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## hyperbolic bowls

### Michael McDowell on wed 29 jan 03

John,

What about doing it in three dimensions out of clay? You could make
some small cones with different angles for the sides easily enough
out of clay on the wheel. Then you could slice them when they are
leather hard and get a pretty good conic section. My dictionary says
that a hyperbolic curve results from slicing at a greater angle than
the sides make with the base. Nothing hard about that. According to
Paul's suggestion, get it right at any size and then blow it up with
a copy machine.

Michael McDowell
Whatcom County, WA
Michael@McDowellPottery.com
http://www.McDowellPottery.com

### John Rodgers on wed 29 jan 03

I have a commission to make a number of bowls that are to be hyperbolic
in form and are/should be nearly mathematically correct. I looked
hyperbola's up in a math book and I'm definitely not up to the
calculations.

Question: Can anyone tell me how to construct with pencil, pins,
string, triangle, and straight edge, hyperbolic curves that I can draw
out on paper or plastic board or something, in order to create and cut
out some profile templates I need for the bowl making?

Would appreciate any help at all on this one.

Thanks,

John Rodgers
Birmingham, AL

### Bryan on wed 29 jan 03

>
> Question: Can anyone tell me how to construct with pencil, pins,
> string, triangle, and straight edge, hyperbolic curves that I can
draw
> out on paper or plastic board or something, in order to create and cut
> out some profile templates I need for the bowl making?
>
> Would appreciate any help at all on this one.
>
Hyperbolic curves represent a vertical cross section of a cone.

You could make a cone, or a few cones with differently angle sides, out
of clay, and then cut vertical slices. If you don't like what you end up
with make a different angle on the cone.

Parabolic bowls would be made if you sliced through the cone at an
angle.

Bryan

### Bob Santerre on wed 29 jan 03

It's super simple for a flat surface, 2 pins, a string and a pencil.
Set the 2 pins at some distance apart form each other, tie the string
into a complete loop that can span the 2 pins (the looser the string the
bigger the hyperbole, the farther apart the pins the greater the long
axis of the hyperbole), loop the string over the pins, place your pencil
firmly inside the perimeter of the string and trace the outline
completely around the pins, always keeping outward tension on the
string. That's a perfect hyperbolic form. Once the pin distance and
the string length are set you can make the same size hyperbolic outline
each time. Of course that only describes the rim or base of the bowl.
Translating that same shape down the tapered walls is another matter.
One way that might work is to make a stack of forms, say with 1/4"
plywood, such that each layer is a bit smaller than the next until you
get to the appropriate base size. Perhaps the easiest way to do this
would be to make the base shape and then for the next layer up trace
around this first form with a thick pencil to expand the size of that
next layer. Keep doing that with each successive layer until you get to
the rim size you want. Now you have a plywood stack of hyberbolic
shapes that together describes a bowl form. Glue them together, cover
the stack with some thin cardboard and you have a mold that you can
drape clay over, cut and shape and dry to leather hard.

Another possibility would be to make the top and bottom shapes out of
plywood, then sandwich a thick piece of construction-grade styrofoam
between them and cut the styrofoam into a hyperbolic bowl-shape using
the edges of the plywood pieces as a guide for the saw or knife.
Styrofoam molds are quite durable.

Good luck, Bob

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

John Rodgers wrote:

> Question: Can anyone tell me how to construct with pencil, pins,
> string, triangle, and straight edge, hyperbolic curves that I can draw
> out on paper or plastic board or something, in order to create and cut
> out some profile templates I need for the bowl making?

### Jim Kasper on wed 29 jan 03

Unfortunately Bob gives a good description of how to draw an ellipse. There is not such a simple method for a hyperbola. The web page below gives one method which looks painful.

http://www.fizziker.com/Wally/

doing the following google search on the web with
"hyperbola drawing" yeilded this method. There is a good chance you could find a better one. If you call your local community college, or perhaps even high school, you might beable to get someone from the math department to plot out for you. It is quite easy to make these shapes on a computer, but with a typical home printer you are constrained to a small size.

Regards,
Jim Kasper
http://zafka.com
>
> From: Bob Santerre
>
>
> It's super simple for a flat surface, 2 pins, a string and a pencil.

### Bill Aycock, W4BSG on wed 29 jan 03

Wrong figure. The shape described by the string gadget is an Ellipse. If
the two pins are at the same place, the figure is a circle.
A parabola can be * Approximated * by putting the pins very far apart and
having the string just barely more than twice the length of the pin separation.
The Circle, the Parabola, the Hyperbola and the Ellipse are all called
'Conic sections' because they can be made by various cuts through a cone.
Bill

At 12:28 PM 1/29/2003 -0500, you wrote:
>It's super simple for a flat surface, 2 pins, a string and a pencil.
>Set the 2 pins at some distance apart form each other, tie the string
>into a complete loop that can span the 2 pins (the looser the string the
>bigger the hyperbole, the farther apart the pins the greater the long
>axis

Bill Aycock - W4BSG
Woodville, Alabama

### Bryan on wed 29 jan 03

http://members.shaw.ca/ron.blond/TLE/QR.EH.APPLET/

This web site lets you change the value of the equation that yields a
hyperbola by moving a slider up and down so that you get a full range of
possible hyperbolas.

Bryan

### Paul Lewing on wed 29 jan 03

on 1/28/03 10:54 PM, John Rodgers at j_rodgers@CHARTER.NET wrote:

> Question: Can anyone tell me how to construct with pencil, pins,
> string, triangle, and straight edge, hyperbolic curves that I can draw
> out on paper or plastic board or something, in order to create and cut
> out some profile templates I need for the bowl making?

John, you don't need to draw a new hyperbola, if you have a picture of one,
of any size, in your book. Take it to the copy place, have it enlarged or
reduced to the proper size and make your template from that.
Good luck, Paul Lewing

### iandol on thu 30 jan 03

Dear John Rodgers,

The first thing to try to understand is what you are dealing with here.

My suggestion is to Jan Gullberg's book, "Mathematics from the Birth of =
Numbers". W. w. Norton and Co. 1997. ISBN 0-393-04002-X. go to page 408 =
and you will find out all about Conic Sections.

There is a subtle curve which can be generated by drawing two lines at =
right angles on a sheet of graph paper. Mark off say, twelve units along =
each of these lines. Join 12 on one to 1 on the other with a straight =
line and go down the axis progressively doing this. This will get an =
approximate curve. This can be refined freehand or using French Curves. =
To the best of my knowledge this is a special form of Hyperbola (not to =
be confused with hyperbole which is a figure of speech, not a =
mathematical figure)

Good definition of Hyperbola on p 564

Have fun,

Best regards,

Ivor Lewis. Redhill, South Australia.

### Michael McDowell on fri 31 jan 03

Phil,

You may wll be right about the definition of a hyperbola. I
certainly claim no expertise in solid geometry. My source was the
entry in the 1986 edition of the Little Oxford Dictionary which is
concise though according to you incorrect.

"However, I think the Hyperbola is 90 degrees to the plane of
the base...whilst the Parabola is some angle other than
90...without it mattters much what the angle of the sides
are of the Cone in question."

Little Oxford:
Hyperbola- curve produced when cone is cut by plane making larger
angle with base than side of cone makes.
Parabola- plane curve formed by intersection of cone with plane
parallel to its side

This is not to hold my diminuitive dictionary up as any sort of
authority in these matters. Just to be forthcoming about the source
for my remarks. Hope you'll do the same now that we've muddied the
waters a bit.

Michael McDowell
Whatcom County, WA
Michael@McDowellPottery.com
http://www.McDowellPottery.com
Feeling bad that it loooks like I will miss yet another NCECA
conference. And I had really thought I would make it this time...

### Philip Poburka on fri 31 jan 03

Yup...I was all wet on my thought...

I had remembered correctly, a bogus notion of it.

Sigh...

Phil
Las Vegas

----- Original Message -----
From: "Dave Finkelnburg"
To:
Sent: Friday, January 31, 2003 2:32 PM
Subject: Re: Hyperbolic Bowls

For John Rogers and others interested in the hyperbola,
Michael McDowell appears to have it right, at least
according to the
source I found. The hyperbola is not a single curve, but a
family of
curves. A vertical slice through a cone would give a
hyperbola, but only
one in an infinite number of possible hyperbolas.
The definition below matches the information Michael
cites (which also
matches my version of Webster's, by the way). For a good
discussion of
hyperbolas, see the site
http://www.xahlee.org/SpecialPlaneCurves_dir/Hyperbola_dir/h
yperbola.html
Isn't plane geometry fun? :-)
Dave Finkelnburg, waiting impatiently....again...as
the kiln
cools....

"....the plane that produces an ellipse is less tilted than
the side of the
cone...; the plane that produces a hyperbola is more tilted
than the side of
the cone ...; and the plane that produces a parabola is
parallel to the side
of the cone...." from
http://mathforum.org/library/drmath/view/53919.html
This is from the Math Forum site of Drexel University.

----- Original Message -----
From: "Michael McDowell"
> You may wll be right about the definition of a hyperbola.
I
> certainly claim no expertise in solid geometry. My source
was the
> entry in the 1986 edition of the Little Oxford Dictionary
which is
> concise though according to you incorrect.
>
> "However, I think the Hyperbola is 90 degrees to the plane
of
> the base...whilst the Parabola is some angle other than
> 90...without it mattters much what the angle of the sides
> are of the Cone in question."
>
> Little Oxford:
> Hyperbola- curve produced when cone is cut by plane making
larger
> angle with base than side of cone makes.
> Parabola- plane curve formed by intersection of cone with
plane
> parallel to its side

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### Gavin Stairs on fri 31 jan 03

There is a family of curves called the conic sections.  They all
represent the boundary curve of the area of intersection of a pair of
solid right circular cones, identical but opposed tip to tip, with a
plane.  I can't go into the whole subject, as it is vast.
There are three sets of curves that are commonly discussed.  The
distinction is the way the plane goes through the cones.  To see the
diagrams below, set your email viewer to a fixed-width font.

Cones:\        /
Ellipse:\        /
Parabola:\        /
Hyperbola:\|      |/

\
/
\
/
\
/
\      /

\
/
\
/
\
/
|\    /

\
/
----\--/----

/
| \  /|

\/
\/
\/
|  \/ |

/\
/\
\/\
|  /\|

/
\            --_

\
/\
| /  \

/
\
/--__\
/  \
|/   |\

/
\
/
\--
/    \
/   |  \

/
\
/
\
/      \
/|   |  \

In the above illustrations, I could not properly represent the skew
lines, so you need to imagine that they are smooth lines which represent
the intersecting planes seen on edge.  The cones have a circular
cross section.

The ellipse is formed by the intersection where the angle of the plane
from the axis of the cones is greater than the opening angle of the
cones. The ellipse so formed is not symmetrical on the short axis unless
the cone angle is zero, meaning the cone is a cylinder: a degenerate form
of the cone.  In this case the ellipses are ovals.  The oval
with angle from the axis of the cylinder a right angle is a circle.
An ellipsoidal mirror has two foci, and a light at one of them is focused
at the other, so it is the mirror form used for condensing optics, as in
a slide projector.

The parabola is formed by any intersection at an angle equal to the cone
opening angle.  The parabola is not a closed curve.  It's arms
keep on opening, but carry on to infinity.  The parabola focuses
light from infinity at its single focus, so it is the mirror used for
astronomical telescopes.

The hyperbola is formed by an intersection angle less than that of the
cone angle.  The complete hyperbola consists of two open figures,
one on top, the other below.  The hyperbola lies entirely within a
pair of straight boundary lines that cross between the upper and lower
curves, and the arms of the curve are asymptotic to these lines.
These lines are the boundaries of the generating cone projected onto the
plane of section.

Gavin

### Dave Finkelnburg on fri 31 jan 03

For John Rogers and others interested in the hyperbola,
Michael McDowell appears to have it right, at least according to =
the
source I found. The hyperbola is not a single curve, but a family of
curves. A vertical slice through a cone would give a hyperbola, but =
only
one in an infinite number of possible hyperbolas.
The definition below matches the information Michael cites (which =
also
matches my version of Webster's, by the way). For a good discussion of
hyperbolas, see the site
http://www.xahlee.org/SpecialPlaneCurves_dir/Hyperbola_dir/hyperbola.html=

Isn't plane geometry fun? :-)
Dave Finkelnburg, waiting impatiently....again...as the kiln
cools....

"....the plane that produces an ellipse is less tilted than the side of =
the
cone...; the plane that produces a hyperbola is more tilted than the =
side of
the cone ...; and the plane that produces a parabola is parallel to the =
side
of the cone...." from =
http://mathforum.org/library/drmath/view/53919.html
This is from the Math Forum site of Drexel University.

----- Original Message -----
From: "Michael McDowell"
> You may wll be right about the definition of a hyperbola. I
> certainly claim no expertise in solid geometry. My source was the
> entry in the 1986 edition of the Little Oxford Dictionary which is
> concise though according to you incorrect.
>
> "However, I think the Hyperbola is 90 degrees to the plane of
> the base...whilst the Parabola is some angle other than
> 90...without it mattters much what the angle of the sides
> are of the Cone in question."
>
> Little Oxford:
> Hyperbola- curve produced when cone is cut by plane making larger
> angle with base than side of cone makes.
> Parabola- plane curve formed by intersection of cone with plane
> parallel to its side

### Logan Oplinger on sat 1 feb 03

John,

My appologies for comming in late on this post. There are a few graphing
calculators you can access online. They will graph hyperbolic as well as
other equations. Doing a search with Google I found several that allow you
to vary the values of an equation and see the results almost
instantaneously!

Two sites that I found that I like are:

http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?
site=mathcom&s1=graphs&s2=equations&s3=basic

and:

http://www.nsci.plu.edu/~jkim/GCalc/resources.html

For the first one, enter the equation in the form x^2-y^2=1.
The "^" caret symbolizes raising x and y to some power, in this case, to
the power of 2. Then set the range for x and y. The best I've found is x
from -10 to 10, and y from 0 to 10. This will give you the upper half of
the graph for this equation. By resetting the range to a lower value, the
magnification will increase, as if moving closer to the bottom of the curve.

Then just print the page, and enlarge the curve portion on a photocopier.

The second site has several graphers that you will have to experiment with.

Logan Oplinger
Another Pacific Island
Latitude: 13.5 Longitude: 144.7

On Wed, 29 Jan 2003 00:54:22 -0600, John Rodgers
wrote:

>I have a commission to make a number of bowls that are to be hyperbolic
>in form and are/should be nearly mathematically correct. I looked
>hyperbola's up in a math book and I'm definitely not up to the
>calculations.
>
>Question: Can anyone tell me how to construct with pencil, pins,
>string, triangle, and straight edge, hyperbolic curves that I can draw
>out on paper or plastic board or something, in order to create and cut
>out some profile templates I need for the bowl making?
>
>Would appreciate any help at all on this one.
>
>Thanks,
>
>John Rodgers
>Birmingham, AL
>
>___________________________________________________________________________
___
>Send postings to clayart@lsv.ceramics.org
>
>You may look at the archives for the list or change your subscription
>settings from http://www.ceramics.org/clayart/
>
>Moderator of the list is Mel Jacobson who may be reached at

### Sabine Wolf on sat 1 feb 03

Hi!

> Question: Can anyone tell me how to construct with pencil, pins,
> string, triangle, and straight edge, hyperbolic curves that I can draw
> out on paper or plastic board or something, in order to create and cut
> out some profile templates I need for the bowl making?

I found a way to draw a hyperbel based on the feature of a hyperbel that the
distances of every point on the curve to two fixed points have a constant
difference. A string is attached to the end of a ruler the other end to one
of the two points. The ruler turns around the other point and a pencil is
hold against the string and the ruler to draw the hyperbel. I scanned a
picture of this: http://www.lythande.de/hyperbeldrawing.jpg .
There are two kinds of 3d hyperboloid objects, the first one can be
constructed by rotating a slanting line and has the form of a cooling tower
of a nuclear plant (http://www.lythande.de/hyperboloid1.jpg ), the second
one is bowl-like (http://www.lythande.de/hyperboloid2.jpg ). My idea is now
for a bowl to form the second kind of hyperboloid with the the help of the
first one.If you put a heap of clay on the wheel, turn the wheel and rotate
slowly over it a straight wire (the wire forms the first kind of
hyperboloid), the wire should cut the second form out of the clay. Now you
can put some plaster over it to cast hyperboloid bowls.
(http://hyperbelcut.gif)

Tschau,
Sabine

### Sabine Wolf on sat 1 feb 03

> (http://hyperbelcut.gif)
http://www.lythande.de/hyperbelcut.jpg of course.