A Study on a Curve and Three Equations

Abstract

The fixed-width curves, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, three differential equations obtained by means of a system of differential equations characterizing fixed-width curves are examined. These differential equations are third order, variable coefficient, homogeneous, linear differential equations based on different variables. Approximate solutions of these equations are calculated with two different polynomial approximations and error analysis is made for the solutions. Thus, the best approach method is determined for the most accurate result. Also, these equations can constitute a characterization for different types of curves according to different frames in different spaces. Therefore, this study is important not only for this curve type but also for the geometry of all curves that can be expressed with similar equations. 

Keywords

• Bernstein polynomial approximation
• Hermite polynomial approximation
• Fixed-width curves
• Frenet frame
• Frenet frame

Bir Eğri ve Üç Denklem Üzerine Bir Çalışma

Abstract

Kinematik, mühendislik, sanat, cam dizayn ve geometri gibi birçok alanda çok özel bir yere sahip olan sabit genişlikli eğriler bu başlık altında özel olarak ele alınmıştır. Bu çalışmada sabit genişlikli eğrileri karakterize eden bir diferansiyel denklem sisteminin vasıtasıyla elde edilen üç diferansiyel denklem irdelenmiştir. Bu diferansiyel denklemler farklı değişkenlere bağlı üçüncü mertebeden, değişken katsayılı, homojen, lineer diferansiyel denklemlerdir. Bu denklemlerin farklı iki polinom yaklaşımı ile yaklaşık çözümleri hesaplanıp hata analizleri yapılmıştır. Elde edilen sonuçlar sayısal bir örnek üzerinden analiz edilerek en iyi sonuç veren yaklaşım metodu tespit edilmiştir. Bu denklemler farklı uzaylarda farklı çatılara göre farklı eğri tipleri için de bir karakterizasyon teşkil edebilmektedir. Dolayısıyla bu çalışma sadece bu eğri tipi için değil benzer denklemlerle ifade edilebilen tüm eğrilerin geometrisi için önem arz etmektedir.

Keywords

• Bernstein polinom yaklaşımı
• Hermite polinom yaklaşımı
• Sabit Genişlikli Eğri
• Frenet Çatısı

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