Demo of some simple cylindersand 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 + y2 = 1;

Right: Three cylindersx = cos(z), y = sin(z), x2 + y2 = 1intersecting each other.

Notice the intersection of the three cylinders is the well-known space curve

helix −→r (t) = 〈 cos(t), sin(t), t 〉

Math 2339, SMU – p. 3/22

Ellipsoidx2

a2+

y2

b2+

z2

c2= 1

Math 2339, SMU – p. 4/22

Ellipsoidx2

a2+

y2

b2+

z2

c2= 1 ; y = b1, (|b1| < |b|)

(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

Ellipsoidx2

a2+

y2

b2+

z2

c2= 1; x = c1, y = c2, z = c3

(The intersection of an ellipsoid with any plane (not necessarily parallel to the coordinate planes)

is anellipse.)

Math 2339, SMU – p. 6/22

Elliptic paraboloid z =x2

a2+

y2

b2+ c

Math 2339, SMU – p. 7/22

Elliptic paraboloid z =x2

a2+

y2

b2+ c; x = c1, y = c2

The intersection of an elliptic paraboloid

(1) with any plane parallel to the z-axis isa parabola;

(2) with any plane not parallel to the z-axis but intersects the paraboloid isan ellipse.Math 2339, SMU – p. 8/22

Hyperbolic paraboloidz =x2

a2−

y2

b2(viewed from different angles)

Notice thehyperbolas(paraboloid intersecting with any planez = c 6= 0),

and theparabolas(paraboloid intersecting with planesx = c1, y = c2, or y = kx.)

Math 2339, SMU – p. 9/22

Hyperbolic paraboloidz =x2

a2−

y2

b2(viewed from different angles)

Notice thehyperbolas(paraboloid intersecting with any planez = c 6= 0),

and theparabolas(paraboloid intersecting with planesx = c1, y = c2, or y = kx.)

Math 2339, SMU – p. 10/22

Hyperbolic paraboloidz =x2

a2−

y2

b2

(Restricted region plot, looks more like asaddleused in real life.

The right figure also plotsz = c, note the intersection is a hyperbola.)

Math 2339, SMU – p. 11/22

Hyperbolic paraboloidz = −x2

a2+y2

b2(Changing the signs of thex2 andy2 terms changes the orientation of the saddle.)

Math 2339, SMU – p. 12/22

Hyperbolic paraboloidz = −x2

a2+y2

b2Notice thehyperbolas(paraboloid intersecting with any planez = c 6= 0),

and theparabolas(paraboloid intersecting with planesx = c1, y = c2, or y = kx.)

Math 2339, SMU – p. 13/22

Hyperboloid of one sheetz2 + c =x2

a2+

y2

b2, (c > 0)

(The right figure plots the one-sheet hyperboloid intersecting with two planesx = c1 andy = c2)

Notice thehyperbolas(hyperboloid intersecting with any plane parallel to the z-axis)

(hyperboloid intersecting with any plane not parallel to the z-axis may be ahyperbolaor anellipse.)

Math 2339, SMU – p. 14/22

Hyperboloid of one sheetz2 + c =x2

a2+

y2

b2, (c > 0)

Notice the hyperbolas (the surface intersecting with any planex = c1, y = c2)

Math 2339, SMU – p. 15/22

Hyperboloid of one sheetz2 + c =x2

a2+

y2

b2, (c > 0)

(with c decreasing to 0, the hyperboloid gradually turns into cone shape.)

Math 2339, SMU – p. 16/22

Cone z2 =x2

a2+

y2

b2The cone intersects with any plane passing the (0,0,0) point(e.g.c1x+ c2y + c3z = 0) in

(1) two straight lines, (2) one straight line, or (3) a singlepoint.

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

(The conez2 =x2

a2+

y2

b2can intersect with any plane not passing the (0,0,0) point in

(1) parabola, (2) ellipse, or (3)hyperbola. That is why these curves are called conic sections.)

(Acknowledgement: The above figure is from Wikipedia.com onconic sections.)

Math 2339, SMU – p. 18/22

Hyperboloid of two sheetsz2 − c =x2

a2+

y2

b2, (c > 0)

(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

z2 − c =x2

a2+

y2

b2, (c > 0); x = c1, y = c2

Notice the hyperbolas of the hyperboloid intersecting withplanes parallel to the z-axis.

Math 2339, SMU – p. 20/22

Comments:The previous plots seem to “favor” the z-axis. That is, the twostandard quadratic forms are written as

(I) Ax2 + By2 + Cz2 + J = 0, ABC 6= 0,

which includes theellipsoid, thehyperboloid(one-sheet & two-sheets), and thecone. Except for the

ellipsoid,C has a different sign fromA andB (in hyperboloid and cone), this “helps” to “favor” the

z-axis. And(II) Ax2 +By2 + Cz = 0, ABC 6= 0,

which includes theelliptic paraboloid, and thehyperbolic paraboloid. The only linear term is assigned to

thez variable, that is why the z-axis is “favored” again.

The above “special treatment” toz can be bestowed to eitherx or y.This will lead to different orientation of the quadratic surfaces, but theessential shapes of the surfaces do not change. (See the nextslide fortwo examples.)

Math 2339, SMU – p. 21/22

Left: Elliptic paraboloid x =y2

b2+

z2

c2− d. (”favor” x)

Right: One-sheet hyperboloidy2 + b =x2

a2+

z2

c2, (b > 0). (”favor” y)

Notice the orientation of each surface.

Math 2339, SMU – p. 22/22