Читайте также:
|
Cylindrical and conic surfaces. Let a system of coordinates 
be given. We say that a curve 
in the space be given parametrically by vector-function 
(or in coordinate form 
where 
are continuous functions of 
defined for 
if 1) for every 
the point lies on ; 2) for every point 
lying on there is 
such that 
holds. Sometimes a curve in the space is given by an equation 
that is obtained by excluding parameter from the coordinate form.
Example. In Cartesian system of coordinates a curve of the second order 
is a line.
Let some curve 
which we will name directing be given.
Let's draw through each point of the directing curve a straight line named generatrix that is parallel to some non-zero vector 
. The set of all points of the space lying on generatrices of a given type is called a cylindrical surface.
cylindrical case conic case
Compose an equation of cylindrical surface in common form. In the introduced notations 
, but by definition of cylindrical surface 
and consequently, an equation of cylindrical surface in vector form: 
. In coordinate form after excluding 
we obtain 
.
Let’s draw through each point of the directing curve a straight line (named generatrix) passing through some fixed non-belonging to the curve point 
(named the vertex). The set of all points of the space lying on generatrices of a given type is called a conic surface.
Compose an equation of conic surface in common form. In the introduced notations , but by definition of conic surface 
and consequently, an equation of conic surface in vector form: 
. In coordinate form after excluding we obtain 
.
An equation of the form 
determines in the space a cylindrical surface at which generatrices are parallel to 
. Analogously, an equation 
determines a cylindrical surface with generatrices that are parallel to 
, and 
– a cylindrical surface with generatrices that are parallel to 
.
Canonic equations of cylinders of the second order are the following:
Elliptic cylinder – 
, hyperbolic cylinder – 
, parabolic cylinder – 
.
Generatrices of all these cylinders defined by these equations are parallel to , and the directing curve is the corresponding curve of the second order (ellipse, hyperbola, parabola) lying in 
. At 
a cone and an elliptic cylinder are called a circular cone and a circular cylinder.
One should remember that a curve in the space can be given as a line of intersecting two surfaces. For example, equations of the directing curve of an elliptic cylinder, i.e. an equation of ellipse in plane has the form 
An equation of cone of the second order with vertex in the origin of coordinates of which the axis is is written in the form 
. Analogously, the equations 
and 
are equations of cones of the second order with vertex in the origin of coordinates of which the axes are 
and 
.
Example. Which surface do the following equations determine in the space: a) 
; b) 
?
Solution: a) The equation determines a parabolic cylinder with generatrices that are parallel to . The directing curve of the cylindrical surface is the parabola 
.
b) The equation can be represented as 
and is decomposed into two equations: 
and 
, i.e. it determines two planes: and the plane passing through 
.
Дата добавления: 2015-10-29; просмотров: 149 | Нарушение авторских прав
| <== предыдущая страница | | | следующая страница ==> |
| Ellipse, hyperbola, parabola and their properties. | | | Surfaces of rotation |