'By Josepli Mac'Donne[~ S.J.
CJ'Iie 9\[ysefius Library 'Disp[ay
'December, 1997
by joseph MacDonnell, S.J.
Mathematics Department
Fairfield University
~u{ed Surfaces 'lJisp{ay
rr'lie 9\Lyse{ius Library
'lJecember, 199 7
Ruled Surfaces
Table of Contents
Introduction to six types of Ruled Surfaces
1 Moebius half twist (unifacial) surfaces
2 Other twist surfaces
2a 2 half twist (Bifacial) surfaces
2 b 3 to 8 half twist surfaces
2 c p/q twist surfaces
3 Saddle surfaces
3 a Saddle in a box, in a cylinder, in a sphere
3 b Triple saddle surfaces
3 c Quadruple saddle surfaces
3d a-saddle surfaces
4 Hyperboloidal ruled surfaces
4 a in a box
4 b in a cylinder
4 c in a sphere
5 Helicoidal ruled surface
5 a in a box
5 b in a cylinder
5 c in a sphere
6 Space curves from cuspoidal surfaces
6 a Twisted cubic
6 b Surfaces generating focal lines
6 c Double helix
6d Two intersecting archemedean spirals
6e The hartenstein surface
Appendix
2
Introduction
What is a ruled surface?
Planes, cones and cylinders are surfaces clearly
generated by straight lines. However, many surfaces with
intricate symmetries that are curved and warped at every
point are also generated by straight lines. A saddle surface
is one example. It is difficult to imagine even one straight
line feeling completely at home in a saddle surface but in
fact the surface is made up entirely of straight lines. Such
surfaces are called ruled surfaces since they are generated
by rulings or straight lines. Computers have made it easier
to accurately compute and plot the pairs of points
determining each line in these surfaces so that their
constructions are rather simple and they can be made as
large and as precise as one pleases. More can, be read about
these fascinating surfaces at the WWW site:
http:/1204.142.194.96/faculty/jmac/rs/sixmodels .htm
Not all surfaces, of course, are ruled (made up of a
family of rulings) . A surface is ruled if and only if through
every point of the surface passes a real ruling which lies
wholly on the surface. That is, the equations for a fixed
family of lines satisfy the equation of the surface, and the
equation of the surface can be represented as a family of
straight lines. For instance, the set of straight-line
equations, x + y = cz and x - y = k/c, represents the
family of lines which satisfy the equation of the saddle
surface, kz = x2 - y2 • This can be shown by eliminating
the constant c from these two equations. There is one value
of c for each generator of the system. This surface,
kz ~ x2 -y 2 = (x+y) (x-y), can be expressed as the
intersection of the 2 planes, cz = x+y and x-y = k/c. 01
the other hand, no family of straight lines will satisfy the
equation x2 + y2 + z2 = k (the sphere), because the sphere
is not a ruled surface.
3
Cartesian, cylindrical and spherical coordinates
Of the many kinds of ruled surfaces, 6 groups are
described here: moebius, "twist ", saddle, hyperboloidal,
helicoidal and cuspoidal surfaces. They were chosen
because of their striking shape and relative simplicity of
construction. The first construction . decision concerns
which coordinate system to use. Spherical , cylindrical and
rectangular coordinates all have their advantages and
disadvantages, depending on the surface in question.
Compare these three different equations for the saddle:
spherical (R, 9, <J> ): R = k cot <1> esc <1> sec 2 9
cylindrical (r, 9, z): kz = r2 cos 29
rectangular (x, y, z) : kz = x2 • y 2
also cz = xy
X
No matter which coordinate system is chosen , the object is
to find the projections of the surface on a plane, a cylinder
or a sphere. Then one finds a sequence of pai rs of po ints
that can be connected by straight lines or rul ings.
.·~v .
1 Moebius half twist (unifacial) surfaces
The Moebius band has always fascinated artists such as
Escher, as well as today's advertising world, because of its
peculiar unifacial property possessing only one edge. It
forms an endless belt around which a column of ants could
march continuously and in the past provided a mechanical
devise for an endless eight-track cassettes tapes. The
surface is usually formed by giving a half twist to a
narrow strip of paper but all its properties are not
obvious. When the strip width increases only a ruled
surface will help visualize the one-sided property because
intersects itself along a straight line . We find that this
y
4
process will not work for a square sheet of paper as . it
refuses to take shape unless it is cut because the moebr~s
surface must self-intersect or it will self-destruct. This
square sheet of paper is a useless model but rulings pass
comfortably and unhindered between each other.
Intersections of these rulings form a straight line as is
proven in The American Mathematical Monthly
(MacDonnell, 1984, p 125).
Two mathematicians, Gaspard Monge (a graduate of the
Jesuit school, Trinite, in Lyons) and Arnold Emch worked
out the necessary mathematical formulas for the fractional
(p/q) twist surfaces. In particular I start with p/q = 1 I 2
for the half twist or "moebius" surface. Monge showed that
ruled surface equation satisfies the partial differential
equation x 2z •• + 2xyz.y + y2zyy = 0 and also that the
solution to this PD equation must take on the algebraic
form z = xF(y/x) + G(y/x)
Then, Emch described such a surface using the z-axis
as C1, the first generator and C2 , a circle of radius a lying
in the xy-plane as the second generator. A ruling passing
through the z-axis at an angle a meets C2 at point m. A
constant angle a would form a cone, but if the angl~ a
varied as the point m moved around the circumference C2
so that a = p8/q, a p/q twist surface results. The equation
for such a surface is given in cylindrical coordinates.
r = a + z tan p/q 8 or z = (r-a) cot p/q 8
Information on the proof for these theorems is presented in
the Appendix.
Using tedious but cunning manipulation of algebra and
trigonometry formulas Emch derives formulas needed to
5
prove his theorems and determine the order of the surface
(the number of times a straight line intersects the
surface) and the degree of the surfaces' equations. He also
studies the curves of self-intersection made by all the p/q
twist surfaces. To verify the theorems and illustrate the
interesting properties he describes, however, the algebra
and calculus needed is daunting and heuristic geometrical
models are not available. Computer programs to study the
p/q twist surfaces such as MATHEMATICA has the power
not only to display these elusive properties, thereby
making these interesting surfaces accessible, but also to
display the surface and the curves of self- intersection on
a screen. MATHEMATICA can also illustrate any number of
rulings for the cylindrical and cartesian coordinates,
thereby facilitating the actual construction of these models
enclosed in a box or in a cylinder. Most of my figures and
diagrams are from MATHEMATICA programs worked out in
conjunction with Dennis Snow of Notre Dame Mathematics
Department.
The moebius surface enclosed in a box
A Moebius surface is generated by a line POP', which
pivots about point 0, making an angle 9/2 with the vertical
z axis as 0 moves around the circumference of circle C of
radius a in the xy plane. Angle xOO = e. This means that 0
moves about the circle one complete revolution, while line
POP' makes only half a rotation about the point 0 .
After one trip around the z
circle, line POP' takes a new P'
position, P'OP, so the surface --T-~
thus generated has undergone
one-half twist. In so doing, it
has intersected itself along a
straight line. Using ele- P / )
mentary analytic geo-metry,
we arrive at the following : p'
equations for the three · • .:
coordinate systems: ex= 112 e
rectangular: y{x 2+y 2+z 2)-2z{x 2+y 2)+2azx-a 2y = 0
cylindrical: z = (r-a) cot 9/2
spherical: R cost <P = cot(9/2)(Rsin <j>-a)
0
6
The rectangular model can be
constructed between two
pairs of planes x = ±X0 and z
= ±Z0 , but the computation is
more easily done using the
cylindrical coordinate on the
plane faces. The projection of
the surface at these planes
using parameters r and 8
results in the equations:
Plane X=X0 : z = (X0 sec 8-a)cot (9/2) and Y=X0 tan e
Plane x=-X0 : z=-(X 0 sec 8+ a)cot(e/2), e
Plane z=±Z0 : r = ±zo tan (9/2)+ a
The curves projected on
planes ±z0 are mirror
images of one another,
while the curves in
planes ±X0 differ
slightly. Points are
marked along the curve
for every 1 0 degrees
difference in e.
After the four planes are assembled, pairs of points are
connected by a ruling starting from the point (a, 0, Z 0 ) ·in
the z = Z0 plane to the point (a, 0, -Z 0 ) in the z = -Z 0
plane. Then the sequence of points in the upper Z0 plane '
moving counterclockwise are connected to the sequence of
points in the lower -Z 0 plane, also moving
counterclockwise, until the surface is completely
constructed.
The moebius surface enclosed in a cylinder
The key to success using a cylindrical frame is the
orientation of the cylinder. If the axis of the cylinder were
along the z axis too many points would be missing since the
tangent of angles around e = + k90° would be too large for
the cylinder of height 4b. So the axis of symmetry chosen
for the cylindrical model is the y axis. Then x = b cos + and
z = b sin e where <jJ is the polar angle in the xz plane. So the
coordinates for the drilling pattern are ( b<jJ ,y) and the
values of the 36 points in these coordinates must be put i n
7
terms of the variable 8. To accomplish this, some rather
complicated variable shifting is required, but since these
equations (4), (5) and (6) can be fed into a computer,
tedious computations are avoided. The major steps in
developing these equations are stated below.
Use of the polar-Cartesian relation r = x sec 8 and the
circle in the xz plane x2 + z2 = b2 produce the equation:
tan 8/2= (x sec 8-a)/..Jb 2-a2 Squaring this, we have
(b 2-a 2 } tan 2 8/2 = x2 sec2 8 - 2ax sec 8 + a2
rearranging the terms would produce the equation
x=(a sec e± tan e/2(b2 sec 9+b 2tan e/2 - a2 ) ·5)/(sec2 9+tan 2 e/2)
which indicates the location of the points for the pattern to
determine the position of the pairs of points to be joined .
. .
.... .. ..
8
The moebius surface enclosed in a sphere
In some ways the construction of a moebius surface in
spherical coordinates is easier. Since the sphere has a
fixed radius R0 , the first requirement is a set of points in
the other two variables (8, <J>). By applying trigonometric
identities to the third form of the moebius equation given
above, the function <1> = f(8) is found:
e = { arctan Ro 2 tan 8/2 ±a ..f[Ro 2{tan2 8/2 + 1 )-a2] }/ (R2 - a2 )
The resulting curve on the spherical surface indicates the
location of the pairs of points to be joined, using each
corresponding <j>-value for 36 values of e between oo and
360°. These points are plotted on 2 hemispheres. Because
of trigonometric symmetries, the curves on the upper and
lower hemispheres are identical, but 180° out of phase. S:>
the curve can be plotted on the two hemispheres, which are
then fastened together after a 180° rotation so that the
curves form spherical mirror images of each other. The
72 holes are paired off by taking the first set of connecting
points (R0 , 0, rc/2) and (R 0 , rt, rt/2), which will be a
diameter, the only horizontal line, and in fact the only line
passing through the center of the sphere. Adjacent pairs of
points are then connected by a line so that each ruling has
only one point on each hemisphere. This is repeated u n t i I
the rulings are completed.
9
2 a 2 half twist (bifacial) surfaces
A bifacial surface is generated in a way similar to the
unifacial (moebius) surface but has two edges and two
sides. This surface has the unusual property that although
each point of the generating circle has one unique
generator, each line has another line parallel to it. This
surface highlights symmetries and other properties
visible when suspended from above and rotated . Here is
shown the pattern and the resulting surface enclosed in a
box and cylinder .
. . .
..· .
. . .
. .
,•
...
2 b 3 to 8 half twist surfaces
Surfaces with the equation z = {r - a) cot p9/ 2
where the numerator p takes on the values: 3, 4 , 5, 6, 7 , 8
have been constructed and enclosed in a sphere. The theory
is an extension of the moebius strip in the first case p= 1 .
This Moebius surface of one half twist occurs when the
angle between generator POP and the vertical is 9/2. If the
angle a was changed to k9/2 where k is an odd integer,
there would be k half twists and the above formulas would
apply if 9/2 were replaced by k9/2 . A longer cylinder is
required : e.g. for k= 3, a cylinder of radius b would have to
be 5b long instead of 4b. But if k were even, the surface
would be two sided and not a moebius surface and instead of
self-intersecting, it would swing around and become
tangent to itself. It is possible to create this model in the
same way but because of the formula, many points would be
missing near 9=90°. This can be avoided by aligning the
axis of the cylinder with the x axis instead of the y axis .
The development of the equations is similar but the points
on the surface of the cylinder would have coordinates ( 9 ,
x) instead of (<P, y) and the 36 values of 9 would be fed into
the formulas: <P=arccos (y/b) x=b cos <P cot 9 and
y=(a esc 0± tan 0/2{ b2 esc 0+ b2 tan 0/2 - a2 ) . 5)/(csc2 0 + tan2 0/2)
The construction of such a surface follows the method
10
described for the surfaces with an odd number of half
twists. Here is the set of all eight half twist patterns
needed for a sphere.
·-· ,._, .. ,
2 c p/q twist surfaces enclosed in a cylinder
Other p/q twist surface, e.g. 1/5, 1/7, 3/4, 2/5, 2 I 3
are made by using the patterns shown, wrapped around a
cylinder as above, to provide the connecting points for the
rulings. From Mathematica programs, some of the patterns
used for case 117 and 5/2 are shown here.
\ \ \-,!::::: ·····
')
•• 0 .':·,·~:: :·:::·:: • • .. ..
• 0 ··:.::.:: :: .. '.:: • • 0
(~ \ '• .... .... ·-·····. ·
·· .... :::::: .... \ \ ...
3 Saddle surfaces
.. .. ..· : ... . . . . ··.· .· .. • 0 • •• ·: : • ...
3 a Saddles in a box, cylinder, sphere
Of the nine real quadric surfaces (which represent
second degree equations), six are ruled: three cylinders,
the cone, the hyperboloid of one sheet, and the hyperbolic
paraboloid. The last two are doubly ruled. The saddle
(Hypar) is generated by two distinct meshes of lines which
are skew but appear parallel when viewed from above. This
surface represents an important point in applied
)
)
)
)
)
}
}
11
mathematics called the "saddle point" found at its center
which is a maximum point in one plane but a minimum
point in another plane , but actually is neither for the given
surface .
Saddles in rectangular coordinates,the equation
kz = x 2 -y 2 , when solved for constant x and constant y ,
produces the following curves , which are the projections
of the surface for the fixed planes:
x =±a ka 2 z=(a2 -y2 ) and
y = ±a k a 2 z = (x2 - a2 ).
These parabolas are plotted on the two pairs of planes ,
x = ±a and y= ±a, using rectangular coordinates (y, z) and
(x, z) respectively. 01 each of the four parabolas, twelve
points are marked corresponding to twelve equidistant
points along the domains. The corresponding pairs of points
are then connected. In connecting these corresponding
points it is helpful to remember that these rulings must be
parallel to either the plane y = x or the plane y = -x.
The saddle surface can also be formed after a 45° rotation
of the x and y axes in the previous model, which yields new
axes X andY where X=(x+y)/;f2, and Y=(-x+y)/;f2 , so
that the hyperbolic paraboloid kz = y 2-x 2 becomes kz =
2XY. The construction consists simply of two isosceles
right triangular planes , ABO and BCD, joined orthogonally
along their hypotenuse BD. This forms the skew
quadrilateral, ABCD. Ore family of rulings for the ruled
surface consists of the lines joining equidistant points
along AB to equidistant points along DC. Corresponding
points from DA to CB constitute the other family of
rulings . All the lines of the first family are parallel to the
x = 0 plane while the lines of the second family are
parallel to the y = 0 plane. This model is the simplest to
construct, since only two mutually perpendicular planes
are required .
12
Saddles in a cylindrical frame
The hyperbolic paraboloid is the above-mentioned
saddle surface, and its most convenient equation occurs in
cylindrical coordinates. The intersection of this surface
with the cylinder r = r0 is the curve kz = r/ cos 28 or
Z = cos 28, where Z = kzlr/ .
This cosine curve, Z = cos 28, can be plotted as a planar
graph, using 8 and Z in place of x, and y as rectangular
coordinates. Its amplitude (for O<Z< I) is r2 /k: that is , z
oscillates between ±r0 2/k. The scale along the 8 axis has to
be adjusted so that the total
distance between the two extreme
positions 8= ±180° measures
27t r 0 , the circumference of the
cylinder. The planar graph can
then be wrapped around the
cylinder with Z=±1(i.e z= ±r0 /k) .
Saddles in a spherical frame
The saddle can be enclosed in a sphere by plotting point
on the spherical surface (R = Ro) as a function of 8 and <j> .
Implicitly this would be sec 28 cot <1> = R. while the
explicit statement for discovering the points would be
<1> = f(8) = cot" 1 ( R o cos 28).
Once the points are marked on the spherical surface
and the holes are drilled threading will take some patience
and a long thin needle greater than the diameter of the
sphere.
2 bTriple saddle surfaces are composed of three
saddles enclosed in a pyramid which intersect in a single
point and this is the saddle point for each of the saddles.
3 c Quadruple saddle surfaces are made up of four
saddles in a rectangular coordinate system. Parabolas are
the generating elements.
3d 8-saddle surfaces are made up of four pairs of
saddles in a rectangular coordinate system. It can be
designed in the shape of a sailboat.
4 Hyperboloidal surfaces in a box, cylinder, sphere
The Hyperboloid is a hyperboloid of one sheet and is
generated by lines connecting points on parallel ellipses .
This surface has the unusual property that it is doubly
ruled - there are not one but two families of rulings . Each
13
family covers the surface .completely. Every straight line
of one family intersects every straight line of the other
(or is parallel to it), but any two lines of the same fa m i I y
are mutually skew. Three skew straight lines d~fine a
hyperboloid of one sheet.
These two families have a surprising property. If this
model were made of lines which intersect in a way which
allows rotation but not of sliding, it would seem that
straight lines fastened in this way would form a rigid
frame. This is not the case, however, the framework is
movable. As the distance between the parallel planes 1t and
1t' increases the angle between the asymptotes of the
generating hyperbolas increases. Two such surfaces are
used in cogwheel transmission.
5 Helicoidal ruled surface in a box, cylinder, sphere
The Helicoid is swept out by a generator which always
intersects a fixed axis at right angles and which rotates
uniformly as its point of intersection moves uniformly
along the axis. It intersects any cylinder concentric with
the axis in a helix. The right helicoid has the shape of the
thread of a screw whose pitch can be varied and mimics a
spiral staircase. The right helicoid is a minimal surface,
which is the kind of surface soap solutions assume to
minimize area for a given configuration, and for this
reason is used in the construction of radio antennae. In
fact, it is the only ruled minimal surface apart from the
plane.
Shown here is the line L is perpendicular to the z axis,
and as its intersection P moves along the z axis, L rotates
about the axis. The intersection of the helicoid with a
14
cylinder produces a helix, which would be equivalent to the
railing of a spiral staircase. In parametric form, the
equation for the helix is: x = rcos 8 y = rsin 8 z = k8.
The equation of the helicoid in the 3 coordinates would be:
rectangular: y = x tan z/k
cylindrical: z = k 8
spherical: R = k 8 sec<j>
The pattern for a cylindrical model is drawn on a plane
graph using z and 8 as the rectangular coordinates. The
domain for 8 is ( -7tr, m) where r is the radius of the
cylinder and the range is one-half the length of the
cylinder. Parallel lines are drawn with slope k. Points are
marked along these lines corresponding to equal distances
along the 8 axis. The pitch of the model is determined by k,
and the number of threads of the screw is determined by
the number of parallel lines drawn. These points are
transferred to the cylinder, and the rulings are made
between pairs of points diametrically opposite, as seen in
the figure. The helicoid has a wide variety of shapes,
depending on the pitch, the proximity of the lines and the
points, and whether the pairs of points connected by
rulings are in fact not quite diametrically opposite.
Another right helicoid is easier to construct because i t
requires only two planes perpendicular to one another.
Taking the z axis as the axis of the right helicoid and
rotating the xz and xy planes through an angle of 45°, we
obtain new coordinates (x, Y, Z) in place of (x, y, z).
The rotation of axes changes the helicoid equation, from
y/x = tan z/x to the equation
Y-Z = -../2 tan (Y+Z)/k-../2,
so that the projections of
the surface on the two planes
will be on the xY plane:
Y = ..J2 x tan(Y/k..J2) ,/
and on the xZ plane:
Z = - -.12 x tan(Z/k..J2)
15
These curves are mirror images of one another so that
patterns can be made for both at the same time by using the
xY or the xZ plane. Use of a computer simplifies
computation of the values of x for equally spaced values of
Y put into the equation x = Y/..J2 cot(Y/k..J2).
6 Space curves from cuspoidal surfaces
6 a The twisted cubic
Two surfaces together constitute a tangential
developable and they meet in a sharp edge called cuspoidal
edge or edge of regression (Hilbert, p. 206). An example is
the twisted cubic: two quadric surfaces will in general
intersect in a curve of degree 4. But if they have a
generator in common then the curve of intersection will
consist of this generator and a
curve of degree 3, called a
twisted cubic which is not a
plane curve. It can be displayed
in a model as the edge of
regression of the surface
formed by its tangents. The
simplest method is to consider
the cubic given by the
parametric equations x: y: z: a
= 83: 82 : 8: 1. The equations for
the tangent at 8 are then x -
28y+82z = 0 and y-28z+82a =
0. The surface is conveniently
generated by rulings between
the three planes in Cartesian
space. (Cundy, p. 185)
Another version of the twisted cubic is constructed
using three quadrics surfaces: a cone, hyperboloid and
cylinder. The sections of the quadric cones are circles
16
which makes construction of this twisted cubic model
possible. The projection of the common generator of
cylinder and hyperboloid can be seen as well as the that of
the cylinder and cone and that of the cone and hyperboloid .
(Cundy, p. 182)
Cone: (x+ 3a/2) 2 + (y-Sa/2) 2 = 9a2 / 2
Cylinder: (x-a} 2+(y-4a) 2 = a2
Hyperboloid: x2 + y2= 10a2
6 b Surfaces generating focal lines
A surface which has the unusual property of containing
two lines perpendicular to each other where the rays come
to a focus in their respective planes. It is a surface of great
importance in optics and is the envelope traced out by the
rays of a pencil reflected or refracted at a spherical
surface. These rays pass through two focal lines orthogonal
to each other. Midway between is the central c i r cuI a r
section called the 'circle of least confusion' lying on the
image plane halfway between the sagittal and tangental
plane, while at the tangental and sagittal image planes, the
ellipse degenerates into a line.
Shown here is the pattern of the points for the
construction of this surface and also how to connect the
points in the subsequent threading. Two congruent ellipses
are mounted in parallel planes with their major axes at
right angles. In each ellipse the major axis is triple the
minor axis. Points on the lower ellipse, is always directed
in diametrically opposite direction to the upper ellipse.
I illustrate the positions of the coordinate axes, the double
lines and the central circle.
17
6 c Double helix
Another cuspoidal surface results from the helicoid
enclosed in a cylinder. Two can be placed in the same
cylinder to form two helices and produce a model of the
now-famous double helix.
6d Two intersecting archemedean spirals
When Two intersecting archemedean spirals are used to
generate surfaces their intersection forms a cusp.
6e The hartenstein surface
The Hartenstein model has interesting geometrical
features such as distinct sheets, double curves and cusps.
The figure shows this developable surface which is
described by Klein (p. 97) and has to do with biquadratic
or quartic
t4 + A.t2 + 11t +v = 0 and depends on the parametric
equations of the points on the surlace:
X = -6W + 2 p t)
y = BW+3pt2 )
z = -3W+4pt3 )
By eliminating p and t we get the equation of the surface
(z+x 2 /12) 3-27[(xz/6)-(y 2 /16)-(x 3 /216) 2 ] = 0
18
Appendix
Theorems concerning p/q Twist surfaces
Ruled surfaces have been studied by geometers such as
Gaspard Monge, who described in his 1850 book
"Application de !'Analysis a Ia Geometrie (pp. 83-89 ) a
particular type of ruled surface useful for sculptors and
architects who could design their curved surfaces while
using a straight edge anchored along a fixed line, such as
the z-axis. He proposed two theorems concerning a surface
whose generator passes through the z-axis which used in
our Partial differential Equations course MA 321 . Monge
has a discursive proof for the first theorem and the proof
for the second is presented here.
Theorem I The Surface equation satisfies the part i a I
differential equation x 2 Zxx + 2xyzxy + y2 zyy = 0 ( 1 )
Theorem II The solution to this equation has the
algebraic form z = xF(y/x) + G(y/x) ( 2 )
Proof of the second Monge theorem
1 . Using the substitutions; u = In x and v = In y and
proceeding in a similar manner to a Cauchy-Euler ODE we
find z. = ZuUx = zjx and in a similar way Zy = Zv Vy = Z/Y
and z,. = zuufx2 - z)x2 and in a similar way Zyy = zvvfY2
- z)y2 and finally z ,~ = Zuv
2 So the equation x Zxx + 2xyz,y + y2zyy = 0 ( 1 )
becomes Zuu + 2Zuv + Zvv - Zu - Zv = 0 (3)
or [D 2u+ 2DuDv + 0 2vD - Du - Dv]z=O
which breaks into two linear factors
L(z) = L 1 (z) L2(z) = [(Du + Dv)(Du + Dv -1 )z] = 0
3 Now for each of these linear operators L1 and L2 use the
Linear POE I {Constant Coefficient} method of solution for
Az. + BZy + Cz = 0 which is: z= e(·CxiA)f(Bx-Ay)
4 So z1 =e0 g(u-v)= g(lnx - lny) = g(ln[y/x]) = G[y/x]
and z2 = e(·u)f(u-v) = ( e(lnx))(lnx - lny) = xF[y/x)
5 The solution: z = z1 + z2 = xF(y/x) + G(y/x) ( 2 )
QED
Emch theorems concerning p/q twist surfaces
In 1920 Arnold Emch in the American Journal of
Mathematics (42 pp. 189-21 0), described such a surface
using the z-axis as C1 , the first generator and C2 , a c i rc I e
19
of radius a lying in the xy-plane as the second generator. A
ruling passing through the z-axis at an angle a meets C2 at
point m. A constant angle a would form a cone, but if the
angle a varied as the point m moved around the
circumference C2 so that a = p8/q, a p/q twist surface
results. The equation for such a surface is given in
cylindrical coordinates.
r = a + z tan p/q 8 or z = (r-a) cot p/q 8
We here show that this cylindrical form is in proper
form according to Monge = xF(y/x) + G(y/x)
Proof:
For these surfaces z = r cot (p 8/q) - a cot (p 8/q)
we can replace cot (p 8/q) by functions of y/x in the
following manner. Take the case of p/q = 1/1
z = r cot (8) - a cot (8)
Since tan t = y/x cot t = x/y and r = x sec t
= x (1 + tan2 t) = x (1 +[y/x]2)
so z = x (1+[y/x]2 ){1/(y/x) } + (-a){1/(y/x)
and z = xF(y/x) + G(y/x)
QED
Note that in some of the old trigonometry books one
finds a meticulous (and boring) examination of "submultiple
angles" e.g. tan A/2 = ( [1 + tan2 A] - 1 )/tan A
and tan A= {(3 tan [A/3] - tan 3 [A/3])/(1-3 tan2 [A/3])} so
all the p/q are not unreachable for the stout hearted, but
there is no need to present the details here for a II
fractions p/q 8 . Then one could use Emch proposes
equations concerning tan rw which is an expansion of
binomial functions. Using w to represent 8 and 8/2 and
using r to represent an assortment of values of p.
In this way it is possible to using Mathematica to
demonstrate Monge's second theorem that
our ruled surfaces z = r cot (p 8/q) - a cot (p 8/q) also
satisfy the predicted form z = xF(y/x) + G(y/x) for a II
cases such as 1/1, 2/1, 3/1, . . 1/2, 3/2, 5/2 .. .
20
References
H. M. Cundy and A. P. Rollett, Mathematical Models.
London: Oxford University Press, 1961
2 Emch, Arnold, American Journal of Mathematics.
42, 1920, p. 189
3 Emch, Arnold, Mathematical Models. Chicago:
Univ. Illinois, 1920
4 Eves, Howard, A Survey of Geometry. Boston:
Allyn and Bacon, 1972
5 D. Hilbert and S . Cohn-Vossen, Geometry and the
Imagination. New York: Chelsea, 1952
6 G. James and R. James, Mathematics Dictionary.
New Jersey: Van Nostrand, 1949
7 Klein, Felix, Elementary Mathematics from an
Advanced Standpoint. New York: Dover, 1948
8 Lipshutz, M. M., Differential Geometry.
New York: McGraw-Hill, 1969
9 MacDonnell, J. F., 1980, International Journal of
Mathematical Education. vol 17, 1986 p. 179.
1 0 MacDonnell, J. F., The American Mathematical Monthly
Feb. 1984 vol. 91 #2, p. 125
1 1 Olmsted, J. Solid Analytic Geometry.
New York: Appleton Century Crofts, 1947
1 2 O'Neill, B. , Elementary Differential Geometry.
New York: Academic Press, 1966
Half twist in shpere Saddle in cylinder
Elliptical Hyperboloid
Focal line
J
'
Multisaddle 8-saddle
Double helicoid
Half twist in cylinder 2 half twists in cylinder
3-saddle 4-saddle