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"This state-of-the-art study of the techniques used for designing curves and surfaces for computer-aided design applications focuses on the principle that fair shapes are always free of unessential features and are simple in design. The authors define fairness mathematically, demonstrate how newly developed curve and surface schemes guarantee fairness, and assist the user in identifying and removing shape aberrations in a surface model without destroying the principal shape characteristics of the model. Aesthetic aspects of geometric modeling are of vital importance in industrial design and modeling, particularly in the automobile and aerospace industries. Any engineer working in computer-aided design, computer-aided manufacturing, or computer-aided engineering will want to add this volume to his or her library. Researchers who have a familiarity with basic techniques in computer-aided graphic design and some knowledge of differential geometry will find this book a helpful reference. It is essential reading for statisticians working on approximation or smoothing of data with mathematical curves or surfaces."

"This book is about modern algebraic geometry. The book starts by explaining this enigmatic answer, algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained."

"Diberikan suatu lapangan yang tertutup secara aljabar k dan bilangan bulat positif n, ruang affine berdimensi-n atas k didefinisikan sebagai himpunan An=((a1,?,an):a1,?,an e k) . Suatu kurva aljabar tak-tereduksi pada bidang affine A2 didefinisikan oleh f(x,y)=0 dimana f adalah polinomial tak-tereduksi yang tidak konstan. Ada kurva-kurva aljabar tak-tereduksi yang dapat diparameterisasi menjadi fungsi-fungsi rasional, dan parameterisasi ini merupakan bentuk yang lebih sederhana dari pemetaan rasional antara dua kurva aljabar tak-tereduksi. Apabila pemetaan rasional tersebut mempunyai pemetaan rasional invers, pemetaan ini menjadi suatu relasi khusus yang disebut birational equivalence dan kedua kurva tersebut dikatakan birational. Pemetaan dan relasi ini juga dapat didefinisikan pada subhimpunan tutup tak-tereduksi dari An yang merupakan bentuk umum dari kurva aljabar tak-tereduksi. Dalam skripsi ini akan dipelajari syarat cukup dan syarat perlu untuk dua kurva aljabar tak-tereduksi, atau secara umum dua himpunan tutup tak-tereduksi, birational dengan membuktikan bahwa untuk himpunan-himpunan tutup tak-tereduksi XcAn dan YcAm, X dan Y birational jika dan hanya jika lapangan fungsi rasional dari keduanya, yaitu k(X) dan k(Y), isomorfik atas k.

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Given an algebraically closed field k and a positive integer n, we define the n-dimensional affine space over k to be the set An= ((a1,?,an):a1,?,ane k) . An irreducible algebraic curve on the affine plane A2 is defined by f(x,y) = 0 where f is a nonconstant irreducible polynomial. Some of these curves can be parameterized as rational functions, and this parameterization is a simpler form of a rational map between two irreducible algebraic curves. If this map has an inverse rational map, it becomes a special relation called birational equivalence and we say that the two curves are birational. This map and relation can also be defined on irreducible closed subsets of An, the generalized form of irreducible algebraic curves. This skripsi studies the sufficient and necessary condition for the two irreducible algebraic curves, or in general two irreducible closed sets, to be birational by exhibiting the proof that for irreducible closed sets XcAn and YcAm, X and Y are birational if and only if their field of rational functions, k(x) and k(y), are isomorphic over k. "