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If a11 coefficients, a22, a33 of one sign, the left part (addresses in zero (a44 = only for x=y=z=0, i.e. to the equation of a surface of S satisfy coordinates only of an edny point. In this case the surface of S is called as an imaginary cone of the second order. If a11 coefficients, a22, a33 have different signs, the surface of S is a material cone of the second order.

The card of a hyperbolic paraboloid gives an idea of its spatial form. As well as in case of an paraboloid, it is possible to be convinced that the paraboloid can be received by parallel movement of a parabola, itself section ­ by Oxz (Oyz) when its moves along the parabola which is the section of a the Oyz (Oxz) plane.

Thus, in noted case we have the elliptic cylinder. In a case, a11 and a22 have various signs, we will receive the hyperbolic cylinder. It is easy to be convinced that the of the hyperbolic cylinder can be brought to a look

If a11 coefficients, a22, a33, a44 one sign, the left part (at any values x, at, z does not address in zero, i.e. to the equation of a surface of S do not satisfy a of any point. In this case the surface of S is called as an imaginary ellipsoid.

Coordinates (x, at, z) any point of M of a straight line of L tx0, ty0, tz0 where t — some number are equal. Substituting these values for x, at and z in the left part (taking out then t2 for a and considering (2, we will be convinced that the M lies on a. Thus, the statement is proved. Idea of a form of a cone can be received by method of sections. It is easy to be convinced that cone sections the planes z = h represent ellipses with half shafts:

The equation (1 defines so-called paraboloids. And if a11 and a22 have an identical sign, the paraboloid is called elliptic. Usually the equation of an elliptic paraboloid is written down in an initial form:

Let S — a noncentral surface of the second order, i.e. a surface for which the invariant of I3 is equal to zero. Let's make standard simplification of an of this surface. As a result the equation of a surface will assume an air

Classification of the central surfaces. Let S — the central surface of the second order. Let's move the beginning of coordinates to the center of this surface, and then we will make simplification of the equation of this surface. In a of the specified operations the equation of a surface will assume an air