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"In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and a topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, “Plane Algebraic Curves” reflects the authorsʼ concern for the student audience through its emphasis on motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles. This text also provides a foundation for the comprehension and exploration of modern work on singularities."

"ABSTRAKTujuan utama dari keamanan komunikasi adalah mencegah data pesan dari akses pihak ketiga. Solusi dari masalah tersebut adalah menerapkan enkripsi End-to-end. Algoritma enkripsi Elliptic Curve Cryptography (ECC) cocok digunakan sebagai metode enkripsi End-to-end pada sistem komunikasi digital termasuk aplikasi chat. ECC adalah jenis kriptografi kunci publik yang mendasarkan keamanannya pada permasalahan matematis dari kurva eliptik, ECC mempunyai tingkat keamanan yang setara dengan RSA menggunakan kunci berukuran lebih kecil. Penelitian ini membahas perancangan dan implementasi dari algoritma enkripsi dan dekripsi yang menerapkan kriptografi ECC sebagai metode enkripsi End-to-end pada aplikasi chat. Algoritma ini menggunakan operasi matematika dan nilai parameter dari kurva eliptik untuk menghasilkan kunci serta ciphertextnya dan juga melakukan dekripsi. Dari hasil ujicoba dan analisa implementasi enkripsi dan dekripsi menggunakan algoritma ECC, algoritma tersebut cocok digunakan sebagai metode enkripsi End-to-end, karena ukuran kunci yang relatif ringan dan operasi matematika yang menggunakan parameter khusus yang berbeda-beda. Hasil pengujian enkripsi dan dekripsi menggunakan parameter kurva eliptik secp192r1 dan secp192k1 yang mempunyai ukuran kunci sekitar 2¹⁹² bit, menghasilkan kunci dan ciphertext paling ringan dan paling cepat diproses, dengan angka nilai keduanya berukuran sepanjang 58 digit.

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ABSTRACTThe main purpose of communication security is to prevent message data from third party access. The solution to this problem is to implement End-to-end encryption. Elliptic Curve Cryptography (ECC) encryption algorithm is suitable for use as an End-to-end encryption method on digital communication systems including chat applications. ECC is a type of public key cryptography that bases its security on mathematical problems from elliptic curves, ECC has a level of security equals to RSA while using smaller keys. This study discusses the design and implementation of encryption and decryption algorithms that apply ECC cryptography as an End-to-end encryption method in chat applications. This algorithm uses mathematical operations and parameter values from the elliptic curve to generate keys and their ciphertext and also decrypt them. From the results of testing and analyzing the implementation of encryption and decryption using the ECC algorithm, the algorithm is suitable for use as an End-to-end encryption method, because of the relatively light key size and mathematical operations that use different parameters. The results of encryption and decryption testing using elliptic curve parameters secp192r1 and secp192k1 which have a key size of about 2¹⁹² bits, produce the lightest and fastest to process ciphertext and keys, with the value of both measuring 58 digits.

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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."

"The book provides an introduction to differential geometry of curves and surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3."