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24 Ansichten42 Seiten1 Letter of the Editor
2 Editorial - Preview
Roland Schröder
3 Building Towers
Josef Böhm
10 Another Game! – Another Pattern?
12 Franz Jauk’s Statistics Problem
Piotr Trebisz
16 Fresnel Integrals and Clothoids
Josef Böhm
23 Natural Equations of Curves
29 A Student’s Problem with Homework files
31 ITERATES Again!
33 Christoph Küderli’s “Error Trapping Error”
David Halprin
36 Two good Turns
39 Christine Kova’s Students

THE DERIVE - NEWSLETTER #90

© © All Rights Reserved

PDF, TXT oder online auf Scribd lesen

1 Letter of the Editor
2 Editorial - Preview
Roland Schröder
3 Building Towers
Josef Böhm
10 Another Game! – Another Pattern?
12 Franz Jauk’s Statistics Problem
Piotr Trebisz
16 Fresnel Integrals and Clothoids
Josef Böhm
23 Natural Equations of Curves
29 A Student’s Problem with Homework files
31 ITERATES Again!
33 Christoph Küderli’s “Error Trapping Error”
David Halprin
36 Two good Turns
39 Christine Kova’s Students

© All Rights Reserved

Als PDF, TXT **herunterladen** oder online auf Scribd lesen

0 Bewertungen0% fanden dieses Dokument nützlich (0 Abstimmungen)

24 Ansichten42 SeitenTHE DERIVE - NEWSLETTER #90

1 Letter of the Editor
2 Editorial - Preview
Roland Schröder
3 Building Towers
Josef Böhm
10 Another Game! – Another Pattern?
12 Franz Jauk’s Statistics Problem
Piotr Trebisz
16 Fresnel Integrals and Clothoids
Josef Böhm
23 Natural Equations of Curves
29 A Student’s Problem with Homework files
31 ITERATES Again!
33 Christoph Küderli’s “Error Trapping Error”
David Halprin
36 Two good Turns
39 Christine Kova’s Students

© All Rights Reserved

Als PDF, TXT **herunterladen** oder online auf Scribd lesen

Sie sind auf Seite 1von 42

ISSN 1990-7079

USER GROUP

C o n t e n t s:

1

Editorial - Preview

Roland Schrder

Building Towers

Josef Bhm

10

12

16

23

29

31

ITERATES Again!

33

Piotr Trebisz

Josef Bhm

David Halprin

36

39

June 2013

D-N-L#90

Information

D-N-L#90

Then you possibly will like

Deutsche Ausgabe: Pasta und Design, Springer Spektrum, ISBN 978-3-8274-2978-0

Buon Appetito!

D-N-L#90

p 1

Liebe DUG-Mitglieder,

begonnen und da will ich gleich allen jenen,

die schon oder demnchst Ferien haben, einen schnen Sommer und erholsame Tage

und Wochen wnschen.

Summer Holidays have started in Austria. Id like to wish all of you already being

in holidays or waiting for them a wonderful

summer and relaxing days and weeks.

Schulden einlsen. Wie so oft haben auch

kurze Anfragen und Beitrge ihre Eigendynamik entwickelt und zu weiteren Untersuchungen angespornt. Aus Building Towers

habe ich die Aufforderung zu eigenen Experimenten ebenso gerne aufgegriffen wie

mich Fresnel Integrals ... an frhere

DNL-Beitrge zur Cornu Spirale und zu den

natrlichen Gleichungen erinnert haben.

Da entwickelte sich gleich ein lebhafter

Briefwechsel mit David Halprin (Two good

Turns). Davids interessante Laufbahn als

Mathematiker verdient einen eigenen Beitrag im nchsten DNL.

Sie finden eine kurze Anmerkung von

Erik van Lantschoot zu seinem Brsseler

Tor in Dendermonde (DNL#89). Krzlich

hat er eine weitere Ausarbeitung basierend

auf seinen intensiven Nachforschungen in

historischen Quellen geschickt.

Christine Kova besuchte vor einigen

Jahren einen DERIVE & GeoGebra Workshop Fadengrafik. Spter schickte sie

Bilder mit Mustern, die ihre Schlerinnen

nicht nur am PC sondern auch mit Holz, Ngeln und Faden erzeugt haben. Krzlich erhielt ich wieder Bilder, nun wagten sich die

Schlerinnen bereits in den 3D-Raum. Ihre

Skulpturen gegossen in Acrywrfel werden

sogar in einem Wiener Museum ausgestellt.

Es ist schn, dass ein MathematikWorkshop so nachhaltige Wirkung zeigen

kann.

Herbert Nieder verdanke ich den Hinweis auf das Pasta-Buch. Obwohl ich kein

besonderer Nudelfan bin, habe ich doch

Appetit bekommen, die Agnolotti und Gnocchi zu modellieren. Im Internet knnen Sie

noch einige Beispiele finden.

How so often short requests and mails develop their own dynamics and inspire for

further investigations. I took seriously Rolands invitation in Building Towers to undertake own experiments and I was reminded on earlier DNL-contributions on the

Cornu Spiral and on Natural Equations

reading Fresnel Integrals .... An animated

communication with David Halprin followed

(Two good Turns). Davids rich career as

mathematician deserves an own contribution in the next DNL.

You can find a short note from Erik van

Lantschoot concerning his Brussels Gate in

Dendermonde (DNL#89). Just recently he

sent another paper based on his intense investigations of historical resources.

Christine Kova attended a DERIVE &

GeoGebra workshop Stitching Patterns

several years ago. Later she sent pictures

with patterns produced by her students not

only by means of mathematics but with

wood, nails and threads. Three weeks ago I

received some new pictures. Her students

transferred the graphics into 3D-space.

Their sculptures were cast in acryl cubes.

Some of their works of (mathematical) art

are exhibited in a Viennese museum. It is

great that a mathematics-workshop can

achieve such a sustained effect.

We owe the note on the Pasta-book to

Herbert Nieder from Hamburg. Although I

am not a noodles fan at all, I got some appetite modelling Agnolotti and Gnocchi. You

can find some more examples in the web.

Viele Gre, Best regards

http://www.austromath.at/dug/

p 2

The DERIVE-NEWSLETTER is the Bulletin of the DERIVE & CAS-TI User Group.

It is published at least four times a year

with a content of 40 pages minimum. The

goals of the DNL are to enable the exchange of experiences made with DERIVE,

TI-CAS and other CAS as well to create a

group to discuss the possibilities of new

methodical and didactical manners in

teaching mathematics.

Editor: Mag. Josef Bhm

DLust 1, A-3042 Wrmla

Austria

Phone:

++43-(0)660 3136365

e-mail:

nojo.boehm@pgv.at

Preview:

D-N-L#90

Contributions:

Please send all contributions to the Editor.

Non-English speakers are encouraged to

write their contributions in English to reinforce the international touch of the DNL. It

must be said, though, that non-English articles will be warmly welcomed nonetheless. Your contributions will be edited but

not assessed. By submitting articles the author gives his consent for reprinting it in

the DNL. The more contributions you will

send, the more lively and richer in contents

the DERIVE & CAS-TI Newsletter will be.

Next issue:

September 2013

Some simulations of Random Experiments, J. Bhm, AUT, Lorenz Kopp, GER

Wonderful World of Pedal Curves, J. Bhm, AUT

Tools for 3D-Problems, P. Lke-Rosendahl, GER

Hill-Encription, J. Bhm, AUT

Simulating a Graphing Calculator in DERIVE, J. Bhm, AUT

Do you know this? Cabri & CAS on PC and Handheld, W. Wegscheider, AUT

An Interesting Problem with a Triangle, Steiner Point, P. Lke-Rosendahl, GER

Graphics World, Currency Change, P. Charland, CAN

Cubics, Quartics Interesting features, T. Koller & J. Bhm, AUT

Logos of Companies as an Inspiration for Math Teaching

Exciting Surfaces in the FAZ / Pierre Charlands Graphics Gallery

BooleanPlots.mth, P. Schofield, UK

Old traditional examples for a CAS whats new? J. Bhm, AUT

Truth Tables on the TI, M. R. Phillips, USA

Where oh Where is It? (GPS with CAS), C. & P. Leinbach, USA

Embroidery Patterns, H. Ludwig, GER

Mandelbrot and Newton with DERIVE, Roman Haek, CZK

Tutorials for the NSpireCAS, G. Herweyers, BEL

Some Projects with Students, R. Schrder, GER

Dirac Algebra, Clifford Algebra, D. R. Lunsford, USA

Treating Differential Equations (M. Beaudin, G. Piccard, Ch. Trottier), CAN

A New Approach to Taylor Series, D. Oertel, GER

Cesar Multiplication, G. Schdl, AUT

Henon & Co; Find your very own Strange Attractor, J. Bhm, AUT

Rational Hooks, J. Lechner, AUT

Simulation of Dynamic Systems with various Tools, J. Bhm, AUT

Blidor, E. v. Lantschoot

and others

Impressum:

Medieninhaber: DERIVE User Group, A-3042 Wrmla, DLust 1, AUSTRIA

Richtung: Fachzeitschrift

Herausgeber: Mag. Josef Bhm

D-N-L#90

p 3

There are many resources dealing with Wythoffs Game. Roland presents this game

together with other ones which show a fascinating connection to the Golden Section.

Building Towers

Roland Schrder, Germany

Which numbers give 1 as sum and -1 as product?

has raised to fame not only in mathematics and got like other famous numbers like and e its own

symbol . We meet the irrational number in many mathematical contexts but also in nature,

architecture, arts and music. There are numerous resources in books and on the web under the keyword Golden Section. If you are lucky you can find a relation between games played with counters

and the number . One of them is Wythoffs Game named after its inventor Willem Abraham

Wythoff (1865 1939). These are the rules:

Two piles (= towers) of different number of counters are offered two players.

The players have to remove counters alternatively according to the following instructions:

- A player is allowed to remove the same number of counters from both piles,

- A player is allowed to remove as many counters as he likes from one pile.

- Winner is the player who removes the last counter.

Wythoff has analysed his game completely by himself. The winner has to know special pairs of

numbers which describe the number of counters forming the two piles which make sure that he will

win the game. Take the pair (1, 2) as an example. If a player is able to leave two piles with 1 and 2

counters then he will win. Wythoff 1 did not only give the pairs of numbers which should be known by

the later winner but he also presented the respective formula:

(FLOOR(n), FLOOR(n2)) with n N

Wythoff said that he had drawn the formula out of the hat. It was proofed by Harold Scott

McDonald Coxeter 2 (1907 2003).

First of all we would like to build the DERIVE-function win_move(x,y) which delivers to a given

pair (x,y) of pile heights the respective pair according to Wythoffss formula which should be achieved

by the later winner to come closer to the final win. There exists to every component x a matching

component y such that [x,y] can be presented in one of the two forms:

[FLOOR(n), FLOOR(n2)] or [FLOOR(n2), FLOOR(n)].

1

W. A. Wythoff, A Modification of the Game of Nim, Nieuw. Archief voor Wiskunde (2), volume 7,

pages 199 202, 1907

2

H.S.M. Coxeter, The Golden Section, Phyllotaxis and Wythoffs Game, Scripta Mathematica, Vol. XIX,

page 142, New York, 1953

p 4

D-N-L#90

Exchanging the components takes into consideration that the order of the piles (towers) is not

important. The fact that to each component x a component y is existing results from Coxeters proof

which will not be repeated in this article. Instead of this we will show an example supported by

DERIVE:

As you can see the left component runs through all natural numbers from 1 to FLOOR(10). It is

important for the tactical analyse of the game that none of these pairs can be reached from another one

performing only one move. The opponent is forced to leave this set of pairs. In case of a pair [x,y]

fulfilling the condition y > x we can reduce y to generate one of the pairs from v(m) with m > x.

The respective (still incomplete) DERIVE-function is:

D-N-L#90

p 5

The third case x/ < y < x is still remaining. In this case we reduce [x,y] in both piles in such a way

that the difference x y equals the difference of a pair from v(m). So we have to substitute for

else ??? the construct:

We remove the zeros and make m dependent on x and y. Function win_move(x,y) is now ready:

If our oppponent leaves piles of heights x and y we run win_move(x,y) in order to learn how to

reduce the number of counters to gain the victory. In case of (13,5) we have to remove 10 counters

from the higher pile and in case of (30,20) our strategy must be removing four counters from each

pile.

Two special cases are missing: in case x = y it is not necessary to find a DERIVE-answer because it is

obvious that one will win with the next move. Case win_move(x,y) = [x,y] is only possible if the

opponent starts the game and possibly knows the strategy. If so then we will loose we can try our

luck and make any move.

Comment of the editor: I remember DNL#45 from 2002 when Richard Schorn wrote an article

about Wythoffs Nim. He closed his ACDC-contribution as follows:

p 6

D-N-L#90

The generation of the two sequences relies on some of the mentioned properties:

Start with 1 as the x-value of the first safe pair. Add this to its position number to

obtain 2 as the y-value. The x-value of the next pair is the smallest positive integer not previously used, it is 3. The y-value is 5, the sum of 3 and the position

number. The next x-value is 4, the smallest integer not yet used.

My program for DERIVE is based on this algorithm in Martin Gardners book

Penrose Tiles to Trapdoor Ciphers. (Richard Schorn)

See the TI-Nspire-version from 2013 which can neither use the UNION

of sets nor the useful MEMBER?-function:

D-N-L#90

p 7

The author of this paper has based on a small DERIVE program discovered two more games with

towers of counters which both are leading to the sequence <FLOOR(n)nN>. Both games are played

by only one player who intends to discover mathematical patterns in his results. Both games are

starting with a sequence of piles of counters according to the figure given below.

The sequence of Countertowers or Cointowerscan be continued to the left as far as one likes. The

positions of the towers are numbered from right to left. When the game starts each tower consists of as

many counters as its number says. For the first game we can assume an infinite number of piles.

Every move consists of taking the last pile (first from right) and distributing its counters piece for

piece on the next piles. We can describe the constellation at the beginning of the game as:

... 7, 6, 5, 4, 3, 2, 1. Then we have:

... 7, 6, 5, 4, 3, 3

... 7, 6, 6, 5, 4

The next moves are left for the reader. (He/she might use the DERIVE program which will be

presented at the end of the article.) Inspecting the piles to be distributed in the next move and counting

its counters we find the sequence <1, 3, 4, 6, ...>. We can conjecture that here again the sequence of

the first components of Wythoffs game appears.

The second game deals with a finite number of towers. As the moves follow the same rules as given in

the first game there will come a moment when no more piles are at the left end of the row to deposit

counters. In this case we proceed by adding one-counter piles in left direction. See an example starting

with 4 towers:

4, 3, 2, 1

4, 3, 3

1, 5, 4

1, 1, 2, 6

1, 1, 1, 2, 2, 3

1, 1, 2, 3, 3

1, 2, 3, 4

Having reached the reverse order all piles are moving to the left without changing the number

sequence. It needed 6 moves to reach <1, 2, 3, 4> from <4, 3, 2, 1>. One can try starting with 5, 6, 7,

... towers and count how many moves are necessary to reach the reverse constellation of the starting

towers. Doing this with paper and pencil (or with the DERIVE-program) you will get the following

table:

Number of towers at the beginning

Number of moves to reach the mirror position

1

1

2

3

3

4

4

6

5

8

This sequence in the 2nd row is again very suspicious to form the sequence of the x-values from

Wythoffs game!

6

9

p 8

D-N-L#90

We will simulate the second game using DERIVE. This simulation will give the results for the infinite

form, too. Lets start with an initial list (vector) representing the towers with heights from 1 to n.

h(z) removes the last element of any sequence of towers (the utmost right tower) and i(z) transforms

this pile of counters in a list of single counters.

(We use the list-oriented commands like REVERSE, FIRST and REST because DERIVE is programmed

in LISP which is based on working internally with lists.)

We rename the longer list as y and the shorter one as x:

The order of towers h(z) and i(z) are compared with respect to the number of towers. The greater

number is y and the smalles one is x. y must be divided in two lists: in f(y) which consists of the same

number of elements as x and g(y) containing the remaining elements.

Now we can add f(y) and x and append g(y) in order to get the next sequence of towers.

For better understanding the procedure we will demonstrate the process starting with a sequence of

six piles:

All steps are collected in one iterative procedure, starting with init(n):

D-N-L#90

p 9

You can find our z from above as 5th element and its follower. It needs 8 moves to come from

[5,4,3,2,1] to [1,2,3,4,5].

Question #1: Can we confirm our conjecture from above about the number of elements until reaching

the reverse constellation? DERIVEs iteration stops when two iterations remain the same.

Question #2: How can we derive the results of the infinite game from these results?

From the above towers(5)-result we can read off the last numbers of the intermediate steps:

[1,3,4,6,8]. Then we would need a 6th and an 7th pile of counters to proceed. The sequence needed is

the increasing partial sequence of the last elements of the towers(n)-results. A short program can

help:

The interested reader might have found out by himself that there might sleep some other exciting

discoverings. It could be worthwhile to change the start conditions. What we showed here are only

results which affirm the conjectures, there are no proofs. The proofs together with other rules can be

found in Roland Schrder, Der Goldene Schnitt in Trmen aus Spielsteinen, in Mathematische

Semesterberichte, Band 60, Heft 1, Springer 2013.

http://scienceindex.com/stories/2251230/Der_Goldene_Schnitt_in_Trmen_aus_Spielsteinen.html

p 10

D-N-L#90

Rolands last paragraph made me considering about other exciting discoveries. What to

discover? So I had the idea to start with towers consisting of odd numbers of counters:

[, 7, 5, 3, 1] and playing the game from above:

We cannot find the reverse order of the initial status. But we notice a cycle of length 7

starting with towers 1, 2, 3, 4, 5, 5, 6.

We can define a function to calculate the length of the period:

How long are the periods for games from 1 to 30 starting piles with odd numbers of counters:

5 piles lead to a period of length 7. I cannot detect any generating rule or pattern, can you?

Next question: How many moves are necessary until having finished the first cycle?

D-N-L#90

p 11

I will check for the game starting with 6 piles form 1, 3, , 11:

This is a strange result, indeed. The period has length 1 (compare with the list of the period

lengths) and the pre-period consists of 22 elements.

You are friendly invited to offer other ideas and games and conjectures and solutions

and proofs, Roland and Josef.

p 12

D-N-L#90

It was during Christmas holidays when my wife Noor and I made walk through

winter wonder world. Suddenly the mobile phone interrupted the silence. At first I

didnt know who was calling. It was a colleague with whom I collaborated many

years ago developing a curriculum for a special type of vocational schools. He

was teacher at a Secondary school for forestry in Styria.

After exchanging some personal small talk about ourselves (family, retirement,

health, ...) he tried to explain his problem. I said, Franz, send an email explaining your request and add one or two examples. I believe to understand you, but

written text is quite better. Coming home from our walk I turned on the notebook

and found his mail:

Dear Josef,

... I have a small problem in earlier times we produced our own programs ...

The problem deals with finding a correlation coefficient. Given are a few pairs of values and their frequencies, e.g. (5,6) 25 times, (4,3) 17 times, etc. I dont want to enter 25 times (5,6) and then 17 times

(4,3), etc. I would like to enter 5,6,25 followed by 4,3,17, etc. That is it.

Slope and intercept of the regression line are not so important. It would be great if I could represent

the regression line but this is not a must for me.

Doing this in Excel I have to enter all pairs into the table and this is boring. At the other hand the

Excel 3D-representation looks fine: the value pairs in the xy-plane and the frequencies in z-direction.

Best regards

Franz

D-N-L#90

p 13

Dear Franz,

... Now to your problem:

Nice to do it with other technologies (btw, it is not too boring working with Excel by copying

down the respective cells). What you are expecting can easily be performed with TI-Nspire. It

can be done without any additional programming.

I am quite sure that you can do it with the older TI-handhelds (TI-92, V 200 and TI-84), too.

But at the moment I am on holidays I dont have these tools at my disposal.

The screen shot below shows the result using one of your data sets. The regression line

could easily be plotted together with the scatter diagram. What cannot be done is a 3Drepresentation.

You can also work with GeoGebra. But here you have to enter the data pairs (copying down

the cells like with Excel). I did not find a way to work with triples (pairs together with their frequencies).

p 14

D-N-L#90

The data set consists of 25 times (0,0), 20 times (1,1), 15 times (0,1) and 5 times (1,0). List

Liste1 was created by a mouse click into the two spreadsheet columns. The correlation coefficient a = 0.57907 was calculated by CorrelationCoefficient(Liste1) and the regression line

b = -13x+24y=8 by FitLine(Liste1) (in German: Korrelationskoeffizient(Liste1) and Trendlinie(Liste1))

It is not possible to have a 3D-representation of the 4 data piles.

Let us turn to DERIVE. Two short programs produce the statistics and a 3D-plot as requested:

(1)

(2)

You are invited to compare with the GeoGebra results from above.

D-N-L#90

(3)

p 15

Back home some days later I could accomplish my proposals working with my good old Voyage 200:

The Stats/List Editor is the appropriate tool on the Voyage 200:

Dear Franz, now it is your choice which tool to use. It was very interesting for me to compare

the several possibilities treating your question. Now I have a question: Why do you need this

tool in order to find the statistics for these special four data pairs?

Josef

p 16

D-N-L#90

Dear Josef,

many thanks for your efforts in your retirement. Your advice was very helpful for me.

You are right, I owe you an explanation for what I am needing all the procedures. Take any population and collect the presence of two certain properties (in German: Merkmale MM).

I have been interested in the influence of the existence of the properties on the correlation. I describe

an example: 100 people form the population. There are some people with both properties (= (1,1)),

with none (= (0,0)) or with only one of them ((1,0) or (0,1)). I wanted to find out the influence of the

frequencies of the pairs on the correlation. 55 persons have both properties, 25 have none, 5 have the

first one only and 15 the second one only.

You know that I was teacher at a special vocational school for forestry. But I have changed the faculty in my retirement. I am very interested in Gender Studies and I need sometimes statistical evidence to underline special conjectures about our social life. To make it short: Id like to know What is

keeping our world together.

Best regards and many thanks again,

Franz

In the DERIVE package is a Utility file called FresnelIntegrals.mth containing definitions

for Fresnel-Integrals. Recently I experienced that the definitions presented there are quite problematic.

After 15 minutes calculation time I found new formulae for the Fresnel Integrals which seem to perform much better. As they might be useful for other DERIVE Users, too, I wouldnt like to deny them

for you.

After treating the logarithmic spirals in great detail earlier I wanted to work on other interesting spirals. And I found the Clothoids or Cornu Spirals a very interesting object. A clothoid is defined as a

curve with its curvature proportional to its arc length. So it is in a certain sense the equivalent to the

logarithmic spiral whose curvature is inverse proportional to its length. Considering its definition we

can easily find the tangent of the clothoid:

t2

t2

cos

,

sin

2a 2

2a 2

with a an arbitrary positive proportionality constant and t the arc length of the curve. Its curvature is

given by

t

= 2.

a

The curve is the defined integral of the tangent from 0 to length s. The resulting integral is a so called

Fresnel integral which unfortunately cannot be calculated analytically. The utility file mentioned

above contains two functions "FRESNEL_COS" and "FRESNEL_SIN" which evaluate these integrals

numerically. Unfortunately the numerical procedure turns out to be unstable for greater values. There

are also power series "FRESNEL_COS_SERIES" and "FRESNEL_SIN_SERIES" available. Both

power series are formally correct but they have some not so nice properties like

a)

b)

D-N-L#90

c)

p 17

they are very susceptible for rounding errors when calculating greater values. So it is necessary evaluating them in exact mode only which causes extra long calculation times very soon.

What to do? Although the tools given in the utility files are not very useful in my opinion, we can use

DERIVE to calculate the integrals requested using another DERIVE feature. Luckily the Fresnel integrals can also be calculated applying the complex error function ERF(z). ERF can be calculated in two

ways: as power series and as continued fraction. The first one converges very fast for small values and

the second one converges fast for large values. And it is implemented as a standard function in

DERIVE. This makes possible calculating Fresnel integrals and clothoids in real time with DERIVE

without any extra programming efforts.

I am sending a short file containing functions for Fresnel integrals and for clothoids. Maybe that one

could produce an updated FresnelIntegrals.mth file.

With best regards

Piotr Trebisz

Comment of the editor: I asked Piotr for possible applications of the Fresnel integrals (except

the clothoid) and he answered:

Unfortunately I cannot offer additional examples at the moment. It was the treatment of the clothoid

which led me to these integrals. I know that they are used in physics for calculation of wave diffraction, they are used in quantum mechanics and I found them in Richard Feymans path integrals. And

as you wrote in your mail clothoids are used in road construction for forming smooth connections between segments of different curvatures and in particle accelerators.

The Fresnel functions are the integrals of cos(a t 2 ) and sin(a t 2 ). a is an arbitrary real number.

Let L the arc length of the clothoid and A its parameter (positive real number). Its curvature is proporL

tional to 2 . Its asymptotic points are

A

A A

A A

,

,

.

and

2

2

2

2

You can plot the clothoid expressions #5 or #8 of the following DERIVE file after introducing a slider

bar for a in real time.

With my best regards

Piotr Trebisz

What follows is Piotrs DERIVE file. As I was very inspired by his mail and mth-file I did some

research in my books and in the web. The list of my findings is given in the references at the

end of this constribution, Josef

p 18

D-N-L#90

See now Piotrs improved Fresnel integral functions together with their application defining

the spiral in another way:

D-N-L#90

p 19

p 20

D-N-L#90

It is interesting comparing the velocity plotting the spiral when using the DERIVE Utility functions FR(z) and Piotrs functions FRESNEL(z) with -2 z 2.

the plot of the slope of the Cornu Spiral:

differential geometry is Alfred Grays

Modern Differential Geometry of Curves

and Surfaces.

Gray presents a generalization of this

spiral with exchanged x- and y-coordinates. cloth(1,a) is the clothoid

D-N-L#90

p 21

There is one chapter in Grays book treating Plotting plane curves with given curvature. Gray

uses MATHEMATICA and offers a special tool to find the natural or intrinsic equation[] of

curves solving a system of DEs.

I cannot transfer this tool directly to DERIVE but I can reproduce a nice Clothoid Flower:

[]

Find more about the Natural or Intrinsic Equation in DNLs#29 and #30 (David Halprins

articles). See also the rich list of references at the end of this contribution, Josef

p 22

D-N-L#90

References

Books:

[1] Eric Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press 1999

[2] Abramowitz & Stegun, Handbook of Mathematicsl Functions, Dover

[3] Vladimir Rovenski, Geometry of Curves and Surfaces with MAPLE, Birkhuser 2000

[4] Alfred Gray, Modern Differential Geometry of Curves and Surfaces, CRC Press 1994

[5] Alfred Gray, Differentialgeometrie, Spektrum Akademischer Verlag, 1994

[6] Georg Glaeser, Geometrie und ihre Anwendungen in Kunst, Natur und Technik, Spektrum 2005

Websites:

[7] Fresnel Integrals

http://functions.wolfram.com/GammaBetaErf/FresnelS/introductions/FresnelIntegrals/ShowAll.html

[8] How to evaluate Fresnel Integrals

http://www.thomasbeatty.com/MATH%20PAGES/ARCHIVES%20%20NOTES/Complex%20Variables/How%20to%20evaluate%20Fresnel%20Integrals.pdf

[9] The Euler Spiral: A mathematical history

http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.pdf

[10] Eigenschaften ebener und rumlicher Kurven

http://page.mi.fu-berlin.de/sfroehli/ss2007/vorlesung02.pdf

[11] Natrliche Gleichung einer Kurve (Hans Walser)

http://jones.math.unibas.ch/~walser/Miniaturen/N/Natuerliche_Gleichung/Natuerliche_Gleichung.pdf

[12] Krmmung von Kurven und Flchen

http://www.soedernet.de/math/1samstage/05/vortrag.pdf

[13] Parameterisierung nach der Bogenlnge und Krmmung

http://www.blaesius-anne.de/arbeit_kurven/kurvenDb.html

[14] The elastica: a mathematical history

http://levien.com/phd/elastica_hist.pdf

[15] Formeln zur Klothoide

http://ww3.cad.de/foren/ubb/uploads/Clayton/Klothoide-Formeln.pdf

[16] Klothoide (GeoGebra-Applet)

http://www.projekt.didaktik.mathematik.uni-wuerzburg.de/klothoide/Klothoide.html

[17] Vorlesung Differrentialgeometrie (Matthias Bergner)

http://www.mathematik.uni-ulm.de/analysis/lehre/diffgeo_ss07/dg06.pdf

[18] David Halprin, Super Duper Osculants, DNLs #29, #30 and #32

[19] Ibrahim de Crdoba, Evaluation of Fresnel Integrals, DNL #39

[20] David Halprin, River Meander and Elastica, DNL #39

I was so much inspired by some of the references especially by [4] and [11] that I tried to

find an easy to use DERIVE implementation for plotting curves with given functions of their

curvature dependent on their curve length. Josef

D-N-L#90

p 23

Josef Bhm, Wrmla, Austria

Hans Walser presents in his paper [11] a TurtleGraph method using MuPad to find curves with given

initial point, given start direction and a function (s) describing the change of direction of the tangent

dependent on the arc length s. (s) is the curvature, its reciprocal value is the radius of the osculation

circle.

Walser starts with an introductory example plotting a pentagon:

We perform n moves (turning n times to the left by angle a and proceeding the distance l in the now

new given direction. start gives the x and ycoordinates of the starting point and as third component

the initial direction. Repeating turning and proceeding 5 times results in a pentagon

We start at any position and move to position s where we have to change our direction according a

function of s. Performing a numerical approximation we can do this only executing very small steps

ds. It turns out to be something like a numerical approximation of a differential equation:

p 24

D-N-L#90

(s) is the curvature function of the arc length, l is the total distance on the curve, ds is the step

width and start are the initial values as explained above.

curvature = 2 in all of its points? I am quite sure

that you know this:

change the direction after each step in the same

constant way, i.e. by the value 2. This gives a

circle with radius 0.5. You see the result of

nat_equ and superimposed the respective circle

(in grey colour). My approximation fits pretty

well! I am satisfied.

We started with Piotrs Clothoid. The curvature is

proportinal to the distance made:

curve like

D-N-L#90

p 25

nat_equ() invites to reproduce some of Grays examples and to experiment with own functions (s).

Our reward are very strange looking figures as can be seen on the next page.

Do you remember Piotrs explanation that the

Clothoid is the equivalent to the logarithmic

spiral?

p 26

D-N-L#90

Many thanks to Piotr Trebisz for his mail about improving the Fresnel Integrals, Josef.

Can we do this with TINspire, too?

D-N-L#90

p 27

It is not too difficult to transfer the DERIVEtreatment on the TINspire environment. What

we cannot do is defining the Fresnel integrals supported by the error function because it is

not implemented there.

We cannot find the asymptotic center of the spiral by calculating the limit. But we see that

the curve approaches (0.5, 0.5). Plotting the clothoid needs some time but it works on the

PC. I didnt try on the handheld.

p 28

D-N-L#90

The program to plot the natural or intrinsic equation is quite identical to the DERIVE

procedure. The lists of the coordinates are stored as lx and ly. I stored them under new

names in order to produce a plot of three curves on same axes. This works in a very

reasonable time.

D-N-L#90

p 29

... My name is Michael Klamecki. My teacher, Prof. Haller recommended to contact you because of

two nasty DERIVE problems which I came across just recently.

My first problem:

I am able to open DERIVE on my notebook only as administrator. Opening it just the normal way

and trying to solve any equation it does not work at all: the window becomes blue and I am unable to

perform any mouse click.

My second problem:

It happened when I loaded a previously saved DERIVE file. In a small pop-up window an error message appeared reporting that there is an error parsing expression #7. After confirming the Ok-button

another error was reported and finally the file was opened.

It would be great if you could give any advice.

with best regards

Michael

DNL:

Dear Michael,

What concerns your administrator-problem, I asked Gnter Schdl, the DERIVE-expert.

This is his answer:

This is a problem of the administrator rights. I know this problem it appears sometimes when

students install DERIVE on their laptops. My standard advice is:

Install again (right click ... perform as administrator).

You can also try to change the rights in the DERIVE folder for All Users.

If this does not work:

Look for another laptop where DERIVE works as you would like, remove DERIVE from your

device and then copy it back (using an USB-stick) from the other laptop. This is how we

could resolve this nasty problem.

I dont know the reason for this problem. Probably it is connected with the security settings of

some computers. I didnt investigate the reason, I have solved the problem in most cases.

Hope this helps, many thanks to Gnter.

I cannot give any advice for your second problem (the parsing error) without knowing the respective file. So please send the file.

Regards

Josef

p 30

D-N-L#90

... I uninstalled DERIVE manually and removed all respective files (except my homework files). Then

I reinstalled DERIVE but the same problem occurred. Ok, I will work with DERIVE as administrator

in the future.

I attach two screenshots and the DERIVE file causing the parsing error in expressions #7 and #8.

I noticed that the fractions in #8 are printed in an unusual way and the AND-operator is written as

AND ().

Best regards

Michael

DNL:

Dear Michael,

the error was easy to find if one knows how to detect it.

Look at expression #5: You use variable k for the slope in the linear function g(x). This

should read g(x) := kx + d. But #5: g(x) := k(x) + d. Something must have happened.

Copy the wrong expressions #7 and #8 into the Entry Line, then you will notice that they are

under quotes, i.e. they are not mathematical expressions but have changed to strings. This is

the reason that the fractions are looking strange. See my screenshot below:

Why that? Go to expression #31! What have you done there?

D-N-L#90

ITERATES Again!

p 31

You defined a quadratic function k(x). This is ok at this position and you will not face any

problems saving the file.

Reloading this file DERIVE finds k(x) among the stored functions. Performing then all commands from the very beginning DERIVE substitutes kx by k(x) which leads immediately to

problems when solving the system in #7.

I renamed k(x) as k1(x), stored the file and everything works. So take care in the future naming variables and functions. I recommend using more-letters-function names, like curve(x),

mo_way(x), ku1(x), ku2(x), etc.

I wish much success in mathematics and much fun with CAS.

Best regards to you and to my colleague Wilhelm Haller,

Josef

Dear Josef, as you showed in the panel discussion in Estonia, the iterates function of Derive works

well when exported to Nspire CAS (in fact, the first export was at the Bonn conference as you wrote

in DNL). But it seems to be less general than the one Derive has.

For example, in Derive, iterates(diff(f,x), f, x^10, 3) returns [x^10, 10x^9, 90x^8, 720x^7]. In Nspire

CAS, the SAME command gives [x^10, 0, 0, 0]. It seems that Nspire CAS computes diff(f, x) BEFORE

replacing f by x^10. Do you have an explanation?

The reason for my question is the following : I have defined many functions in Nspire CAS and, in

some cases, I was unable to use the iterates function. So I have used recursion which becomes slow.

With iterates, it would be so fast! But I am not a good programmer compared to you. So, maybe

you can help me!

Best regards and hope to see you and Noor soon,

Michel

p 32

ITERATES Again!

D-N-L#90

Dear Michel,

this seems to be really difficult.

I tried several approaches. The problem is as you mentioned the immediate evaluation of

the differential operator. When we call dif(f,x) Nspire evaluates to zero.

We see the bold d which cannot be transferred into the program as unevaluated operator.

I have a very weak solution writing the differential quotient into the program.

Maybe that experts like Philippe Fortin will provide a more general solution.

From Michel

Thank you dear Josef. Your solution is not so weak and can be useful!

This example shows (again) how Derive was special... And this kind of example (and others)

could make a good presentation for the Derive/Nspire session in Malaga. Is there any chance you at

tend ACA 2013? If not, let us promise to have a good beer together in Austria in 2014!

Michel

D-N-L#90

p 33

In a TI-NspireCAS program I tried to trap the error Division by 0 by the construct

Try...Else...EndTry. This does not work because the Else-block will not be executed.

I informed in the online manual and didnt find any advice and I had no success with a Google research. Trapping other errors e.g. dimension error is possible applying Try ... Else ... EndTry.

Can you please forward this deficiency to TI? I dont find any possibility to do this by myself.

See the program without (above) and with the non working error trapping routine (below).

p 34

D-N-L#90

Dear Philippe,

How are you? Here I am again with a Nspire-question (provided by a DUG-member):

Is it possible to catch a Division by Zero error using the Try - EndTry procedure?

So many errors can be handled by Try - EndTry. How to do with this one?

I attach the respective program.

It is no problem to circumvent the problem using an IF-contruction.

Best regards

Josef

This was Philippes answer:

In fact, computations as 1/0 or 0/0 doesnt cause an error we do get a result, equal to

undef. There must be some reasons for this choice (but I am not sure of which ones !).

The question is then : is it possible to check that p2 contains the special value undef.

It is not possible to use

if p2=undef then

If p2 is actually equal to undef, the result of this test will be undef = undef, which is not true

or false

It would be possible to use something like

If when(p2 = undef, false, false, true) then

Another solution is to test the equivalent string value

If string(p2) = "undef" then

Last, since the expected result should be a number, it is possible here to test

If getType(p2) = "EXPR" then

But this would not work if p2 could possibly contains values as sqrt(2) which also have the

type EXPR

None of these solutions is perfect (all are using odd constructions and the last one is not

correct for all situations). It would be better to have a function directly testing for a undef

value.

Of course, a Try EndTry including the test statement if p2>p then would also be a possi

ble approach

Best regards,

Philippe

D-N-L#90

p 35

Dear Philippe,

many thanks for your extended answer.

Like you I tried many approaches but not the last one. It works.

I will publish your answer + solution of the problem in the next DNL.

It would be interesting to learn why Division by 0 is a special error which needs an extra

treatment.

Best regards and thank you once more,

Josef

And here is Philippes final comment:

Dear Joseph,

Just a thought concerning the use of undef (which is not an Error). It could be related to

limit computation

When entering

lim(1/x,x,0)

it is rather a good point to get undef instead of an error message

by the way, you probably noticed that

1/0 returns undef

1/0^2 returns infinity

This is also consistent with limit computation

(this could be included in the answer you plan to publish !)

Best regards,

Philippe

The Brussels Gate does not end fascinating me; at first I wanted to learn more about the Dutch

engineers who built all three gates (Brussels, Mechelen and Gent the last one has been destroyed) in

1822. In the meanwhile I found out that the head of the engineers was a certain Cornelis Alewyn.

Then I wanted to lear from where C. A. has got his experience. Who knows history of technics knows

that one has to do some research in France because the succeeders of Vauban started applying

algebraic methods for calculation and planning military buildings. And I was lucky!! The inventor of

the draw bridge with a counterweight on a curved track is Bernard Forest de Blidor (1697-1761) who

gave the equation of the track as a formula. Applications can be found in the United States (two

double line road bridges) and in four or five French towns and in Switzerland. I will prepare a

respective paper investigatig the difference between this curve and the parabola which I calculated in

Dendermonde.

Best regards

EvL

p 36

D-N-L#90

David Halprin, Australia

This was written in 1992 and made use of Mathcad 2.0. forgive this transgression.

There are many traps for the unwary when attempting to find a functional relationship between

two variables so as to represent a plane curve in some coordinate system.

Since the Cartesian framework is the usual default coordinate system let us examine a common

pitfall. Many curves have a symmetry about one axis, (at least), and if one requires the simplest equation to represent the curve, then perhaps one arc of the curve will be preferable, since the symmetrical

arc will follow with ease.

A trivial example is the circle. The simplest equation in Cartesian Coordinates is for a semicircle, where we have y = f(x), whereas for the whole circle we have to combine it with y = -f(x) to

have the expression [y - f(x)][y + f(x)] = 0 which gives us the usual quadratic form. One should note

well that:1)

Such a quadratic form allows 2 values for y for any value of x as it does also allow 2 values for

x for any value of y.

2

2

x + y = c2.

2)

The circle is a composite curve, when expressed as y = f(x). If we had chosen a Polar Coordinate System then we would have a much simpler functional relationship between two variables

and the circle is a simple curve of one arc, r = c

One can manipulate symbols in such a way that two different curves, or two arches, or branches,

of related curves, (reflected, rotated and/or translated), can be combined in one equation to make it

look like one curve, when it may be a simpler exercise, and much simpler equation, to analyse it as a

composite curve. For instance any curve which lies above the X axis may be conjoined with its reflection beneath the X axis and the resultant equation may be encompassing the two parts as though they

are two branches of the same curve, but the equation will be of a higher degree, and it is far easier to

look at each branch separately.

In the problem of the Railway Transition Curve, one can fall into the trap of assuming that the

entire curve is best represented:1)

2)

in one equation.

However, staying with Cartesian Coordinates, and looking for only one of the symmetrical arcs,

one would have obtained a far simpler solution, even more significant than in the case of the circle example.

One necessary quality of a point of inflection is that it has no curvature at the point itself, but on

either side of that point the curvature takes opposite directions, meaning that a concave arc changes to

a convex arc.

The force on the flange of the train's wheels has to be constant as it rounds the bend. The only

way to state this with exactitude, is to talk about the relationship of the curvature to the arc length,

namely:

D-N-L#90

p 37

A Railway Transition Curve, which is also used by road builders (road engineers), who want it

to be such that the radius of curvature does not change discontinuously, since this would involve discontinuities in the centrifugal (centripetal) forces between the rails and wheel flanges of the trains, or

the tyres and road for cars and trucks. So what is required is that the radius of curvature is a continuous function of the arc length.

The simplest and most natural functional relationship is that curvature (the reciprocal of the radius of curvature) is proportional to the arc length.

1

s.

This equation uses two intrinsic variables, which are not to an external framework, as are the extrinsic Cartesian (x,y) or the extrinsic Polar (r,). There are many advantages to be gained from familiarity with the intrinsic variables, of which there are three, from which we may choose any pair.

Radius of Curvature, rho ()

Arc length, (s)

Tangential Angle, phi (), where

ds dx

dy

dx

dy

,

= cos ,

= sin ,

= cos ,

= sin .

d ds

ds

d

d

This curve has been studied for some time under various and has also proven to have been useful in optics. It is known as Klothoid, Clothoid, Cornu's Spiral, Euler's Spiral and Railway Transition

Curve.

In addition to the use of a pair of variables in a functional relationship to define a plane curve,

such an equation can be split into two equations by the use of a parameter, which is often referred to as

a dummy variable. When one eliminates the parameter from the pair of equations, one is back to the

original equation in two variables. However, if one looks for a geometrical image for the parameter, it

is often an interesting finding. Commonly, when transforming coordinates from one system to another,

one is left with two parametric equations and the parameter is one of the variables associated with the

coordinate system, from which one is transforming.

e.g. Going from polar to Cartesian, the polar angle, is commonly the parameter, and it is a very

convenient one, especially when instructing a plotting program to draw the curve, since one tells it to

range from 0 to 2 and the curve is drawn with great facility.

However, there are serious pitfalls to acceptance without inspection of any and every such parameter, that may appear when transforming, as described, depending on the particular curve and the

geometrical meaning of the variable, pertaining to that particular curve.

e.g. When transforming from Intrinsic to Cartesian Coordinates, there are s, and . Usually

there is no problem with s and , since with most curves one can range their values fairly well, but the

radius of curvature may not range, and hence one may obtain, at most, a small arc of the curve

and, at worst, an impossible representation of the original curve.

There are certain differential equations, involving elliptic integrals, called Fresnel Integrals, which

have no closed form of solution, but which can be approximated with a strange looking series. These

integrals can be used as parametric pair of equations to define the Clothoid.

p 38

D-N-L#90

In fact, since it is the clothoid and can be specified very in the two variables, s (arclength) and

(radius of curvature), it is only a matter of numerical approximation, which will find the point of inflection. Further, the curve is compounded from two specially selected arcs of the clothoid, in the first

and second quadrants respectively, and the omitted parts of the curve from the diagram are useless to

railway engineers, since they are spirals about asymptotic points. The curve, as you will see below, is

made up of two spirals, which are interconnected through a point of inflection, so the arc chosen is

anywhere from the point of inflection up to as much of the spiral as it suits the engineers to use.

1.

d

1

==

= 2 Ks

ds

2.

= K s2 + L

3.

4.

5.

6.

(Let s = 0 at = 0 L = 0)

K

ds

1

=

=

d 2 K

s=

dx

dy

= cos = cos( K s 2 ) and

= sin = sin( K s 2 )

ds

ds

cos

dx

dy

sin

= cos =

and

= sin =

d

ds

2 K

2 K

In Levies paper ([14] on page 22) I found this pretty figure presenting osculating cubic,

clothoid and elastica. Can you reproduce this figure? The parameter form of the elastica is

given below, Josef.

See: David Halprin, River Meander and Elastica

in DNL#39.

D-N-L#90

This is one of the examples we produced in

a workshop (DERIVE & GeoGebra) several

years ago.

One can underlie background pictures of

ready made stitching (scans from books,

pictures from the web, or own designs).

Students learn to apply parameter form and

families of segments and arcs.

Students works:

p 39

p 40

D-N-L#90

This is a more complex pattern. I took it as background picture in DERIVE and you can see

left below the first step modelling the figure. Josef