توصيفگر ها :
ژئودزي , هندسه ديفرانسيل , درون يابي , خم , رويه
چكيده فارسي :
ژئودزي‌ها يكي از مفاهيم مهم در هندسه ديفرانسيل هستند كه به واسطه كاربردهاي فراوان آن‌ها در بسياري علوم نظير فيزيك، مكانيك، زمين‌شناسي و غيره مورد توجه محققان در ديگر رشته‌ها نيز قرار دارد. اما اگر با خمي مواجه باشيم كه در پي آن باشيم كه اين خم به عنوان ژئودزي روي يك رويه محسوب شود، چه بايد كرد؟ مبحث درون‌يابي رويه‌اي به دنبال پاسخ براي چنين سوالي است. درون‌يابي با توجه به ابزار پايه در هندسه ديفرانسيل و با استفاده از روش‌هايي مثل معادلات ديفرانسيل، جواب‌هايي براي اين سوال ارائه مي‌دهد. رسم شكل‌ها به كمك نرم‌افزارهايي چون ميپل، درك چگونگي وقوع خم مفروض روي رويه‌ي درون‌يابي پيشنهادي را ميسر مي‌سازد.
در اين پايان‌نامه با معرفي ابزار لازم از هندسه ديفرانسيل، به تشريح درون‌يابي رويه‌ها در حالت‌هاي ساده و خاص مي‌پردازيم. به علاوه روش خاص درون‌يابي رويه‌اي با نام كونز براي حالت معيني مطرح مي‌شود كه چهار رويه‌ي مرزي داده شده‌اند. در اين حالت لازم است اين چهار خم براي رويه‌ي مفروض ژئودزي هم باشند.
چكيده انگليسي :
Geodesics are one of the most important concepts in differential geometry, which due to their many applications in many sciences such as physics, mechanics, geology, etc., are of interest to researchers in other fields. The topic of interpolation is a procedure that seeks to answer such a question. Interpolation provides answers to this question by using the basic tools in differential geometry and applying methods such as differential equations. Draw shapes with the help of software such as Maple makes it possible to understand how the assumed bend occurs on the proposed interpolation procedure. A geodesic between two points on a surface is defined as a curve on the surface that connects these two points with the shortest length. A geodesic can also be defined as a curve with zero geodesic curvature. The geodesic curvature of a curve on a surface at a point is equal to the curvature of the vertical image of curve on the tangent plane of surface at point. These studies include making umbrellas (tabs), cutting a painted path, producing fabric, and fiberglass twisted for tape, which plays an important role in pipe production. In 2004, Wang and colleagues first addressed the issue of finding a family of procedures involving a given particular curve. Procedures are then constructed with geodesics and presumptive data $C^0$-Hermit. To do this perform, a procedural interpolation associated with a particular transient bend at $m$ point in three-dimensional Euclidean space E^3 will be defined. In addition, procedures are presented in terms of polynomial scaling scale functions with a number of examples. In detail, at first we study the necessary and sufficient condition for the scaling scale functions and their derivatives for a parametric procedure such that their basic bend is. It will be observed that these conditions are very simple for separable scalable functions in terms of variables. Also the existence and uniformity of the procedure $C^0$-Hermit interpolator for a $C^0$-Hermit data is proved when the scaling functions of the polynomial are assumed. The following are examples of these procedures with a number of polynomials. Introduction of delineated procedures is another goal of this dissertation. In this path, a for a lined procedure whose conductor bend is a geodesic. In addition, for each point on a hypothetical routing procedure with a geodesic guide we conclude a sufficient condition for the interpolation of the $C^0$-Hermit procedure. Finally, after introducing a classification procedures, an example for a transient procedure is given at two points of E^3. This dissertation also examines the existence of conditions for the interpolation of procedures for the four supposed boundary bends in such a way that these four bends are for the geodesic procedure. The contents of this dissertation are organized in four chapters. The first chapter is dedicated to the definitions and introductory concepts. In the second chapter, we discuss, the interpolation of procedures by use some examples. The interpolation of procedures in the form of delineated procedures constitutes the content of the third chapter, and finally the fourth chapter is devoted to a special mode of interpolation of procedures. There is also an appendix at the end of the chapters for the Maple command procedures in the dissertation.
