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Bullet-nose Curve
In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation :a^2y^2-b^2x^2=x^2y^2 \, The bullet curve has three double points in the real projective plane, at and , and , and and , and is therefore a unicursal (rational) curve of genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ... zero. If :f(z) = \sum_^ z^ = z+2z^3+6z^5+20z^7+\cdots then :y = f\left(\frac\right)\pm 2b\ are the two branches of the bullet curve at the origin. References * Plane curves Algebraic curves {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bullet Nose Curve
A bullet is a kinetic projectile, a component of firearm ammunition that is shot from a gun barrel. Bullets are made of a variety of materials, such as copper, lead, steel, polymer, rubber and even wax. Bullets are made in various shapes and constructions (depending on the intended applications), including specialized functions such as hunting, target shooting, training and combat. Bullets are often tapered, making them more aerodynamic. Bullet sizes are expressed by their weights and diameters (referred to as "calibers") in both imperial and metric measurement systems. For example: 55 grain .223 caliber bullets are of the same weight and caliber as 3.56 gram 5.56mm caliber bullets. Bullets do not normally contain explosives but strike or damage the intended target by transferring kinetic energy upon impact and penetration. Bullets shot by firearms can be used for target practice or hunting. Description The term ''bullet'' is from Middle French, originating as the diminutive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflection Point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (''f'', its first derivative ''f, and its second derivative ''f'''', exist and are continuous), the condition ''f'' = 0'' can also be used to find an inflection point since a point of ''f'' = 0'' must be passed to change ''f'''' from a positive value (concave upward) to a negative value (concave downward) or vice versa as ''f'''' is continuous; an inflection point of the curve is where ''f'' = 0'' and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Point
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curves in the plane may be defined as the set of points satisfying an equation of the form f(x,y) = 0, where is a polynomial function If is expanded as f = a_0 + b_0 x + b_1 y + c_0 x^2 + 2c_1 xy + c_2 y^2 + \cdots If the origin is on the curve then . If then the implicit function theorem guarantees there is a smooth function so that the curve has the form near the origin. Similarly, if then there is a smooth function so that the curve has the form near the origin. In either case, there is a smooth map from to the plane which defines the curve in the neighborhood of the origin. Note that at the origin b_0 = \frac, \; b_1 = \frac, so the curve is non-singular or ''regular'' at the origin if at least one of the partial derivatives o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of : then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant of a sequence of invariants called the plurigenera. Case of curves In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curves
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
