# Special Plane Curves: What's New 1996 – 2007

## 2007-10

Much update on my plane curve website: Cross Curve, Bezier Curve, Astroid, Deltoid. Each with several new constructions in GeoGebra.

Egadz, this year is a decade anniversary of my plane curve website. The site started in 1996, with most time put in 1996 and 1997. The Mathematica programs, which makes up most graphics of the site, is written in about 1994 – 1996.

As i start to rework the site now… i'm starting to add proofs of many of the properties. Finding proofs using synthetic geometry on many of the curve's constructions and Euclidean properties won't be easy.

I was re-working the astroid curve construction. For example, it can be defined as a circle of radius 1/4 rolling inside a circle of radius 1, and also as a trammel. I was trying to find a synthetic geometry proof of the connection of the two. But after 1 or so hour of eyeballing, am fruitless. I'm guessing it'd be more difficult, to find a synthetic proof of the construction of its center of tangent circle. Offhand am not sure it can be done, since the concept of curvature is not that much in the realm of Euclidean constructions.

Was trying to create a GeoGebra file for the parallel ray caustic of deltoid. (which envelopes the astroid) However, i noticed that GeoGebra lacks the feature of make envelope of lines. (i.e. draw ALL lines without user having to drag things.) GSP can do it.

also needs to do a generic proof on all hypocycloids and epicycloids, relating to their construction properties, in one shot.

## 2006-08

There are about 7 Mathematica packages, and there are about 95 Mathematica notebooks on this site. I've made them all freely available. The condition to use is that if you use it, please give credit and link back to the site. Thanks.

You can download the dozen or so Mathematica packages here: Mathematica, and you can download the 90 or so notebooks on various pages on this site.

If you would like to donate, please go to paypal.com and send money to xah@xahlee.org. Any amount counts.

## 2006-07

updated two sections: Coordinate Systems and Coordinate Transformation and Vectors.

## 2006-06

Major facelift of the site. Now, the Front Page has icons for each curve.

Also, created a [http://xahlee.org/xahlee_org/SpecialPlaneCurves_dir/img_index.html Image Index] that has thumbnails to almost all illustrations on the site. (the page is 5 megabytes and may take a while to load.)

Added a new construction to the ellipse page

Added Thales's theorem and others to the circle page.

## 2005-12

Added a photo of spiral at the: spiral photos page.

## 2005-08

Added a photo of a cathedral showing its use of geometry, at the curlicue page

## 2005-06

A large update. The following pages of surfaces have been updated, mostly added Mathematica notebook, and Graphing Calc files, and also cleaned up writings a bit: helicoid-catenoid, Scherk, Boy Bryant-Kusner surface, cross-cap, Steiner surface, Kummer surface

## 2005-05

Updated: coordinate system

Added a movie file and gcf file to the Barth Sextic surface.

added Mathematica and Graphing Calc files to: Kuen surface, breather surface

## 2005-04

2005-04-24: Added a Photo gallery of reflecting Disks, linked at the bottom of Parabola page.

2005-04-24: Added a Mathematica file and Graphing Calculator file to the Breather surface of constant curvature. (K=-1)

## 2005-03

Since 2005, there are about 3000 unique visitors each week. Yay!
(with this command:
`grep 'GET /SpecialPlaneCurves_dir' weekly.log | grep 'html HTTP' | awk '{print $1}' | sort | uniq | wc -l`

2005-03-17: added some photos of curlicue. See curlicue page.

2005-03-14: added a theorem and 3 illustrations to Parallel Curves page. That is: the locus of cusps of parallel curves is the original curve's evolute.

2005-03-01: a new page on Curvature

Slight style change on how the site looks. HTML <center> tags are removed for simpler code and look. The HTML DOCTYPE has upgraded from version 3.2 to 4.01. Style may be enhanced with style-sheets in the future.

## 2005-02

2005-02-26: a new page on Cusp

2005-02-17: Each day, I'll post snippets of info or commentary on curves and surfaces in the Curves ＆ Surfaces mailing list

## 2005-01

• 2005-01-23: Mike William sent me several parametric formulas for modeling various seashells. See updated Seashell page.

• Added a draft version of Polynomial page.

## 2004-12

Added construction for Lemniscate of Gerono.

2004-12-17: Mathematica notebooks with graphics and animations can be downloaded at: Mathematica Packages page.

Added draft version of: hyperbolic trig functions, Lissajous.

2004-12-13: Added: string-art animation at at the bottom of Parabola page. Added trifolium parallel movie and some artistic images at the bottom of parallel page. Added real_roses.mov and Cotes's Spiral at Rose page.

Added two new fun animations of "spiral-graph" at the bottom of: hypotrochoid page

2004-12-07: now you can order the over 100 Geometer's Sketchpad files. Each one is self-contained. You can also order separately the 8 Mathematica packages and 80 notebooks on plane curves. Please see the order page for detail.

2004-12-07: Added dynamic plane geometry java applet for: Bezier Curve, Cross Curve, Piriform, Quadratrix of Hippias. Look for the link right under the construction illustration where the link starts with _z_. Click on that will open a new window with a Java applet that you can trace the curve or see the family of the curve dynamically.

2004-12-06: added the Bezier Curve

2004-12-03: added the Cross Curve, Piriform. Also updated many other pages.

2004-12-03: The CD-ROM version of Robert Yates's book *Curves and Their Properties* are updated with better scans and HTML table of contents.

This website is beginning to use a new software for plane geometry instead of Geometer's Sketch Pad. The new software is Ruler and Compass. It is written in Java. It is free. It is available for Mac, MS Windows, or Linux. This will make readers of this site more readily enjoy mathematics without needing to buy extra software. The software can be downloaded at http://mathsrv.ku-eichstaett.de/MGF/homes/grothmann/java/zirkel/index.html

## 2004-11

2004-11-25: added draft versions of Isoptic and Orthoptic, Radial, Roulette and Glissette, Line,

2004-11-24: updated the astroid page, with proof of the various equations, graphing calculator files, some curve construction properties, and PDF file.

2004-11-24: cleaned up the scans *Curves and their Properties*. Now each page is scanned by itself, fits the screen, much readable, and smaller file size. See Curves and their Properties Sample Pages

NEW! all 77 QuickTime animations on this site is now made available free.

Please consider making a donation, please use paypal.com to send me money at “paypal at xahlee dot org”. Or, send money directly to:

Xah Lee P. O. Box 390595 Mountain View, CA 94042-0290

I appreciate your help.

If you have emailed to me in the past months and didn't get reply, probably because I missed it. I'm getting over a hundred junk emails a day. Please send again with subject line "plane curves".

Added a new curve: Clothoid.

Added material for Cassinian Oval. In particular, the parameters corresponding the torus cut.

Modified the catenary page, added explanations about two surfaces being isometric.

Added a new curve: Folium of Descartes. A draft version.

2004-11-08: added a new Seashells photo exhibition. Annotation will be coming in the coming months.

Now this website has a mailing list discussion forum. Please join in to discuss all matters related to curves and surfaces. To subscribe, send a email to curves_surfaces-subscribe@yahoogroups.com.

In the next 2 months, there will be a update at least once per week. And, each curve will come with a Adobe PDF file, and there will be also Graphing Calculator files added to the site.

## 2004-04

2004-04-20: Nothing new, but there's now a math Surface Gallery with about 60 surfaces, most of them being surfaces in differential geometry. Each surface can be rotated in the browser.

## 2003-06

2002-06-18: New! Now for CD-ROM , you can choose to download them instead.

## 2003-05

2002-05-26: Added and updated a bunch of photos on: catenary, conic sections, parabola pages.

The Geometer's Sketchpad, Cabri Geometry, QuickTime movies files, and my Mathematica notebok files are back on the conic sections, hyperbola, ellipse, parabola pages. On other pages, these files are in the CD-ROM version only.

## 2002-08

2002-08-25: The Geometer's Sketch Pad files, Mathematica notebooks, QuickTime movies files has been added to the following pages now: conic sections, cycloid, deltoid, ellipse.

A subscription based system should soon come to this site. Much addition is in the planning..

2002-08-08: A new book: Apollonius of Perga; Conics, Books I-III. Published by Green Lion Press in 1998, edited by Dana Densmore. This one is only $23, much cheaper than the $160 one (C.J. Toomer, 1990, Springer Verlag).

## 2002-07

2002-07-28. This site has moved from one machine to another. If you find missing files, please let me know. (write to me at xah@xahlee.org. Thanks.)

## 2002-05

more photos. catenary page has many new photos of the St. Louis Gateway Arch. caustics page has new photos of cups under light. mathematical material will be coming to the inversion page soon.

## 2002-04

2002-04-30: Adde 3 scenes from the movie “Sex, Lies and Videotape” (1989) illustrating projection, at the Conic Sections page.

2002-04-21: Few more edits. The sine curve page has a photo of a form modeled after the surface Sin[x]*Sin[y], as well as a new graphics at the top. The Seashell page is again updated with new photos. A draft of coordinate systems page is up. That page should talk about the basics of different coordinate systems, and how to convert and apply transformations among the different systems. For example, given a formula for a curve in some coordinate system, how to scale, rotate, translate the curve. Also added a photo of a cup of milk showing the light rays reflection at the bottom of caustics page. The hyperbola page has a photo of a rotating dice.

2002-04-08: a new catenary-helicoid morph movie added to the catenary page. (3.6 MB)

2002-04-02

I now offer price money for math problem solvers! See Seashell

This week only, 50% discount on my site CD. See CD-ROM page.

More photos added. Look at this spiral page: Spiral

For those who have bought CD-ROM of my site, just $10+shipping for the latest CD.

I'm starting to collect quality photos of math related buildings or things. Right now i need:

+ Quality photo of St. Loui's Gate. (See: catenary) + Quality photo of Windmill, may it be classic style in Holland, or the modern windmill electricity farm. see http://www.windmillworld.com/ + Quality photo of whirlpool. + Quality photo of tornado. + Quality photo of violin's neck. See Spiral. + Quality photo of a violin's sound hole. (shaped like a integral symbol)

## 2002-03

Ah, this month seems to be the photo shootout month. I added quite a lot curve related photos.

2002-03-30. Added a brand new curve page: spiric Sections. No graphics so far. Much work needs to be done there.

2002-03-29: The **Cassinian Oval** has been updated. Added several Mathematica notebooks. Yes, you can download them without the File Not Found. Also updated the graphics on that page.

2002-03-29: Added a Infinity symbol to the bottom of Lemniscate of Bernoulli page.

2002-03-19: Added a complete **Seashell** page. This page is pending to be cleaned up, with explanations.

2002-03-09: add several new graphics to the Archimedean spiral page. A photo of architecture's column decoration using the lituus curve will be added soon.

2002-03-09: add a several spiral of nature photos at the bottom of equiangular spiral page. Including a spiral cauliflower and a fern. (thanks to Dror Bar-Natan)

2002-03: added a catenary chain toy model photos to the bottom of catenary page. (thanks to Dror Bar-Natan)

2002-03-09: added a Radio Dish photos to the bottom of parabola page. (thanks to Don Baccus' photo gallery: http://donb.photo.net/random_walks/vla.html)

2002-03-19: added the Wired Magazine May 2000 issue cover, in this page below. (scroll down to see it.)

2002-03-19: also updated the cassinian oval page. Added how the Cartesian equation is derived.

## 2002-02

Added a link to the cycloid page. Joseph Portney, A somewhat detailed exposition on the property of cycloid being the curve of fastest decent. (called brachistochrone) http://www.navworld.com/navcerebrations/millennium/millennium.htm

02-20: Added two more new books to the references page page. They are: Geometric Constructions by George Martin, and Plane Algebraic Curves by Gerd Fischer.

## 2002-01

Good new year.

My new year resolution is to get back into mathematics, and bring this site alive with quality mathematics. May it be theorems, curve properties, architectural photos, or illustrations. I'm commencing back into studying geometry on my spare time. Right now complex analysis.

Photos of hyperboloid architectural structures has been added to the bottom of the hyperbola page.

added a book recommendation _Complex Numbers and Geometry_ by Liang-shin Hahn, on the references page. Also new is a website that contains two Java Applets on conic sections to the bottom of Conic Sections page. (search for "Slavomir")

This site has been free of charge since early 1996, without any advertisement, and has been in today's state since early 1998. Since then, development has stopped for lack of time as well as lack of commercial support. In order to bring you more quality material for this site, i would like to obtain commercial support to pay the costs. The main materials of the site are still free for many poor students, but i now have removed supporting files from the online version. They are the 115 Geometer's Sketch Pad files, 23 Cabri II files, 71 Mathematica notebooks, and 82 QuickTime movie files. In balance, i'm slashing $20 off the full featured CD-ROM version.

If you are wondering why so many of the websites are no-longer free these days, please see this comic by Scott McCloud for background: http://www.scottmccloud.com/comics/icst/icst-5/icst-5.html or this article on HowStuffWorks.com: http://www.howstuffworks.com/penny-per-page.htm/printable. I have a daily job. I still love math for the sake of it. However, there are many other things i enjoy doing and are rewarding to me other than creating this site for public for free.

## 2001-12

Eric W. Weisstein and Wolfram Research Inc's MathWorld.com is back! For those who are not familiar with it, it is one of the largest, and most easy-to-read college level encyclopedia of mathematics, and is now available back on the web free! This is great news for the world.

For those who know the story of Eric Weisstein's work, Eric has wrote a detailed account here:
~~http://mathworld.wolfram.com/erics_commentary.html~~
. Notice the grand philanthropic tone. I like to make a few quick notes here: Eric's write-up is the sour grapes syndrome. Note that a site like Mathwold.com is tantamount to astronomical amounts of advertising dollar for Wolfram Research Inc., and if it isn't CRC Press's help of publishing it in the first place, then Wolfram Research may not have hired its author, nor as much people would help. Eric mentioned none of these. In the end, according to Eric's write-up, Wolfram Research paid CRC for a settlement outside of court, as well as giving CRC certain rights. It is good to see justice done. This is a story of greed and lies and ignorance. Weisstein and Wolfram wants to rob the name off of CRC Press, CRC Press is perhaps incompetent and ignorant of internet nature (according to Eric's article), and the greedy populace demand it be back free of charge, all in the name of altruism. It is also a good thing that America is a country of freedom, allowing conflicts to settle in some civilized and fair way. At the end, we consumers benefit from capitalism.

When i have some time, i'll may write a more detailed account of it.

## 2001-11

Much expanded the references page, to include about 7 more undergraduate text books on curves and geometry.

The CD-ROM of this website is reduced in price till November 25th.

The hyperbola page has been updated with a photo of a basket made of sticks, forming a hyperboloid. (bottom)

## 2001-07

A photo of a car's headlight added to the optical section of Parabola page.

I found that the entire content of 〈The CRC Concise Encyclopedia of Mathematics〉 by Eric W. Weisstein amazon available for download on the web. Dying students and unscrupulous academics, go search! Meanwhile, let's not forgive the greed and ignorance of the mob, who would reflexively cry greed and point fingers at some apparent big-name publisher if they can no longer get something that was given to them freely by some source.

07-12: PS the complete CRC book downloadable bootleg mentioned above
is gone. It was at [censored by request]. Although the CRC Concise
Encyclopedia of Mathematics bootlet on the site is gone, you can enjoy
many math related online books linked from that site, as well as the
benevolence of CRC press who gave permission for the entire content of
〈Handbook of Applied Cryptography〉 to be downloadable freely
online. Meanwhile, a mirror of Eric's “encyclopedia” is still
available on the web, badly maintained with lots of broken images. For
those of you who are interested in issues of morality and altruism, i
would recommend the book
〈Basic Economics〉 amazon, by Thomas Sowell. Failing on money, read the website
http://www.capitalismmagazine.com/.
If you are concerned about CRC's lawsuit case, study the documents available at
~~http://mathworld.com/docs/legal.html~~
and tell those totally oblivious to stop crying “greed of the big corps”.

Back in 1996, a demo version of Geometer's Sketchpad is readily available for download from the publisher's site. The demo version does not allow saving files, export, or printing, but is otherwise complete. I began to use it massively on this site. The idea is that i wanted to use something that everyone can readily benefit, even those who may not have money to pay for a full version of such software. I noticed today that the demo version is no longer readily available. This may have happened few months back. One has to be a faculty member to request a demo download, and the use of the software other than evaluation purposes is prohibited. This ruined my original plan, as now great many people won't be able to use the over hundred GSP files on this site. If you are interested, you might want to write to the publisher of Geometer's Sketchpad, to persuade them to make their demo version readily available for the masses, even introducing some 15 minute limit per session. I might strike a deal and become a re-distbributor of GSP along with the sale of my CD, but still a free demo version could benefit all parties involved.

## 2001-04

Thank you for your support. I've added sample scanned pages of Robert C Yates' book *Curves and Their Properties*. Including the complete chapter on Conic Section and Trig functions. See curves_r_yates_sample.html

Added one photo and a illustration of a solar energy dish at the Parabola page's related web sites section.

## 2001-02

This site is finally on CD! See details at my CD-ROM sell page.

Added two books to the printed references page. Geometry of Curves by John W. Rutter, and Elementary Geometry of Algebraic Curves by Chris G. Gibson.

## 2000-12

This site will be pressed on CD in the next 2 months. The CD will include the complete site and Mathematica programs, and also the complete electronic copy of Robert Yates' book 〈Curves and Their Properties〉 amazon (1952). Send me email if interested. The site itself will always be available free on the web.

The commercial right to use LiveGraphics3D (See: http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/index.html) is owned by Wolfram Research Inc. I have removed its use on this site because at this point I do not wish to use it commercially.

Wolfram Research Inc.'s MathWorld.com has been shutdown around October because of copyright issues with the original publisher CRC Press. It is bad news for math community because it is a significant high quality math resource online. Hopefully the issue can be resolved soon and the website back online. See more commentary from me on my related websites page.

## 2000-09

Added the saga on the history of mathworld.com and me. Find it here: related websites. Added book cover pictures to the printed references page.

## 2000-04

This site now has its own domain name “xahlee.org”. This means that the URL starts with 〔http://www.xahlee.org/〕 instead of the old 〔http://www.best.com/~xah/〕. Since this February, this site has been hosted by http://www.mathforum.com/. mathforum.com is a commercial organization dedicated to improving math education. They are hosting my site free of charge. They are perhaps the best math service and community on the net. One of their excellent service is their comprehensive annotated math site index. Check them out.

Other than the above news, this site did not change much. The usual maintenance has been lagged for a couple of months. URL has not been validated. I'll bring them up to date around the end of this month. A downloadable version is also coming soon. You need to email me if you really want it.

## 1999-12

Added a new curve Catenary that includes a dynamic model of catenoid. Added a dynamic model of the pseudo sphere to the tractrix page, and a dynamic model of the Dandelin Sphere for a ellipse cut to the conic sections page.

More small sites on curves.

- Hollister David. Images of moiré; patterns of circles and lines simulating conics. Including a nice small animation that simulate conics with changing eccentricity. The site includes some other very nice mathematical arts. http://clowder.net/hop/index.html
- Frederic Leymarie (leymarie@lems.brown.edu). A bit mathematical background on the "Ellipse seen as a circle" property. http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/Curves.html.
- Henrik Wann Jensen. Some computer simulated images of caustics. (lights reflected or refracted through glasses.) (1996) http://www.gk.dtu.dk/home/hwj/pictures/caustics.html and http://www.gk.dtu.dk/home/hwj/papers/gi96.html.
- Works by Sir William Rowan Hamilton (1805 – 1865) on optics and other math subjects. (a very large and serious site) Site created by David R. Wilkins (dwilkins@maths.tcd.ie) of Trinity College, Dublin. http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Optics.html.

## 1999-11

**News! 1999-11-21** I just spent several hours working on the astroid page. Added two sketchpad files and one diagram. This is about the first time in the past year that I worked on curves. Cheer for me and wait for more updates.

This month, a very import event took place in education: The complete Encyclopedia Britannica is on-line free, available at http://www.britannica.com/. The Geometry section of this encyclopedia along are perhaps better in both scope and quality than all geometry materials on the internet combined as of today. If you have read and understood all the geometry articles in Britannica, that would be equivalent to several post graduate degrees in various fields of geometry. The same thing can be said for other subject areas. In short, Encyclopedia Britannica is a summary of all human knowledge (minus only current research of last few years). Britannica contains a huge section on Geometry, and a section dedicated to special plane curves. Browse their site and use their search feature to find what you want. Here are some other new links:

- http://chronomath.irem.univ-mrs.fr/. This new site in French contains about 71 curves. Its author is Serge MEHL (szmehl@wanadoo.fr).
- I found this Italian page http://www.museo.unimo.it/theatrum/macchine/_00biblio.htm which is part of the The Virtual Museum of Mathematical Machines. This Italian page contains over a hundred animations that anyone can understand.
- This site in Italian is dedicated to conic sections. http://atene.provincia.parma.it/%7essrondan/coniche/index.htm It's about 30 pages and contains Cabri Geometry files and photographs or drawings or related history and links. Part of it contains Italian translations of my conics sections pages.
- http://www.ping.be/math/ is a fairly large collection of calculus tutorials. Many sections are related to plane curves. This particular page http://www.ping.be/math/curves.htm lightly discuss envelope of curves.
- A graduate student's work of 30 or so JavaSketchpad applets on various curves. A very interesting supplement to the materials on my site. http://oz.nthu.edu.tw/%7eg863204/GC/Dygeo.html.
- This site http://www.brown.edu/Students/OHJC/hm4/k.htm is a essay on the history of curvature. It is written by a student but well researched.
- Jan Wassenaar curve pages now has its own domain at http://www.2dcurves.com/. (it used to be at http://www.mirajan.demon.nl/curves/curves.htm). This site is perhaps 10 pages covering basic information on about 30 curves, grouped by degree.
- An excellent site with many illustrations on the applications of conic sections. http://www.camosun.bc.ca/~jbritton/jbconics.htm.
- Mathematical Association of America's MathLand article by Ivars Peterson, musing on billiard game played on a elliptical table, with illustrations. http://www.maa.org/mathland/mathland_3_3.html.

This site is getting about 12000 hits and 60 mega bytes of data transfer per day. My whole site is about 25 Mega bytes in disk space. Overall, it's hitting my Internet Service Provider's quotas. I am planning to register my own domain name, or possibly looking for a free host. I've been thinking to offer this whole site as one single compressed file to download for offline readings, or facilitate organizations that wishes to mirror. (again, please write to me before any public mirroring. Check my copyright page for detail.)

Here is a page linkToMySite.html that you can check out the 15000 links to my site. Thank you my readers for your support. Elementary mathematics (such as Calculus) is a very practical science. But I hope that more young students will appreciate the beauty of math. This must begin with the teachers in their attitude and enjoyment of their profession. As we mature into the info age, those cold, exotic, advanced, beautifully abstract, and far-reaching modern mathematics should become more popular.

November 5th was the Burn All GIFs Day. On that day, I had the following notice on my front page. I will convert all the gifs on this site to PNG format as soon as perhaps half of browsers will read that format. I'll perhaps put up a archived GIF version for download.

NOTICE: Today is the It is a special internet event where all freedom conscious developers are supposed to change all the GIF format graphics images on their site to some other format that's not GIF. I, the author of this site, am a freedom conscious software developer. I didn't have time to change the hundreds of GIF images on my site to other format Unisys (blue line on chart), once a well-known computer company, holds a patent on a method of data compression called LZW. Other, much better, methods of data compression are not covered by any patent. LZW is used in an obsolete graphic format called GIF, which many web sites use in order to be compatible with old web browsers. (http://corp2.unisys.com/LeadStory/lzw-license.html) Unisys is now demanding that web sites pay them $5000 or more to use these now-obsolete GIF graphics if the software originally used to create the GIFs was not covered by an appropriate Unisys license. The catch is that it appears to be difficult or impossible to get a Unisys license to use LZW in free software that complies with the Open Source Definition (http://opensource.org/osd.html) or in low-volume proprietary software. gd is an example of a package that can no longer support GIF because of Unisys's LZW licensing terms. The fact that Unisys was able to patent LZW is due to a flaw in
the US patent system that makes even
pencil-and-paper calculations
patent-able. You could violate some US patents just doing the
story problems in a math or computer science textbook!
The League for Programming Freedom
is working to fix the US patent system to harmonize it with those
of other, more sensible, countries. In the meantime, Unisys's actions
are "What does it have to do with me?" you ask? This may not be a relevant issue to your life. But all human affairs has something to do with you one way or another. You are a human being, after all. Awareness is a form of knowledge, and the beginning of understanding. If one day you live in a freedomless society, it would be too late. |

## 1999-07

Just a note so that you know this site is not stale. I've added some few minor links on some pages. They are:

- Wilson Stothers. Wilson's Inversive Geometry. (Good) http://www.maths.gla.ac.uk/~wilson/cabripages/inversive/inversive0.html.
- Richard Barnes (student). Some analysis on n-ellipses. (the locus of points whose sums of distances to n fixed points are equal.) (dead: http://www.mindspring.com/~bifurcation/html/index.html) Permanent link added to the Ellipse page.
- Aidan Cully. A Java program that dynamically shows a the intersection of a plane and a cone. http://www.jalacy.com/conics/java.shtml
- This page contains some animation of mechanical devices tracing out some curves. http://personal.atl.bellsouth.net/atl/e/d/edwin222/
- A pretty java animation based on cardioid. Source www.bekkoame.ne.jp

Two curve pages that has been minor updated in August 1998 but not uploaded. My mistake. They are the Equiangular Spiral and Sinusoid and Trig functions pages. The minor changes are trivially expanded history section on the spiral and a Mathematica v.3 notebook, and two links on the Trig page.

## 1999-03

Again, I didn't have time to add much materials on the site. There will be a CD that contains the complete site. The CD is very much delayed. For up-to-date info, check (dead: http://rsp.math.brandeis.edu/MathCD.html).

Now the conic sections page is updated so that all Mathematica files are in version 3.0 notebook format.

This month introduces the use of LiveGraphics3D. It is a Java applet that allows dynamic rotation of 3 dimensional models. Conics Dynamic Models is the first page that uses it. For more info on LiveGraphics3D, see it's intro page liveGraphics3DIntro.html.

200411: note: LiveGraphics3D is no longer used on this site.

## 1998-08

I've spend two weeks working on curves. There are three new pages and minor updates to the whole site. Again, I'm seeking feedback. Please send it to me. Here are the additions:

- A new page: epicycloid and hypocycloid.
- A new page: rose. Also, there's this large sized image gallery epiHypocycloidEpiPedal.png (1101x1801, 130 k) showing 40 epi/hypocycloids and roses, where the latter is the a pedal curve of the former.
- A new page: conchoid of Nicomedes. This is one of the oldest curve.

Possibly more additions this month.

## 1998-07

For those waiting for the CD, it's still in the making. I'll announce it as soon as it is out. At times there may be no news for months, but all external links are validated by machine regularly. Old links are removed as appropriate. New web resources I found are added in. If the new site is not large, there won't be announcement here. Please let me know if you find a new web resource, even a single page.

I've been running this website for 2 and half years now. I have not been actively involved for over a year, but recently I've been doing more work on it. On the whole, I have rough ideas of what I want to add to the project. However, I would very much appreciate your specific feedback. What do you like to see? Do you want in-depth info or more curves? Do you want more mathematical substance? Do you want more accurate historical info? Do you really want a search form? Do you want more Mathematica notebooks, sketchpad/Cabri, or QuickTime movies? Do you want suggested exercise? How are you using this site? I'd very much like to hear from you. Note that even a trivial remark is a valuable statistical data for me. All comments, including negative ones, are welcome. Send it to me at xah@xahlee.org. If you have time, tell me a bit about your background. (➢ for example: teacher or student? at what level?)

There are few updates:

- A new curve page: cissoid.
- Updated the Cissoid of Diocles page. Including somewhat expanding notes on history and a paragraph on using cissoid to double the cube.
- A new version of my plane tiling package. This new version has some minor bug fixes and additions. Recall that the package's functionality is to facilitate design and generation of any periodic plane pattern. Get it at my Mathematica Packages page.
- As I've mentioned before, all version 2 Mathematica notebooks on this site is slowly being replaced by version 3 notebooks, I've put all my version 2 notebooks on this site into a compressed archive for those v.2 users, downloadable at my Mathematica Packages page. Notebook on some curve pages are already in version 3 format. (i.e. ending with .nb)
- In the related websites page, I've added a section on curve plotting softwares.

## 1998-04

This site will be published on Compact Discs by two publishers. One is a CD prepared for mathematicians. This CD collects past issues of various math journals, useful math softwares, and other relevant materials. This CD will also include my wallpaper groups exposition (The Discontinuous Groups of Rotation and Translation in the Plane) and Math Graphics Gallery (Mathematics Graphics Gallery). For further info on this CD, please see
~~http://rsp.math.brandeis.edu/MathCD.html~~.

Another CD that will contain this site is Cryptosystems Journal (http://ourworld.compuserve.com/homepages/crypto/), privately published by Tony Patti. Right now I don't have much info on either. I will announce more info a few months later. I expect the availability date to be sometimes late in 1998.

All my Mathematica packages are now converted to v.3.0 format. The older version are no longer maintained but can still be downloaded by those who use v.2.x. Check out the new versions at Mathematica Packages page.

Here are two minor links added in March.

- Euler's Spiral, American Math Monthly v.25 (1918). Available at http://ourworld.compuserve.com/homepages/crypto/amm.htm.
- Simon Plouffe. On caustics, cardioids, and modular arithmetic. Significant mathematical info. http://www.lacim.uqam.ca/plouffe/cercles/cercles.html (the link is dead as of 1999-10)

## 1998-02

This site is now under maintenance mode. There won't be any significant additions. I will keep adding or updating links, and add minor paragraphs or files. I may revert to development mode sometimes in the future. If I do, I will probably extend materials on existing pages of curves instead of adding new curves, because I believe depth is much valuable than breadth. There are already many popular math books and resources that introduce these curves.

My Mathematica Packages page will be continuously updated with new versions of packages or maybe even new packages. Check back few times a year.

Few new sites on curves are:

- Afred Gray's Curves and Surfaces Gallery 〔
~~http://math.cl.uh.edu/%7Egray/~~〕, from his second edition of 〈Modern Differential Geometry of Curves and Surfaces with Mathematica〉. This site is about 6 long pages of images of various curves and surfaces. - A personal page of Jan Wassenaar (postmaster@mirajan.demon.nl) at http://www.mirajan.demon.nl/curves/curves.htm. This site is about 5 pages covering basic information on about 30 curves, grouped by degree.
- Introducing Curves, by Mathematician C.T.J. Dodson. (dodson@indigo.ps.umist.ac.uk) This site is about 6 pages. It is a mathematical introduction to curves. A background of multivariable calculus is required. Not for general readers.
- 〈Earliest Known Uses of Some of the Words of Mathematics〉
~~http://members.aol.com/jeff570/mathword.html~~, a personal site maintained by Jeff Miller (JeffM@sanctum.com). This site is 5 long pages. You can find brief historical backgrounds on the naming of most curves. - The Virtual Museum of Mathematical Machines, from University of Modena and other organizations. This sites comes in English and Italian version. It contains photographs and diagrams of mechanical devices that traces various curves.

Besides web awards, over the past two years I've received many encouraging notes and kudos from readers. Here I wish to thank you all. I will be adding more geometry related subjects to my web pages.

## 1996-02

This plane curve project was put on the web.