On a certain surface, called Globoïd

Citation for published version (APA):Meiden, van der, W. (1980). On a certain surface, called Globoïd. (Eindhoven University of Technology : Dept ofMathematics : memorandum; Vol. 8005). Technische Hogeschool Eindhoven.

Document status and date:Published: 01/01/1980

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EINDIIOVEN UNIVERSl'ry OF TECHNOLOGY

Departme nt o f Hathematics

Memorandum 1980-05

March 1980

On a ce ~tai n s ur face , c all ed Globold

by

W. van de r Meiden

AMS subj ec t c l assifica t ion (1 979 ) 53 A 05

, ·r· .... ~.' .... U~ .. ~·

On a certain surface, cal led GloLoid

:3 1. In affine threedimensiona l Euclidean s pace E with companion vector space

fig. 1

"",3 =.' a circle y with radius e and centre 0 is given in the xz-plane of an orthogonal frame Oxy~. A pl a ne IT revolves, always parallel to the y-axis,

touching y; its intersect ion with the xz-plane, clearly a tangent line of y,

is fixed in 11 and bears the point P, fixe d in 11, at distance a from the

tangent point.

If the angle between tangent line and x-axis is denoted ~ then the position

vector of P is

(1. 1) T !? ;;; [a cos Cjl - e sin Ip, 0, a sin Cjl + e cos Ip] •

From P a line h is drawn i n 11, llaving an angle a with the tangent line; 8 and Cjl are related Ly

(1. 2) Cjl = ka

h is taken as the axi s of a c i rc ul ar cylindre ra with radius b. The GloboId is the enveloping SUL- td c e o f the system {r a} a. This note contains some investigations into the nature of this envelope. We use notation, terminology

and theorems o f r J wjthout further refere~ce.

- 2 -

2. The triple of vectors

(2. 1 ) T S := [-cos ~ cos a, sin a, -sin ~ cos eJ

(2.2) r:"" [cos ~ sin a, cos a, sin !jl sin aJT

(2.3) s:-sxr ]T

[sin !jl, 0, -cos !jl

is a positively oriented or.thonormal frame in

h and ~ being normal to IT.

From (1.1) and (2. 1,2,3) we easily obtain

]T (2.4) ~ = {9,~,~}[-a cos a, a sin a, -e

S being the direction of

For the cylindre fa is an obvious parametric representation

(2.5) ~ = e + AS + b~ cos $ + b~ sin $ ,

from which, by elimination of A and $ follows as an equation

(2.6) r a

To derive an equation o r a parametric representation for f . involves

differentiation of ~, 9, rand 5; we concentrate on this detail first.

3. Derivatives of an orthonormal frame with respect to the parameter 8, to be

denoted with a prime I, can be expressed linearly in that . frame by the skew so-called Cartanmatrix K, specifically

.-with

(3.2) K =

[. 0 -1 -k cos a

0 k sin a

cos a -k sin e 0

Through (2.4) we conclud e

(:3 • :3) ~' {gl~>~} eke () , f =: cos -ke sin fj , -ka .

; ""

. 1 ' 1",'

. : " ,!

.)

"

- 3 -

4. By differentiation o t (L. 6 ) and S0me manipulation we have

(4. 1) r I 8

The inner products in this expression can be calculated with (2.5),

(3.1,2,3) to give

bcosljl{eksin8-bksinO sinljl-A}+bsinljl{ak-kAcos8+bksin9 cos ljI}- 0

and consequently

(4.2) a - A cos e cot IjI = k ~,--_---------1\ ek sin 8

From engineering considerations we may suppose

(4.3) ek sin e < A < a / Icos 81 ,

whence cot IjI > a and ~ t 1jI, IjI arise, with IjI £ [0,

blades rand 1',

71 371 [0, 2J U [71, TJ. From one pair (8,A) two values 1T 2J . The envelope consists accordingly of two

(4.4) r x p + Aq of br cos tjJ + b s si.n IjJ -

(4.5) r x p + Aq of b r cos tjJ + bs si.n lji

Another way to arrive at (4.2) could have been considering (2.5) as a

function of three variables and equalling its Jacobian to zero.

5. We denote derivatives of functions of several of the variables a, A, IjI by subscripts 1, 2, 3 respectively; the derivatives of x in (2.5) are Dl~'

D2~' D3~; those of ~ in (4.4), where ~ depends on 8 and A according to , .

(4.2), are ~1 and ~2; those of tjJ are 1jI1 and 1j12'

We have

from which rather easily fo llo ws

confirming that rand r 0 are t angent along r n r 8' as is well known. The latter fact implies that

(5.3) ~l x ~2 ~(r cos lji + s sin lji)

Il depending on 8 and A. Thus

. ,. ~- .

.. "

- 4 -

(5.4) ~ ~ det[~l'~2' 1: cos \jJ + s sin lji]

Moreover, from (4.2) it follows that

(5.5) (A - ek ~in elcot lji '" k(a - A cos 0) ,

(5.6) IjJ 1 (A - ek sin 2

6)/sin \jJ -k(A sin 6 + e cos e cot Iji) ,

(5. 7) 1JJ2

(A - ek sin 8)/Sin2

lji :. cot lji + k cos 8 .

According to the last remark in § 4 the dependancy of Iji as a function of e and A implies that Dl~' D2~ and D3~ are linearly dependant; or, functions v

and 1 of 8 and A exist so that Dl~ ~ VD2~ + 1D3~'

Since

(5.8) D1~ = E' + Ag' + b~'cos lji + bs'sin ljJ

.j

(S· ! · :; ) f r ke cos :r lk 0 -1 -k cos ~] [; A :]]" sin 1 0 k s~n ' e : cos -ke ll-ka cos 8 -k sin e sin (g'!'''lke a s f:J b cos lji - bk cos 8 sin 1JJl

-ke sin 8 + A + bk sin 8 sin ljJ j -ka + Ak cos e - bk sin e cos ljJ (5.9) D2~ S ' D3~ -br sin tjJ + bs cos lji

and

(5.10) (r ..

cos tjJ + s sin lji) )( D3~ bg -

we derive that

(5.11) v ~ ke cos 8 - b cos tjJ - Lk cos e sin tjJ ,

moreover

(-ke sin e + A + bk sin e sin tjJ): + (-ka + Ak cos 8 - bk sin e cos tjJ)~ =

1 (-br sin tjJ + b~ cos ~Jl ,

hence

' . I

- l ' .

- 5 -

(5.12 ) beT + k sin e)sin w -\ + ke si.n e

-beT + k sin G)cos ~ ka Ak cos e

which confirms (4.2) and leads to

(5. 13) -A + ke sin 0 --------~--- - k b sin 1jJ

sin e k(\ cos 8 - a) _ k ~in e • b cos ~)

Substituting these results in (5.4) entails

6. We borrow that (see [ ] 3.5)

(6.1)

Since N = ~1 x ~2/1~1 x ~21 it follows from (5.3) that

(6.2) N = ~ I ~ ,-1 (!:: cos ~ + S s in ~) .

Writing for the moment £ -= ~I~I-l we have

hence

Now g

(6.4)

-~ g R.

2

£(r' cos Ij! + s'

£ det[ r cos

~ and

9,

sin

~1jJ2 sin 1jJ (cot 1jJ + k cos 8) .

S ince

(6.5) cot lji + k cos 8 k •

- 6 -

a - ek ~in e cos 8 A - ek sin 8

it is seen from (4.3) a nd (5 . 7 ) that

(6.6) 5gn t :; sgn(1J sin lji) •

Literature

Meiden, W. van der, Meetkunde en Kinematica;

syllabus 2212, THE, 1980.

., .