Dual spherical motions and, the dual ruled rose and ellipse surfaces

Bu sunumda gül ve elips eğrilerinin birim küredeki karşılıkları, bu karşılıklarla ilgili dual yüzeyler, detaylı yazım ve çizimlerle verildi. Gül eğrilerinin doğal hareketlerle oluşan düzlemsel eğriler oldukları iyi bilinmektedir. Eğrilerin ve yüzeylerin Euclidean uzaylardan birim küre yüzeylerine aktarılmaları genel olarak yazılmıştır. Sonuç olarak, dual rulet yüzeyler veya açılabilir dual rulet yüzeyler, düzlemsel veya yüksek boyutlu uzayların eğri veya yüzeylerinin birim küredeki karşılıkları bulunarak yazılmıştır. Kullandığımız yazım, çizim ve yöntemleri kullanarak herhangi boyutlu uzaylarda tanımlı eğrileri veya yüzeyleri birim küreye aktarabileceğimi ve bu aktarımların dual rulet yüzeylerini yazıp çizebileceğimi söyleyebilirim.

Dual küresel hareketler, dual açılabilir rulet gül ve elips yüzeyler

The spherical representations of the rose curves being generated by (hy(t;n,m,r,$alpha$) and ep(t;n,m,r,$alpha$), and the developable ruled dual rose and ellipse surfaces and their graphs are given. It is well known that the roses are generated by natural mechanism on the plane. Translation operations of curves and surfaces in generally from any Euclidean space $^n$ to the real sphere are given originally in this paper. Finally, the dual ruled or developable ruled surfaces are obtained by the unit sphere representations of the planar or any dimensional Euclidean space curves and surfaces.

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