1) Write the generic equation of the ellissoide and to describe of varying:

ellissoide is called why if we cut it along axis z = 0, x = 0, y = the 0 sections are ellipses.

varying is:

to) if to according to member there is the -1 then ellissoide is imaginary

b) if the famous term is 0 the ellissoide is reduced to a point

 

2) Write the generic equation of the iperboloide to one stratum and to describe of varying:

iperboloide to a stratum is called why if we cut it along axis y = 0 and x = 0 the sections are hyperbolas while if we cut along plans parallels to it to the plan xy we obtain of the ellipses.

it does not have varying:

 

3) Write the generic equation of the iperboloide to two stratums and to describe of varying:

iperboloide to two stratums is called why if we cut it along axis y = 0 and x = 0 the sections are hyperbolas while if we cut along plans parallels to it to the plan xy we only obtain of the ellipses for advanced values of |c|

it does not have varying:

 

4) Write the generic equation of the cone and to describe of varying:

This is a homogenous equation in how much appears all elements of the same degree wants to say that it can be constructed only using a bundle of straight. From a mathematical point of view it coincides with the norm.

varying is:

to) if the coefficient of z² is positive then the cone is imaginary.

 

5) Write the equations of the paraboloidi and to describe of the characteristics:

is a elliptic paraboloide, is obvious that its sections are of the ellipses, is not that a convex goblet, for values of z smaller of 0 comes described an imaginary elliptic paraboloide.

is a hyperbolic paraboloide or saddleback, is obvious that its sections long plans parallels to the plan xy are of the hyperbolas while sezioni parallels to the plan yz are of the concave parabolas while the sections parallels to the plan zx are of the convex parabolas.

 

6) Write the equations of the cylinders and to describe of the characteristics:

is a elliptic cylinder

is an imaginary elliptic cylinder

is a hyperbolic cylinder

is a parabolic cylinder

 

7) For which reductions of quadriche they are obtained of the plans:

a) 2 autovalori and correspondents null linear terms

x² = tn ž if tn > 0 2 real plans are had parallels

if tn = 0 2 real plans are had coinciding

if tn < 0 2 complex plans are had parallels

b) 1 autovalore 0 and others 2 of sign discorde, tn = 0

x² - y² = 0 ž 2 plans incidents

c) 1 autovalore the 0 and others 2 of Concorde sign, tn = 0

x² y² = 0 ž 2 complex plans incidents whose intersection is the real axis z.

 

8) Which orthogonal matrix concurs to reduce to one single variable linear shape lx² hy jz = k?