Bijnens Doom

(1)

Bijnens Doom

Citation for published version (APA):

Meiden, van der, W. (1974). Bijnens Doom. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7412). Technische Hogeschool Eindhoven.

Document status and date: Gepubliceerd: 01/01/1974

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--

..

TECHNISCHE HOGESCHOOL EINDHOVEN Onderafdeling der Wiskunde

Memorandum no. 1974-12 17 oktober, 1974

BIJNENS DOOM door

W. van der Meiden

Onderafdeling der Wiskunde Technische Hogeschool Eindhoven Postbus 513, Eindhoven,

(3)

BIJNENS DOOM

W. van der Meiden

§ 1. Geometrie.

Zij van een regelmatig 20-vlak de ribbe a, de straal van de omgeschreven bol

p, de straal van de omgeschreven cirkel van een paralleldoorsnede door 5

hoekpunten R. Dan is ([IJ, p.59)

(1) a

=

iRvlO -

2rs

terwijl toepassing van de stelling van Pythagoras in liMA

2•0A3•0 (die gelijk-benig is) geeft

(2) p

=

iRrs

zodat

(3) sin

e

=

!

=

1 15

p 5 ' cos

e

==

~ 15 ,

tan

a

=

2 •

We nemen de oorsprong van het coordinatenstelsel in M, en drukken de coordi-naten van _{AO.O - A3•}_{0 uit in}R. We gaan uit van A_{2 •0}

=

(R,O,~).

Om A

2• 1 te vinden moeten we draaien over 72°. Nu zien we in li A3•0 A2 •0 A2• 1

dat

sin 36°. a/2R

=

iV10 - 215

waaruit m.b.v. elementaire goniometrie

cos 36

°

=

H

15

+ I) ,

cos 72 °

=

H

15 -

1) , sin 144°

=

iv'lO -

2rs

cos 144°

= -

H

rs

+ 1) •

(4)

Op een schaal van R :- J geeft dit voop de cOQrdinaten van _{AO.O - A3•0:} A x y 0.0 0 0 1.0 -) 0 J • 1

t(1 -

15)

_{- !/IO}

+

21.5

1.2

HI5

+ 1)

- iVIO - 215

1.3

HI5

+ ])

iv'1O -

215

1.4

,(1

-15)

n/10

+

215

2.0 0 2. I HI.5-I)

,/10

+

2/5

2.2 -HI5+1)

!v'1O -

215

2.3 -H/5+1)

- ,110 -

21.5

2.4 HI.5 - 1)

- 1VIO

+

215

3.0 0 0

§ 2. De verde ling van de driehoe~en.

Bij gegeven A, B, C vindt men in het vZak

ABC de punten A' , ••• ,C" door

A.' ..

~

(2B + C), A" ..

~

(B + 2C) en cyclische verwisseling, en 1 D ..

"3

(A + B + C) • Dat betekent: z

-1 15

- l

-

~

-

~

- !

!

I

!

l

!

!/5

c.

A

---~---~---B

(5)

I xC) xA' == -3 (2x B + I + _YC) YA, .. - (2y 3 B I zC) zA' :: - (2z ₃ _B+ enzovoorts.

De corresponderende punten op de boZ vindt men dan als

enzovoor ts •

Literatuur.

[IJ Neut, D.N. van der, en A. Holwerda, Meetkunde met de beginselen der Goniometrie III, ge herz. druk; Wolters, Groningen, 1959.

(6)

y

4

--~~!:!..-~--/1

A:z.o

A~.o

x

,

I iii'

,

",' I , , iIII'

I , '

A

0.0

(7)

§ 3. PrograIDpl8 'REAL"PROCEDU~E'LENGTH(V);'ARRAV'V[*];

LENGTH::SQRT(V[112+V[2]2+V[31**2"

'PROCEDURE'WEIGHTED

MEAN(U,V,W,A);'VALUE'A;'REAL'A;·AR~A~'U.V.W[*);

'BEGIN"INTEGER'l;

tFOR'I::l,2.3 'OO'U[Il:=AV[I]+(l-A)W[Il

'ENO'WEIGHTED

~EAN; 'PROCEOURE'POI~T(V,fAC);'VAlUElfAC;'R~ALIFAC;'ARRAV'V{*J;

'BEGIN"INTEGER'I;

'FOR'I:=1,2,3'OO'Vlll:=V[Jl*FAC

'END'POINT;

'PROCEDURE'PARTITION(A.8,C.Al.A2.BI.Bl.Cl,C2,D.AHO):

'VAL~E'RHO;'REAL'RHO;'ARRAY'A.A.C.Al.A2.Bl.B2.Cl,C2.0(*J;

'BEGIN"INTEGER'I;

wEIGHT~D

MEANCAl,C.B.1/3);

~OINT(Al,RHO/LENGTH(Al»;

wEIGHTED MEAN(A2.B.C,1/3';

POINT(A2,R~O/LE~GTH(A2»;

wEIGHT£O MEANCBl,A,C,1/3);

POINT(31.R~O/LENGTH(Bl»;

.EIGHT£O MEAN(B2,C,A,1/3);

~OINT(~2.R~O/LENGTH(B2»)' wEIGHT~O

MEAN(Cl.B,A.1/3);

~OINT(Cl,RHO/LENGTH(Cl»;

wEIGHT£D MEAN(C2,A.B,1/3);

POINT(C2,R~O/LENGTH(C2»;

tFOR'I:=1,2.3'DO'O(I):=(A[I]+Bll)+C11)/3;

POINTC),RHO/LENGTH(D»

'END'POINT;

'PROCEOURE'OUT~UT(A.g.C.RHO);'VAlUE'RHO;'REAL'RHO;'ARRAY'A,B.Cr*];

'BEGIN'

tAR~AY'Al'A2,81,B2,Cl,C2,O(1:3)1

.RITE(PkINT,<I.I»;

wR IT E (Pj:HNT.

<I,

"'_=It, 3F 12.6>. A ( 1 ] ,A (2] • A ( 3] ) ;

wRITEC PRINT.<I,ng_=".3f12.6>,B( 1),8(2).8(3));

NR I

TE cPR

INT.

<I,

"C_=", 3F 12.6>, C [

1 ] •

C [2] ,C ( 3] ) ;

PAQTITION(',8,C,AI.A2,Bl,B2,Cl,C2,D.RHO);

(8)

wRITf(::>RINT,<I,UA2=",3fI2.6>.AZ(1),A2(Z],AZ[31);

wR I TE (:lR I N T

t <I, uIH:",

3f 12.6>, Bll1

}t

B

l (

2),81 ( 3)) ;

«R

I

TE (:>R INT,

<I '''~2=U t

3f 12.6>.82

r

1 )

,B2( 2 1.132 [ 3) ) ;

wR I TE PR

I

NT,

<I

t"C 1="

t

3f 12.6>.C

HI ]

,C

l ( 2)

,C

I [

3}) ,

IIJRITE(JRINT,<I."C2=".3f12.6>.C2[11,C2l2J.C2(3]);

~~ITE(::>RtNT.<I'''D_=''.3F12.6>.D(lJtO[2).D[3));

'END'OUTPUT;

'ARRAY'A,B,C[1:3);

IREAL'R.RH3,~ELTA; 'INT~GER' I "H

RHO:=R*SQRT(S'/2;

DELTA:=8*A~CTA~(1)/S;

A(11:=A£21:=OIA(31:=-R*SQRT(S)/2;B[31:=C[31:=-R/2;

'fOR'I:=O'STEP'1'UNTIL'4'DO'

'9EGIN'B(ll:=-R*COS(I*DELTA):R[2):=-R~SIN(I*~ELTA); Crl):=-R*COS(I+l)*OELTA);C[2J:=-R*SI~«1+1'*DELTA);

OUTJUTCA,R,C,RHO'

'END' ;

A(3):=R*SQ~T(5)/?IB(3):=C(31:=R/2;

'fOR'I:=O'STEP'l'UNTIL'4'DO'

'AEGIN'R(l]:= RCOS(IDELTA);BI21:=

R~SrN(l*OELTA);

, E \10' ;

C(l):= RCOS«I+I)DFLTA);Cl2):=

R*SI~«I+l)*DELTA);

OUTPUT(A,B.C,RHO'

Ar3):=-R/2CA(31:=C(3):=R/2:

'fO~tI:=O'STEPtl'UNTIL'4tDO' 'REGIN'A[11:=-R*COS(I~DELTA);A(21:=-R~SIN(I*OELTA);

8(1):= RCOS«1+2)OELTA';B(21:=

R*SI~«I+2)*DELTA);

ClI):=

~*COS«I+3)*DELTA);C{2}:= R*SI~(I+3)*DELTA);

OUTOU1(A,8.C,QHO)

t P.jl')' ;

Al31:=R/2;3(3J:=Cl31:=-R/2;

'fOR'I:=O'ST~PlltUNTILt~'DO'

t8E5I~tA[11:=

RCOSCIOELTA';A(2):=

P.~SIN(I*DELTA);

~(11:=-~*COS«J+2)*DELTA);B[21:=-R*SI~«I+2)*OELT~); C{11:=-R*C05«I+3)*OELTA);C'2):=-R*SI~«I+3)*DELTA);

OUTOUT(A,R,C,RHO)

t

t::ND

f ; t

FNO' •

(9)

§ 4. Output bij R

:=

1. tl) A = 0.000000 0.000000 -1.118034 A(O.O) (= AO.O) q = -1.000000 -0.000000 -O.SOOOOO A(l.O) C = -0.309011 -0.951051 -1).500000 A(l.l) AI= -0.fJ86194 -0.365013 -0.515696 A2= -0.620991 -0.130026 -0.575696 AI= -0.231200 -0.130026 -0.812895 R2= -0.1l8flOO -0.365013 -1.050095 Cl= -0.383791 O.OOOOO() -1.050095 C2= -0.167594 0.000000 -O.AI2895 D = -0.549093 -0.39893Q -0.888451 6.2 A = 0.000000 0.000000 -1.1Ib034 A(O.O) 8 = -0.309011 -0.'151051 -0.500000 A( l. I)

c

= 0.809011 -0.587785 -0.500000 A(I.2) Al= 0.073299 -0.955')16 -0.575696 A2= 0.502397 -0.AI6193 -0.5"15696 Al= 0.620991 -0.451181 -0.812895 32= 0.310498 -0.2255-10 -1.05()095 Cl= -0.118600 -0.365013 -1.0:;'0095 C2= -0.237200 -0.130026 -0.812895 D = 0.209135 -0.645497 -0.888451 6.3 A = 0.000000 0.000000 -1.11d034 A(O.O)

a

= 0.809011 -0.587785 -0.500000 A( 1.2) C = 0.809011 0.587185 -0.500000 A(I.3) Al= 0.931495 -0.225590 -0.575696 A2= 0.931495 0.225590 -0.575696 81= 0.620991 0.451181 -0.812895 82= 0.310498 0.?25590 -1.050095 Cl= 0.310498 -0.225591) -1.0:'0095 C2= 0.620991 -0.451181 -0.8]2895 D = 0.678116 0.000000 -0.8884~1 6.4 A = 0.000000 0.000000 -1.118034 A(O.O) 8 = 0.809011 0.t:;91185 -0.500000 A( I. 3) C = -0.309011 0.951057 -0.500000 A(J.4) AI= 0.502391 0.816193 -0.575696 A2= 0.013299 0.955616 -0.515696 81= -0.231200 0.130026 -0.R12895 82= -0.118600 0.365013 -1.050095 Cl= 0.310498 0.225590 -1.0:;0095 C2= 0.620991 0.451181 -0.812895 0 = 0.209135 0.645497 -0.888451 6.5 A = 0.000000 0.000000 -1.118034 A(O.O)

a

= -0.309011 0.951057 -0.500000 A( 1. 4) C = -1.000000 -0.000000 -0.500000 A(l. I) AI= -0.620991 0.130026 -0.515696 A2= -0.886l94 0.36"013 -0.515696 91= -0.161594 -0.000000 -0.812895 92= -0.383191 -0.000000 -1.050095 Cl= -0.118600 0.365013 -1 .. 050095 C2= -0.231200 0.1 -0 .. 812895

o

= -0.549093 0.395;939

",v ..

888451

(10)

8 -A6 A = 0.000000 0.000000 1.118034 A(3.0) 8 = 1.000000 0.000000 0.500(100 A(2.0) C :: 0.309017 0.951057 0.500000 A(2.l) Al= 0.886194 0.365013 0.575696 A2= 0.620997 0.130026 0.575696 RI= 0.237200 0.730026 0.812895 '32= 0.118600 0.365013 l.tiS0095 Cl= 0.383791 0.000000 1.050095 C2= 0.767594 0.000000 0.812895 D = 0.549093 0.398939 0.888451 A7 A = 0.000000 0.000000 1.118034 A(3.0) B = 0.309017 0.951057 0.500000 A(2.1) C :: -0.809011 O.'5~178c) 0.500000 A(2.2) Al= -0.013299 0.955616 6.515696 A2= -0.502397 0.A16193 0.575696 81= -0.620991 0.451181 0.cH299s 82= -0.310498 0.225590 ).0500<:)5 Cl= 0.118600 0.365013 1.0~OO95 C2= 0.231200 0.130026 0.812895 D = -0.209735 0.645497 0.888451 A8 A = 0.000000 0.000000 1.118034 A(3.0) B = -0.809011 0.587785 0.'500000 A(2.2) C = -0.809017 -0.587185

O.SuOOOO

A(2.3) Al= -0.931495 0.?25590 0.575696 A2= -0.931495 -0.225590 0.515696 131= -0.620997 -0.451181 0.812895 >:12= -0.310498 -0.225590 1.050095 Cl= -0.310498 0.?25590 1.050095 C2= -0.620991 0.451181 0.812895 D = -0.678716 -0.000000 0.888451 A9 A = 0.000000 0.000000 1.118034 A(3.0) q = -0.809017 -0.587785 0.500000 A(2.3) C = 0.309011 -0.951051 0.50000G A(2.4) Al= -0.502397 -0.816193 0.575696 A2= -0.073299 -0.955616 0.575~96 Bl= 0.231200 -0.730026 0.812895 82= 0.118600 -0.365013 1.050095 Cl= -0.310498 -0.225590 1.050095 C2= -0.620997 -0.1_.51181 _0.A12895 0 = -0.209735 -0.645497 0.888451 A10 A = 0.000000 0.000000 1.118034 A(3.0) B = 0.309011 -0.951057 0.50000e A(2.4) C = 1.000000

o.oeoooo

0.500000 A(2.0) Al= 0.620997 -0.730026 0.515696 A2= 0.886194 -0.365013 0.575696 Bl= 0.767594 0.000000 0.812895 B2= 0.383197 O.flOOOCO 1.,)50095 Cl= 0.118600 _-0.365013 _1.050095 C2= 0.237200 -0111'730026 0 .. 812895 D = 0.549093 -0 .. 398939 (1",888451

(11)

611 A = -1.000000 -0.000000 -0.500000 A(1.0) 8 = -0.809017 0.587785 0.500000 A(2.2) C = -0.809017 -0.587785 0.500000 A(2.3) AI= -0.931495 0.225590 0.515696 A2= -0.931495 -0.225590 0.~75696 81= -1.004794 -0.451181 0.191899 82= -1.078093 -0.225590 -0.191899 Cl= -1.078093 0.225590 -0.191899 C2= -1.004794 0.451181 0.191899 0 = -1.098185 -0.000000 0.209735 612 A = -0.309017 -0.951057 -0.500000 A(1. 1) '3 = -0.809017 -0.587765 0.500000 A(2.2) C = 0.309017 -0.951057 0.500000 A(2.4) AI= -0.502397 -0.816193 0.575696 A2= -0.073299 -0.955516 0.575696 81= 0.118600 -1.095038 0.191899 82= -0.118600 -1.09503£1 -0.191899 Cl= -0.547698 -0.955516 -0.191899 C2= -0.739597 -0.816193 0.191899 D = -0.339358 -1.044436 0.209135 613 A = 0.809017 -0.587785 -0.500000 A(J .2) B = 0.309017 -0.951057 0.500000 A(2.4) C = 1.000000 1).000000 0.500000 A(2.0) Al= 0.620997 -0.730026 0.575696 A2= 0.886194 -0.365013 0.575696 '31= 1.078093 -0.225590 0.191899 82= 1.004794 -0.451181 -0.191899 Cl= 0.739597 -0.816193 -0.191899 C2= 0.547698 -0.955616 0.191899 0 = 0.888451 -0.645'+97 0.209735 614 A = 0.809017 0.587785 -0.500000 A(l.3) ., = _1.000000 _0.001)000 _0.500eoo _A(2.0) C = 0.309017 0.951057 0.500000 A(2.1) Al= 0.886194 0.365013 (\.515696 A2= 0.620997 0.730026 0.575696 81= 0.547698 0.955616 0.191899 B2= 0.739597 0.816193 -0.191899 Cl= 1.004794 0.451181 -0.191899 C2= 1.078093 0.225590 0.191899 0 = 0.888451 0.645497 0.209735 615 A = -0.309017 0.951057 -0.500000 A(1.4) ., = _0.309017 _0.951057 _0.500000 A(2.1) C = -0.809017 0.587785 0.500000 A(2.2) Al= -0.013299 0.Q55616 0.575696 A2= -0.502397 0.A16193 0.575696 Al= -0.739597 0.816193 0.191899 92= -0.547698 0.955516 -0.191899 Cl= -0.118600 1.OQSQ38 -0.191899 C2= 0.118600 1 .. 095038 0.191899 D = -0.339358 1 0 0.209135

(12)

Al6 A

=

1.000000 0.000000 0.500000 A(2.0) B

=

0.809017 -O.5tH785 -0.500000 A(l.2) C

=

0.809017 0.S8718~ -0.500000 A(1.3) A1= 0.931495 -0.225590 -0.575696 A2= 0.931495 0.225590 -0.575696 Al= 1.004194 0.451181 -0.191899 82= 1.078093 0.225590 u.191899

Cl=

1.078093 -0.225590 0.191899

C2=

1.004794 -0.451181 -0.191899 0 = 1.098185 0.000000 -0.209735 A17 A = 0.309017 0.Q51057 0.500000 A(2. 1) B = 0.809017 0.587785 -0.500000 A(I.3) C = -0.309017 0.951057 -0.500000 A(J.4) Al= 0.502397 0.816193 -0.575696 A2= 0.073299 0.955616 -0.57569" Bl= -0.118600 1.095038 -0.191899 82= 0.118600 1.095038 0.191899

Cl=

0.547698 0.955616 0.191899 C2= 0.739591 0.816193 ... O.1918~9 0 = 0.339358 1.044436 -0.209735 A18 A = -0.809017 0.587785 0.500000 A(2.2) 8 = -0.309017 0.951057 -0.500000 A(I.4) c = -1.000000 -0.000000 -0.500000 A(I.O) A1= -0.620997 0.730026 -0.575696 A2= -0.8b6194 0.365013 -0.515696 B1= -1.078093 O.2255'iO -0.191899 92= -1.004794 0.451181 0.191899 C1= -0.739597 0.8161'13 0.191899 C2= -0.547698 0.95561b -0.191899 0 = -0.888451 0.645497 ... 0.209735 AI9 A = -0.809017 -0.587785 0.500000 A(2.3) B = -1.000000 -0.000000 -0.500000 A(1.0) C = -0.309017 -0.951057 -0.500000 A( 1.1) Al= -0.886194 -0.365013 -0.575696 A2= -0.620997 -0.730026 -0.575696 91= -0.547698 -O.955b16 -0.191899 82= -0.739r;91 -0.816193 0.191899

C1=

-1.004794 -0.451181 0.1-11899 C2= -).018093 -0.225590 -0.1918Q9 0 = -0.1\88451 -0.6c.S497 -0.209735

A20 A = 0.309017 -O.qSl057 0.500000 A(2.4) 8 = -0.309017 -0.951057 -0.500000 A(l. 1) C = 0.809017 -0.'>87785 -0.500000 A( 1. 2) Al= 0.073299 -0.955616 -0.575696 A2= 0.502397 -0.816193 -0.575696 81= 0.739591 -0.AI6193 -0.191899 B2= 0.547698 -0.Q5561b 0.191899 Cl= 0.118600 -1.095038 0.191A99 C2= -0.118600 -1.095038 -0.1:j1899 o = 0.339358 -1.044436 -0,.209135

(13)

)(.

-~~~--:"'---I

A

2.0 11