“algebraic surface”的意思、由来-百科全书

algebraic surface

in three-dimensional space, a surface the equation of which is f(x,y,z)=0, with f(x,y,z) a polynomial in x, y, z. The order of the surface is the degree of the polynomial equation. If the surface is of the first order, it is a plane. If the surface is of order two, it is called a quadric surface. By rotating the surface, its equation can be put in the form

Ax² + By² + Cz² + Dx+ Ey + Fz= G.

If A, B, C are all not zero, the equation can generally be simplified to the form

ax²+by²+cz²=1.

This surface is called an ellipsoid if a, b, and c are positive. If one of the coefficients is negative, the surface is a hyperboloid of one sheet; if two of the coefficients are negative, the surface is a hyperboloid of two sheets. A hyperboloid of one sheet has a saddle point (a point on a curved surface shaped like a saddle at which the curvatures in two mutually perpendicular planes are of opposite signs, just like a saddle is curved up in one direction and down in another).

If A, B, C are possibly zero, then cylinders, cones, planes, and elliptic or hyperbolic paraboloids may be produced. Examples of the latter are z=x²+y² and z=x²−y², respectively. Through every point of a quadric pass two straight lines lying on the surface. A cubic surface is one of order three. It has the property that 27 lines lie on it, each one meeting 10 others. In general, a surface of order four or more contains no straight lines.

词条	algebraic surface
释义	algebraic surface in three-dimensional space, a surface the equation of which is f(x,y,z)=0, with f(x,y,z) a polynomial in x, y, z. The order of the surface is the degree of the polynomial equation. If the surface is of the first order, it is a plane. If the surface is of order two, it is called a quadric surface. By rotating the surface, its equation can be put in the form Ax² + By² + Cz² + Dx+ Ey + Fz= G. If A, B, C are all not zero, the equation can generally be simplified to the form ax²+by²+cz²=1. This surface is called an ellipsoid if a, b, and c are positive. If one of the coefficients is negative, the surface is a hyperboloid of one sheet; if two of the coefficients are negative, the surface is a hyperboloid of two sheets. A hyperboloid of one sheet has a saddle point (a point on a curved surface shaped like a saddle at which the curvatures in two mutually perpendicular planes are of opposite signs, just like a saddle is curved up in one direction and down in another). If A, B, C are possibly zero, then cylinders, cones, planes, and elliptic or hyperbolic paraboloids may be produced. Examples of the latter are z=x²+y² and z=x²−y², respectively. Through every point of a quadric pass two straight lines lying on the surface. A cubic surface is one of order three. It has the property that 27 lines lie on it, each one meeting 10 others. In general, a surface of order four or more contains no straight lines.
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