Demo of some simple cylinders
and quadratic surfaces
Yunkai Zhou
Department of Mathematics
Southern Methodist University
(Prepared for Calculus-III, Math 2339)
Acknowledgement: The very nice free software K3dSurf was used for the plots.
Math 2339, SMU – p. 1/22
Left: Cylinder x = cos(z);
Right: Cylinder y = sin(z)
Math 2339, SMU – p. 2/22
Left: Cylinder x2 + y 2 = 1;
Right: Three cylinders x = cos(z), y = sin(z), x2 + y 2 = 1
intersecting each other.
Notice the intersection of the three cylinders is the well-known space curve
→
helix −
r (t) = h cos(t), sin(t), t i
Math 2339, SMU – p. 3/22
x2 y 2 z 2
Ellipsoid 2 + 2 + 2 = 1
a
b
c
Math 2339, SMU – p. 4/22
x2 y 2 z 2
Ellipsoid 2 + 2 + 2 = 1 ; y = b1 , (|b1 | < |b|)
a
b
c
(Notice the intersection is an ellipse.
In fact the intersection of an ellipsoid with any plane that intersects with it is an ellipse.)
Math 2339, SMU – p. 5/22
x2 y 2 z 2
Ellipsoid 2 + 2 + 2 = 1; x = c1 , y = c2 , z = c3
a
b
c
(The intersection of an ellipsoid with any plane (not necessarily parallel to the coordinate planes)
is an ellipse.)
Math 2339, SMU – p. 6/22
x2 y 2
Elliptic paraboloid z = 2 + 2 + c
a
b
Math 2339, SMU – p. 7/22
x2 y 2
Elliptic paraboloid z = 2 + 2 + c; x = c1 , y = c2
a
b
The intersection of an elliptic paraboloid
(1) with any plane parallel to the z-axis is a parabola;
(2) with any plane not parallel to the z-axis but intersects the paraboloid is an ellipse.
Math 2339, SMU – p. 8/22
x2 y 2
Hyperbolic paraboloid z = 2 − 2
a
b
(viewed from different angles)
Notice the hyperbolas (paraboloid intersecting with any plane z = c 6= 0),
and the parabolas (paraboloid intersecting with planes x = c1 , y = c2 , or y = kx.)
Math 2339, SMU – p. 9/22
x2 y 2
Hyperbolic paraboloid z = 2 − 2
a
b
(viewed from different angles)
Notice the hyperbolas (paraboloid intersecting with any plane z = c 6= 0),
and the parabolas (paraboloid intersecting with planes x = c1 , y = c2 , or y = kx.)
Math 2339, SMU – p. 10/22
x2 y 2
Hyperbolic paraboloid z = 2 − 2
a
b
(Restricted region plot, looks more like a saddle used in real life.
The right figure also plots z = c, note the intersection is a hyperbola.)
Math 2339, SMU – p. 11/22
x2 y 2
Hyperbolic paraboloid z = − 2 + 2
a b
(Changing the signs of the x2 and y 2 terms changes the orientation of the saddle.)
Math 2339, SMU – p. 12/22
x2 y 2
Hyperbolic paraboloid z = − 2 + 2
a b
Notice the hyperbolas (paraboloid intersecting with any plane z = c 6= 0),
and the parabolas (paraboloid intersecting with planes x = c1 , y = c2 , or y = kx.)
Math 2339, SMU – p. 13/22
2
2
y
x
Hyperboloid of one sheet z 2 + c = 2 + 2 , (c > 0)
a
b
(The right figure plots the one-sheet hyperboloid intersecting with two planes x = c1 and y = c2 )
Notice the hyperbolas (hyperboloid intersecting with any plane parallel to the z-axis)
(hyperboloid intersecting with any plane not parallel to the z-axis may be a hyperbola or an ellipse.)
Math 2339, SMU – p. 14/22
2
2
y
x
Hyperboloid of one sheet z 2 + c = 2 + 2 , (c > 0)
a
b
Notice the hyperbolas (the surface intersecting with any plane x = c1 , y = c2 )
Math 2339, SMU – p. 15/22
2
2
y
x
Hyperboloid of one sheet z 2 + c = 2 + 2 , (c > 0)
a
b
(with c decreasing to 0, the hyperboloid gradually turns into cone shape.)
Math 2339, SMU – p. 16/22
2
2
x
y
Cone z 2 = 2 + 2
a
b
The cone intersects with any plane passing the (0,0,0) point (e.g. c1 x + c2 y + c3 z = 0) in
(1) two straight lines, (2) one straight line, or (3) a single point.
The cone can intersect with any plane not passing the (0,0,0) point in
(1) parabola, (2) ellipse, or (3) hyperbola.
Math 2339, SMU – p. 17/22
x2
y2
(The cone
= 2 + 2 can intersect with any plane not passing the (0,0,0) point in
a
b
(1) parabola, (2) ellipse, or (3) hyperbola. That is why these curves are called conic sections.)
z2
(Acknowledgement: The above figure is from Wikipedia.com on conic sections.)
Math 2339, SMU – p. 18/22
2
2
y
x
Hyperboloid of two sheets z 2 − c = 2 + 2 , (c > 0)
a
b
(with c increasing from 0, the hyperboloid turns from cone shape into more obvious two sheets.
Math 2339, SMU – p. 19/22
Hyperboloid of two sheets
2
2
y
x
z 2 − c = 2 + 2 , (c > 0); x = c1 , y = c2
a
b
Notice the hyperbolas of the hyperboloid intersecting with planes parallel to the z-axis.
Math 2339, SMU – p. 20/22
Comments:
The previous plots seem to “favor” the z-axis. That is, the two
standard quadratic forms are written as
(I) Ax2 + By 2 + Cz 2 + J = 0, ABC 6= 0,
which includes the ellipsoid, the hyperboloid (one-sheet & two-sheets), and the cone. Except for the
ellipsoid, C has a different sign from A and B (in hyperboloid and cone), this “helps” to “favor” the
z-axis.
And
(II)
Ax2 + By 2 + Cz = 0,
ABC 6= 0,
which includes the elliptic paraboloid, and the hyperbolic paraboloid. The only linear term is assigned to
the z variable, that is why the z-axis is “favored” again.
The above “special treatment” to z can be bestowed to either x or y.
This will lead to different orientation of the quadratic surfaces, but the
essential shapes of the surfaces do not change. (See the next slide for
two examples.)
Math 2339, SMU – p. 21/22
y2 z2
Left: Elliptic paraboloid x = 2 + 2 − d. (”favor” x)
b
c
2
2
x
z
Right: One-sheet hyperboloid y 2 + b = 2 + 2 , (b > 0). (”favor” y)
a
c
Notice the orientation of each surface.
Math 2339, SMU – p. 22/22