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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.
., .
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