Assist. Lecturer Hasan Al-Assady
Cylinders and Quadric Surfaces
Cylinders
A cylinder is a surface that is generated by moving a straight line along a given
planar curve while holding the line parallel to a given fixed line.
- The curve is called a generating curve for the cylinder
Example
Find an equation for the cylinder made by the lines parallel to the z-axis that pass
through the parabola y = x2, z = 0
solution:
The point P(a, a2, 0) lies on the parabola y = x2 in the xyplane. Then, for any value of z,
the point Q(a, a2, z) lies on the cylinder because it lies on
the line x = a, y = a2 through P parallel to the z-axis.
Therefore, the points on the surface are the points whose
coordinates satisfy the equation y = x2
This makes y = x2 be an equation for the cylinder, so we call the cylinder y = x2.
QUADRIC SURFACES
A quadric surface is the graph in space of a second-degree equation in x, y, and z.
We fucus on the special equation (Ax2 +By2 + Cz2 + Dz = E)
where A, B, C, D, and E are constants.
- The basic quadric surfaces are ellipsoids, paraboloids, elliptical cones, and
hyperboloid.
- Spheres are special cases of ellipsoids
1- Ellipsoid:
The equation of the ellipsoid is:
Example :
Sketch the ellipsoid
solution:
choose y = 0 and z = 0
x = a, -a
(a,0,0) and (-a, 0, 0) cuts the coordinates
choose x = 0 and z = 0
y = b, -b
(0, b ,0) and (0, -b, 0) cuts the coordinates
choose x = 0 and y = 0
z = c, -c
(0, 0, c) and (0, 0, -c) cuts the coordinates
therefore,
Assist. Lecturer Hasan Al-Assady
Cylinders and Quadric Surfaces
if z = 0 , then
ellipse between x,y-planes
if y = 0 , then
ellipse between x,z-planes
if x = 0 , then
ellipse between y,z-planes
so the graph is :
2- The Elliptical Paraboloid :
The equation of the elliptical paraboloid is:
Example :
Sketch the elliptical paraboloid
solution
choose y = 0 and z = 0
x = a, -a
(a,0,0) and (-a, 0, 0) cut the coordinates
choose x = 0 and z = 0
y = b, -b
(0, b ,0) and (0, -b, 0) cut the coordinates
choose x = 0 and y = 0
z= c
(0, 0, c) cuts the coordinates
therefore,
if x = 0 , then
parabola between y, z-axes
open in the positive of z-axis
if y = 0, then
parabola between x, z-axes
open in the negative of z-axis
choose z = c, then
graph is above:
elliptic between x, y-axis, so the
3- The Elliptical Cone :
The equation of the elliptical cone is:
Example :
Assist. Lecturer Hasan Al-Assady
Cylinders and Quadric Surfaces
Sketch the ellipsoid the hyperbolic paraboloid
solution:
choose y = 0 and z = 0
x = a, -a
(a,0,0) and (-a, 0, 0) cut the coordinates
choose x = 0 and z = 0
y = b, -b
(0, b ,0) and (0, -b, 0) cut the coordinates
choose x = 0 and y = 0
z = c, -c
(0, 0, c) and (0, 0, -c) cuts the coordinates
therefore,
if x = 0 , then
line between y, z-axes
if y = 0, then
line between x, z-axes
choose z = c, then
ellipsoid between x, y-axis,
so the graph is above:
4- Hyperboloid Of One Sheet :
The equation of the hyperboloid of one sheet is
Example :
Sketch the ellipsoid the hyperbolic paraboloid
Solution:
choose y = 0 and z = 0
x = a, -a
(a,0,0) and (-a, 0, 0) cut the coordinates
choose x = 0 and z = 0
y = b, -b
(0, b ,0) and (0, -b, 0) cut the coordinates
choose x = 0 and y = 0
z = c, -c
(0, 0, c) and (0, 0, -c) cut the coordinates
therefore,
if x = 0 , then
hyperbolic between y, z-axes
if y = 0, then
hyperbolic between x, z-axes
choose z = 0, then
ellipsoid between x, y-axis,
so the graph is above:
Assist. Lecturer Hasan Al-Assady
Cylinders and Quadric Surfaces
5- The Hyperboloid of two sheets :
The equation of the hyperboloid of two sheet is
Example :
Sketch the Hyperboloid of two sheets
Solution:
if x = 0 , then
hyperbolic between y, z-axes
if y = 0, then
hyperbolic between x, z-axes
choose z = 0, then - (
negative of ellipsoid between
x, y-axis, so the graph is above:
6- The hyperbolic paraboloid :
The equation of the hyperbolic paraboloid is
Example :
Sketch the ellipsoid the hyperbolic paraboloid
solution:
choose y = 0 and z = 0
x = a, -a
(a,0,0) and (-a, 0, 0) cut the coordinates
choose x = 0 and z = 0
y = b, -b
(0, b ,0) and (0, -b, 0) cut the coordinates
choose x = 0 and y = 0
z= c
(0, 0, c) cuts the coordinates
therefore,
if x = 0 , then
parabola between y, z-axes
open in the positive of z-axis
if y = 0, then
parabola between x, z-axes
open in the negative of z-axis
choose z = c, then
hyperbolic between x, y-axis, so the graph is above: