA1-939 misc GBTB basis functions misc Shape control of developable GBT-Bézier curve misc Developable GBT-Bézier surfaces misc Duality misc Enveloping developable GBT-Bézier surfaces misc Spine curve developable GBT-Bézier surfaces misc Mathematics
topic_browse	misc QA1-939 misc GBTB basis functions misc Shape control of developable GBT-Bézier curve misc Developable GBT-Bézier surfaces misc Duality misc Enveloping developable GBT-Bézier surfaces misc Spine curve developable GBT-Bézier surfaces misc Mathematics
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title	Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications
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title_full	Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications
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author_browse	Sidra Maqsood Muhammad Abbas Kenjiro T. Miura Abdul Majeed Gang Hu Tahir Nazir
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title_sort	shape-adjustable developable generalized blended trigonometric bézier surfaces and their applications
callnumber	QA1-939
title_auth	Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications
abstract	Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
abstractGer	Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
abstract_unstemmed	Abstract Developable surfaces have a vital part in geometric modeling, architectural design, and material manufacturing. Developable Bézier surfaces are the important tools in the construction of developable surfaces, but due to polynomial depiction and having no shape parameter, they cannot describe conics exactly and can only handle a few shapes. To tackle these issues, two straightforward techniques are proposed to the computer-aided design of developable generalized blended trigonometric Bézier surfaces (for short, developable GBT-Bézier surfaces) with shape parameters. A developable GBT-Bézier surface is established by making a collection of control planes with generalized blended trigonometric Bernstein-like (for short, GBTB) basis functions on duality principle among points and planes in 4D projective space. By changing the values of shape parameters, a group of developable GBT-Bézier surfaces that preserves the features of the developable GBT-Bézier surfaces can be generated. Furthermore, for a continuous connection among these developable GBT-Bézier surfaces, the necessary and sufficient G 1 $G^{1}$ and G 2 $G^{2}$ (Farin–Boehm and beta) continuity conditions are also defined. Some geometric designs of developable GBT-Bézier surfaces are illustrated to show that the suggested scheme can settle the shape and position adjustment problem of developable Bézier surfaces in a better way than other existing schemes. Hence, the suggested scheme has not only all geometric features of current curve design schemes but surpasses their imperfections which are usually faced in engineering.
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title_short	Shape-adjustable developable generalized blended trigonometric Bézier surfaces and their applications
url	https://doi.org/10.1186/s13662-021-03614-3 https://doaj.org/article/3f1a8938d0754e438617dd7333ae1ec4 https://doaj.org/toc/1687-1847
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author2	Muhammad Abbas Kenjiro T. Miura Abdul Majeed Gang Hu Tahir Nazir
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