Bilgisayar destekli geometrik tasarım ve hareket geometrisi
Küçük Resim Yok
Tarih
1998
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Ege Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
ÖZET BİLGİSAYAR DESTEKLİ GEOMETRİK TASARIM VE HAREKET GEOMETRİSİ TANTAY3ahadır Doktora Tezi, Matematik Bölümü Tez Yöneticisi: Prof. Dr. Ali Çalışkan Kasım 1998, 107 sayfa Bu tez çalışmasında diferensiyel geometri ve hareket geometrisinin temel kavramları, bilgisayar destekli geometrik tasarımda çok kullanılan eğri tiplerinden olan Bezler eğrilerine uygulandı. Bir nesnenin şeklinde olduğu gibi, matematik temsilinin de eksenlerden ve öteleme, dönme gibi afin dönüşümlerden bağımsız olması gereği açıktır. Bu çalışmada, projeksiyonlar, ölçekleme gibi dönüşümler ve n-boyutlu uzayda genel D dönüşümü için invaryantlık şartlan çıkarıldı. Bunun yanında bir eğrinin parametrik gösteriminin bilgisayar destekli üretime getireceği avantajlara bir yenisi daha eklenerek verimliliğin arttırılması hedeflendi. Ayrıca, lineer ve homografik parametre dönüşümleri eğriye uygulandığında elde edilecek yeni eğrinin 3-boyutta ve n-boyutta genel ifadeleri elde edilerek dönüşümden önceki ve sonraki teğet vektörler arasındaki bazı ilişkiler 3-boyutlu ve n-boyutlu uzay için elde edildi. Bu çalışmada ayrıca sonlu fark ifâdeleri eğri çiziminde kullanılarak herbiri diğerinin geliştirilmişi olan dört eğri çizim algoritmasısunuldu. Bilgisayar grafiklerinde çok kullanılan eğrilerden olan Bezier eğrilerinin iki farklı tipini elips çiziminde kullanarak ikisi arası kuvvet kıyaslaması yapıldı. Bir eğrinin ofset eğrisi üzerinde durarak, ofset eğri ve eğrinin kendisinin eğrilikleri, burulmaları, teğet vektörleri, bu vektörler arasındaki açı ve ofset uzaklıktan cinsinden çeşitli bağıntılar elde edildi. Bir eğriye 2. mertebeden değen bir polinom eğrisi olup olmadığı araştırıldı ve bunun bir parabol eğrisi olduğunu görülerek buna bir eğrinin oskülatör parabolü adı verildi. Burada, 2. ve 3. dereceden Bezier eğrilerinin hareketleri incelenerek, mutlak hız, relatif hız, sürüklenme hızı, pol noktaları ve pol eğrisi ifadeleri, kontrol noktalan ve hodografları cinsinden elde edildi. Ayrıca, x(t) parametrik kübik hareketinin geometrik karakterizasyonunda relatif hız kullanıldı. Bunun için relatif hız eğrisinin düzlemi ayırdığı çeşitli bölgeler incelendi ve hareketi karakterize etme problemi bir nokta-konum problemine dönüştürülerek hareket analiz edildi ve bu, bir algoritma ile verildi. Anahtar sözcükler: Hareket geometrisi, diferensiyel geometri, bilgisayar destekli geometrik tasarım, hodograf, Bezier eğrileri
vn ABSTRACT COMPUTER AIDED GEOMETRIC DESIGN AND KINEMATICS TANTAY3ahadir Ph.D. in Mathematics Supervisor: Prof. Dr. Ali Çahşkan November 1998, 107 pages In this thesis, fundamental concepts of differential geometry and kinematics have been applied to Bezier curve segments which are frequently used in computer aided geometric design. It is clear that, like the shape of an object, its mathematical representation should be independent of axes and affine transformations such as translation and rotation. In this study, conditions of invariance for transformations such as projections and scaling and general transformation D in n-dimensional space has been obtained. Moreover, increasing of efficiency has been aimed by adding a new advantage of parametric representation to computer aided manufacturing. Besides, after obtaining expressions of a curve in 3 and n dimension, which is obtained by applying linear and nomographic parameter transformations to the curve, some relations between tangent vectors before and after transformation has been obtained for 3 and n-dimensional space.vm In this study, by using finite differences in curve drawing, 4 different algorithms each of which is the improved one of the previous have been introduced. By using two different types of Bezier curves, frequently used curves in computer graphics, in ellipse drawing, force comparison between them has been done. Explaining offset curve of any curve, some relations among curvatures, torsions, tangent vectors, angle between these vectors and offset distance of both curves and its offset curve have been obtained. The existence of a polynomial curve which has second order contact with a curve has been examined, determining that it is a parabola it has been called osculating parabola of the curve. By examining second and third degree Bezier curves, expressions for absolute, relative and dragging velocities and poll points have been obtained in terms of control points and hodographs. Besides, relative velocity in the characterization of x(t) parametric cubic motion has been used. For this, various regions that relative velocity curve separates the plane have been examined and by transforming a motion characterization problem to a point location problem the motion has been analyzed and this has been given with an algorithm. Key words: Kinematics, differential geometry, computer aided geometric design, hodograph, Bezier curves.
vn ABSTRACT COMPUTER AIDED GEOMETRIC DESIGN AND KINEMATICS TANTAY3ahadir Ph.D. in Mathematics Supervisor: Prof. Dr. Ali Çahşkan November 1998, 107 pages In this thesis, fundamental concepts of differential geometry and kinematics have been applied to Bezier curve segments which are frequently used in computer aided geometric design. It is clear that, like the shape of an object, its mathematical representation should be independent of axes and affine transformations such as translation and rotation. In this study, conditions of invariance for transformations such as projections and scaling and general transformation D in n-dimensional space has been obtained. Moreover, increasing of efficiency has been aimed by adding a new advantage of parametric representation to computer aided manufacturing. Besides, after obtaining expressions of a curve in 3 and n dimension, which is obtained by applying linear and nomographic parameter transformations to the curve, some relations between tangent vectors before and after transformation has been obtained for 3 and n-dimensional space.vm In this study, by using finite differences in curve drawing, 4 different algorithms each of which is the improved one of the previous have been introduced. By using two different types of Bezier curves, frequently used curves in computer graphics, in ellipse drawing, force comparison between them has been done. Explaining offset curve of any curve, some relations among curvatures, torsions, tangent vectors, angle between these vectors and offset distance of both curves and its offset curve have been obtained. The existence of a polynomial curve which has second order contact with a curve has been examined, determining that it is a parabola it has been called osculating parabola of the curve. By examining second and third degree Bezier curves, expressions for absolute, relative and dragging velocities and poll points have been obtained in terms of control points and hodographs. Besides, relative velocity in the characterization of x(t) parametric cubic motion has been used. For this, various regions that relative velocity curve separates the plane have been examined and by transforming a motion characterization problem to a point location problem the motion has been analyzed and this has been given with an algorithm. Key words: Kinematics, differential geometry, computer aided geometric design, hodograph, Bezier curves.
Açıklama
Anahtar Kelimeler
Matematik, Mathematics, Bilgisayar destekli tasarım, Computer aided design, Diferensiyel geometri, Differential geometry, Geometri, Geometry
