- published: 01 Mar 2014
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In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old.
In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus and the geometric genus turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. Let's summarize the results. (in detail, for each kind of surfaces refer to each redirections)
Subscribe Now: http://www.youtube.com/subscription_center?add_user=ehoweducation Watch More: http://www.youtube.com/ehoweducation An algebraic surface is a mathematical object of complex dimension two, since it's a surface, and because it's complex this means that it can contain real and imaginary numbers. Find out about the classification of algebraic surfaces with help from an educator with years of experience in this free video clip. Expert: Walter Unglaub Filmmaker: bjorn wilde Series Description: Understanding physics will require you to understand a number of very important concepts at the subject's core. Find out about physics and calculus with help from an educator with years of experience in this free video series.
This is a demonstration of the program asxp for visualization of algebraic surfaces in real affine 3-space. The surfaces can depend on (two) parameters, which are set by moving the mouse over the display window. Also it is possible to turn the surface by giving the eulerian angles relative to the center of display. The projection is orthogonal, fixed viewport, looking from z=100, x=y=0 to the origin of coordinates. The surface is cut with a ball of radius 20 and then displayed. The equation in the video is f(x,y,z,a,b) = -10 + a * x^2 + b * y^2 + a * x^2 * z + b * y^3 * z + a * x * z^2 a,b are the parameters set by moving the mouse over the display window. There is nothing special about this equation, I concocted it randomly when I wrote the program, but in the meantime I have come...
From equation to 3d-printable object in 37 minutes. This is surface `himmel_und_holle` from the algebraic surface gallery, by herwig hauser: http://homepage.univie.ac.at/herwig.hauser/gallery.html software used: * bertini * bertini_real * blender * simplify3d * 3mf converter * matlab * the terminal * sublime text 3 * microsoft's online fixer: https://tools3d.azurewebsites.net/ screen recorded using quicktime player. please recommend a better software for this task if you can.
In Blender, I split a piece off an algebraic surface which was computed using Bertini_real, and processed through Matlab into an .stl file. The surface, Trichter, is from Herwig Hauser's algebraic surface gallery: http://homepage.univie.ac.at/herwig.hauser/gallery.html. This video is several steps into a process I use for 3D printing algebraic surfaces. I had already split one of the pieces off the triangulation, this was the second split of two. At the end, things got a little screwy because I got interrupted by a student asking a question ;)
We extend our approach to curvature to general algebraic surfaces. The formulas get involved, but they have pleasant symmetry and are quite powerful.
With the program asxp_npr (npr stands for non-photorealistic rendering) it is now possible to plot integral curves to a direction crossfield made of extremal sectional curvature directions (that is the eigenvectors of the Weingarten-endomorphism on the tangent space).
Designing with algebraic surfaces as the basis of cyberspace architecture For more visionary architectural illustrations and art novelties, visit www.oanaunciuleanu.com and subscribe to Oana Unciuleanu Art & Architecture on FB.
Coloquio del Instituto de Matemáticas Martes 8 de marzo de 2016 "Algebraic surfaces: main problems and recent developments" Xavier Roulleau
We extend our approach to curvature to general algebraic surfaces. The formulas get involved, but they have pleasant symmetry and are quite powerful.
We introduce surfaces, which are the main objects of interest in differential geometry. After a brief introduction, we mention the key notion of orientability, and then discuss the division in the subject between algebraic surfaces and parametrized surfaces. It is very important to have a balanced view between these two aspects; most texts are oriented, following Gauss, to the parametrical side: we will at least initially compensate by providing more detail on the algebraic surfaces. Important examples of surfaces include quadrics, such as spheres, or more generally ellipsoids, or hyperboloids. We also mention some more unusual examples, including the Oloid, discovered by Paul Schatz in 1929.