Presented and Prepared Byrashmi.M
Cylinders and quadric surfaces1
Vectors and geometry 18
Presented and Prepared byRashmi.M
content edited by Nandakumar.M
cylinders
Objectives
1.Introduces the concept of cylinders and quadric surfaces
2.Familiarise different type of cylinders through examples
Introduction
In this session we introduce the idea of cylinders ,not only the idea but discuss in detail different cylinders through examples and figures. After that we move on to the quadric
surfaces there we examine certain quadric surfaces like ellipsoid ,hyperboloid ,paraboloid etc
definition :A cylinder is a surface generated by a line which is always parallel to a fixed line and passes through a given curve.
That is acylinderis the surface composed of all lines that
(1)lie parallel to a given line in space and
(2)pass through a given plane curve.
The fixed line is called the axisof the cylinder and the given plane curve is called a guiding curve or generating curve
Note 1
- If the guiding curve is a circle, the cylinder is called a right circular
- cylinder.
- Since the generator is a straight line, it extends on either side infinitely. As such, a cylinder is an infinite surface.
- The degree of the equation of a cylinder depends on the degree of the equation of the guiding curve.
- A cylinder whose equation is of second degree, is called a quadric cylinder.
Definition(Enveloping cylinder):The locus of the lines drawn in a given direction or parallel to a given surface is called the Enveloping cylinder of the surface
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Equation of the enveloping cylinder of a sphere
To find the equation of the enveloping cylinder of the sphere
Whose generators are parallel to the line
Let be any point on the surface of the cylinder so that the equations of the
Generator through this point are
Any point on this line is ,if it lies on the given sphere ,we have
The roots of the equation is
This is the required equation of the enveloping cylinder of the given sphere
Note 2:
In solid geometry, where cylinder means circular cylinder, the generating curves are circles, but here generating curves are of any kindsee fig (1)
When graphing a cylinder or other surface by hand or analyzing one generated by a computer, it helps to look at the curves formed by intersecting the surface with planes parallel to the coordinate planes. These curves are called cross sections or traces.
Example 1 (The parabolic cylinder)
Find an equation for the cylinder made by the lines parallel to the -axis that pass through the parabola, (see Fig.2).
Solution
Suppose that the point lies on the parabola in the xy-plane. Then, for any value of z, the point will lie on the cylinder because it lies on the line through parallel to the z-axis. Conversely any point whose y-coordinate is the square of its x-coordinate lie on the cylinder
because it lies on the line through parallel to the (see Fig.3).it is known as the “cylinder y=x2”
Example 2 The equation defines the circular cylinder made by the lines parallel to the z-axis that pass through the circle in the xy-plane.
Example 3 The equation (i.e., the equation ) defines the elliptical cylinder made by the lines parallel to the z-axis that passes through the ellipse in the xy-plane.
In a similar way, we have the following :
- As Example 1 suggests, any curve in the xy-plane defines a cylinder parallel to the z-axis whose equation is also .
- Any curve in the
xz-plane defines a cylinder parallel to the y-axis whose space equation is also.
- Any curvedefines a cylinder parallel to the x-axis whose space equation is also.
Summarizing the above we have
An equation in any two of the three Cartesian coordinates defines a cylinder parallel to the axis of third coordinate.
Example 4 The equation defines surface made by the lines parallel to the x-axis that passes through the hyperbola in the
plane (see Fig. 4).
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Quadric Surfaces
A quadric surface is the graph in space of a second-degree equation in x, y and z. The most general form is
,
where A, B, C, D,E,F,G,H,J,K are constants, but the equation can be simplified by translation and rotation, as in the two-dimensional case. We will study only the simpler equations. Although the definition did not require it, the cylinders considered here are also example of quadric surfaces. We now examine ellipsoids ,paraboloids, cone, and hyperboloids.
Example 5 The ellipsoid (see Fig. 5)
…(1)
cuts the coordinate axes at, and. It lies within the rectangular box defined by the inequalities and. The surface is symmetric with respect to each of the coordinate planes because the variables in the defining equation are squared.
The curves in which the three coordinate planes cut the surface are ellipses. They are
when
when
when
The section cut from the surface by the plane , is the ellipse
…(2)
.
Example 6 The elliptic paraboloid (Fig. 6)
… (3)
is symmetric with respect to the planes and as the variables and in the defining equation are squared. The only intercept on the axes is the origin Except for this point, the surface lies above or entirely below the xy-plane, depending on the sign of . The sections cut by the coordinate planes are
the parabola
:the parabola
: the point .…(4)
Each plane above thexy-plane cuts the surface in the ellipse
.
Example 7 The circular paraboloid or paraboloid of revolution…(5)
is obtained by taking by in Eq.(3) for the elliptic paraboloid. The cross sections of the surface by planes perpendicular to the z-axis are circles centered on the z-axis. The cross sections by planes containing the z-axis are congruent parabolas with a common focus at the point.
Application : Shapes cut from circular paraboloids are used for antennas in radio telescopes, satellite trackers, and microwave radio links.
We have already discussed that , A cone is a surface generated by lines all of which pass through a fixed point (called vertex) and
(i) all the lines intersect a given curve (called guiding curve)
or (ii) all the lines touch a given surface
or (iii) all the lines are equally inclined to a fixed line through the fixed point.
The moving lines which generate a cone are known as its generators.
When the moving lines satisfy condition (ii) in the definition of a cone, we term the cone as enveloping cone.
In Fig. (a) all the generators pass through the vertex and cut the curve , while Fig. (b), all the generators pass through the vertex and touch the given sphere .
Example 8 The elliptic cone
…(6)
is symmetric with respect to the three coordinate planes Fig.(7). The sections cut by the coordinate planes are
the lines …(7)
the lines …(8)
:the point .
The sections cut by planes above and below the xy-plane are ellipses whose centers lie on the -axis and whose vertices lie on the lines in Eqs. (7) and (8).
If , the cone is a right circular cone.
Example 9 The hyperboloid of one sheet
…(9)
is symmetric with respect to each of the three coordinate planes (Fig. 8). The sections cut out by the coordinate planes are
the hyperbola
the hyperbola …(10)
the ellipse
The plane cuts the surface in an ellipse with center on the z-axis and vertices on one of the hyperbolas in (10).
If , the hyperboloid is a surface of revolution.
Remark : The surface in Example 9 is connected, meaning that it is possible to travel from one point on it to any other without leaving the surface. For this reason it is said to have one sheet, in contrast to the hyperboloid in the next example, which as two sheets.
Example10 The hyperboloid of two sheets
…(11)
is symmetric with respect to the three coordinate planes (Fig.9). The plane does not intersect the surface; in fact, for a horizontal plane to intersect the surface, we must have . The hyperbolic sections
, ,
have their vertices and foci on the -axis. The surface is separated into two portions, one above the plane and the the other below the plane. This accounts for the name, hyperboloid of two sheets.
Eqs. (9) and (11) have different numbers of negative terms. The number in each case is the same as the number of sheets of the hyperboloid. If we replace the 1 on the right side of either Eq. (9) or Eq.(11) by 0, we obtain the equation
for an elliptic cone (Eq.6). The hyperboloids are asymptotic to this cone in the same way that the hyperbolas
are asymptotic to the lines
in the xy-plane.
Example 11 The hyperbolic paraboloid
`…(12)
has symmetry with respect to the planes and (Fig.10 below).
The sections in these planes are
the parabola …(13)
the parabola …(14)
In the plane, the parabola opens upward from the origin. The parabola in the plane opens downward.
If we cut the surfaces by a plane, the section is a hyperbola,
,…(15)
with its focal axis parallel to the y-axis and its vertices on the parabola in (13).
If is negative, the focal axis is parallel to the x-axis and the vertices lie on the parabola in (14).
Near the origin, the surface is shaped like a saddle. To a person travelling along the surface in the yz-plane, the origin looks like a minimum. To a person traveling in the xz-plane, the origin looks like a maximum. Such a point is called a minimax or saddle point of a surface.
Summary
Now let us summarise the session here we introduced the idea of cylinders followed by so many examples like parabolic cylinder,etc.next we defined the quadric surface and also discussed some of the surfaces together with the diagram like ellipsoid ,hyperboloid etc , but in the coming sessions we discuss the nature and shape of quadric surfaces in detail.
So before moving to the next session let us try to answer these questions
Hope you enjoyed the session see you next time till then bye bye
Assignment
Sketch the surfaces
Cylinders
1. 2.
Ellipsoids
3.
Paraboloid
4.5.
FAQ
1Can you explain the ellipsoid of revolution?
Answer: consider the example 5 in that we discussed the ellipsoid , If any two of the semiaxes a, b, and c are equal, the surface is an ellipsoid of revolution. If all three are equal, the surface is a sphere.
2 Example in 1 is called as “the cylinder y=x2”why?
Answer:By the example ,Regardless of the value of z, the points on the surface are the points whose coordinates satisfy the equation. This makes an equation for the cylinder. Because of this, we call the cylinder “the cylinder.”
Quiz
1Surface generated by a straight line which intersects a given circle and is perpendicular to its plane is
(a)right circular cylinder (b) guiding circle (c)enveloping cylinder (d)conicoid
2.The locus of the lines drawn in a given direction or parallel to a given line so as to touch a given surface is called
(a)right circular cylinder (b) guiding circle (c)enveloping cylinder (d)conicoid
3.The general equation of second degree in two dimensions representing a cylinder whose generators are parallel to the z axis
(a)ax2+2hxy+by2+2gx+2fy+c=0 (b)ax2+2hxy+by2+2lx+2fy+c=0
(c)ax2+2hxy+by2+2gx+2ny+c=0 (d)none of these
Answer
1 (a)right circular cylinder
2 (c)enveloping cylinder
3 (a)ax2+2hxy+by2+2gx+2fy+c=0
Glossary
CYLINDER:A cylinder is a surface generated by a line which is always parallel to a fixed line and passes through a given curve.
QUADRIC SURFACE:A quadric surface is the graph in space of a second-degree equation in x, y and z. The most general form is
,
CONE :cone is a surface generated by lines all of which pass through a fixed point (called vertex) andall the lines intersect a given curve (called guiding curve)
References
1S.L.Loney The Elements of Coordinate Geometry ,Macmillian and company, London
2Gorakh Prasad and H.C.Gupta Text book of coordinate geometry, Pothisala pvt ltd Allahabad
3P.K.Mittal Analytic Geometry Vrinda Publication pvt Ltd,Delhi.