The Development of the Oloid

Journal for Geometry and Graphics

Volume No

Hans Dirnbock and Hellmuth Stachel

Institute of Mathematics University Klagenfurt

Universitatsstr A Klagenfurt Austria

Institute of Geometry Vienna University of Technology

Wiedner Hauptstr A Wien Austria

email stachelgeometrietuwienacat

Abstract Let two unit circles k k in p erp endicular planes b e given such that

A B

each circle contains the center of the other Then the convex hull of these circles is

called Oloid In the following some geometric prop erties of the Oloid are treated

analyticallyItisproved that the development of the b ounding torse leads to

elementary functions only Therefore it is p ossible to express the rolling of the

Oloid on a xed tangent plane explicitly Under this staggering motion whichis

related to the wellknown spatial Turbulamotion also an ellipsoid of revolution

inscrib ed in the Oloid is rolling on We give parameter equations of the curve

of contact in as wellasofitscounterpart on

The surface area of the Oloid is proved to equal the area of the unit sphere Also

the volume of the Oloid is computed

Keywords Oloid Turbulamotion development of torses

MSC N A A

Intro duction

Let k k be two unit circles in p erp endicular planes such that k passes through

A B A

the center M of k and k passes through the center M of k see Fig Thetorse

B B B A A

developable connecting k and k is the enveloping surface of all planes that touch k

A B A

and k simultaneouslyIfanytangentplane contacts k at A and k at B then the line

B A B

AB is a generator of In this case the tangentlineofk at A must intersect the tangent

line of k at B in a nite or innite p oint T on the line of intersection b etween and

see Fig the triangle AB T can also b e found in Fig and Fig

All gures in this pap er are orthogonal views But only in Fig and Fig the sup erscript nisused

to indicate that geometric ob jects have b een pro jected orthogonally into a plane

ISSN Heldermann Verlag

H Dirnbock H Stachel The Development of the Oloid

Figure Circles k k dening the Oloid

A B

Wecho ose the planes as co ordinate planes and the midp oint O of M M as the

A B

origin of a cartesian co ordinate system Then wemay set up the equations of k k as

A B

and z k x y

k y z and x

We parametrize the torse by the arclength t of k with the starting p oint t at U on

Figure Co ordinate system and notation

the negative y axis Then we obtain the co ordinates

A sin t cos t

H Dirnbock H Stachel The Development of the Oloid

Since the p oint T on the y axis is conjugate to A with resp ect to k we get

cos t

In the same way conjugacy b etween T and B with resp ect to k implies

cost cos t

cos t cos t

The upp er sign of the z co ordinate corresp onds to the upp er half of

From and we compute the squared length of the line segment AB as

cos t cos t

sin t cost cos t AB sin t cos t

cos t cos t

which results in

Theorem All line segments AB of the torse are of equal length

This surprising result has already b een proved in But probably also P Schatz was aware

of this result when he to ok out a patent for the Oloid cf in see also Figures

and p

Let u denote the arclength of k starting on the p ositive y axis Then A k and

B A

B k are p oints of the same generator of if and only if the parameters t of A and u of B

ob ey the involutive relation

t u cos t

or cos cos cos u

cos t

For real generators of the condition cos t is necessary By the restriction

t and u

weavoid vanishing denominators It has to b e noted that for the parametrization by t

b ecomes singular at t

In the following we restrict each generator of to the line segment AB Thus we obtain

just the b oundary of the convex hul l of k and k

A B

Development of the Torse

When is develop ed into a plane then the circles k k are isometrically transformed into

A B

d d

planar curves k k resp ectively Itiswellknown from Dierential Geometry see eg

A B

d d

the geo desic p or p that at corresp onding p oints A k and A k

curvatures are equal This can b e expressed in a more geometric way as follows cf p

When is sp ecied as the tangent plane of along the generator AB then the curvature

In the generalization presented in the circles k k are replaced by congruent ellipses with a common

A B axis

H Dirnbock H Stachel The Development of the Oloid

d d

center K of k at A A is lo cated on the curvature axis of k at A which is the axis of

revolution of the curvature circle k see Fig compare Fig Since K k

is aligned with T and B we get for the squared curvature radius

cos t

AK k

cos t

Hence the curvature of k reads

cos t

This is the socalled natural equation of k with arclength t

In order to deduce an explicit representation of k wecho ose as the tangent plane at

the p oint U k with minimal y co ordinate In weintro duce a cartesian co ordinate system

with origin U U and axes I and II see Fig We dene the rst co ordinateaxis I parallel

to the tangentvector of k at U Then due to Riccatis formula see eg p weget

cos t dt I t I

t t dt for

II t II sin t dt

with the sp ecications I II Byintegration of weobtain

tan

sint

t arcsin arcsin

cos t

Theorem In the cartesian co ordinate system I II see Fig the arc length parame

trization of the development k of the circle k reads

p p

cost

cos t cos t arccos I t

cos t

cos tln II t

cos t

Proof The integrals in the left column of could not b e immediately solved with the use

of common computeralgebrasystems We succeeded as follows The integral for II tcan

b e transformed into

tan

sin t

sin arcsin II t arcsin dt

cos t

p p

t t

cos t cos t sin tan

cos t

and this gives rise to the second equation in From

dII

sin t sin

dt cos t

The sign of the z co ordinate is equal to that of B in

Note but This proves that at U there is exactly a fourp ointcontact b etween

k and its curvature circle see Fig or Fig A

H Dirnbock H Stachel The Development of the Oloid to tin elopmen Figure Axonometric view of the Oloid and its dev

H Dirnbock H Stachel The Development of the Oloid

due to we obtain

dI dII cos t

A A

cos

dt dt

cos t

Then the integration can b e carried out using the substitution The rst quarter t tan

of the develop ed curve k ends at

p p

I II ln

A A

There is an analogous representation of the develop ed image k of the circle k in terms

d d

of its arclength u The curves k and k are congruent since halfturns ab out the axes

A B

x z y interchange k and k while the Oloid is transformed into itself However

A B

based on and due to the curve k can also b e parametrized in the form

cos e I t I t

AB B A

II t II t sine

B A AB

d d

Here the angle e see Fig denes the direction of the develop ed generator A B

Angle has already b een computed in and is the angle made by the generator

AB of and the tangentvector

v cost sin t

of k at A The dot pro duct of v and the vector AB according to and gives

A A

sin t sin t cos t

cos v AB sin t cos t sint sint cos t

cos t cos t

Elementary trigonometry leads to

cos t

sin

cos t

and nally to

cos t cos t

sin e

cos t

cos t cos t cos t

cos e

cos t

We substitute these formulas in Then due to we obtain

Theorem In the cartesian co ordinate system I II of see Fig or Fig the devel

of the circle k has the parametrization with resp ect to the arclength t of k as opment k

B A

follows

p p

cos t cos t

cost

I t arccos

cos t

ln II t

cos t cos t

H Dirnbock H Stachel The Development of the Oloid d A k of d A k e olute together with the ev B k of d B k and A k of d A k t of the Oloid with the images elopmen Figure The dev

H Dirnbock H Stachel The Development of the Oloid

Figure Detail of Fig with the image k of the center curve k

under orthogonal pro jection into

In a similar way also the evolute e d of k see Fig or Fig can b e computed The

parameter representation

I t I t sin

K A

II t II t cos

K A

of e d makes use of the curvature radius according to From and we obtain

cos t cos t

sin

cost

cos cos t

H Dirnbock H Stachel The Development of the Oloid

and nally as parametrization of the evolute e d of k

k A

p p

cos t

cost

I t arccos

cos t

ln II t

cos t

The evolute e d obviously see Fig do es not pass through the cusps of k the curvature

k A

radius tends to innity This reveals that these cuspidal p oints are not ordinaryFor k

the Taylorseries expansion of the parameter representation at t is

I t t O t II t t O t

B B

d d

Therefore the singularities of k and k are of order and class German Ruc kkehrach

A B

punkte

Motions Related to the Oloid

According to Fig we assume that the Oloid is rolling on the upp er side of In the following

we therefore cho ose for p oint B in the negative z co ordinate For the sakeofbrevitywe

substitute

c s s c s sin t and c cos t with

In order to describ e the rolling of on weintro duce a moving frame of with origin

A k The rst vector of this frame is the tangentvector v according to The second

A A

vector w p erp endicular to v is sp ecied in the tangent plane We dene

A A

p p

AB v cos s w c c c

A A

sin

The vector

c n v w s c

A A A

p erp endicular to completes this cartesian frame n is p ointing to the interior of

While the Oloid is rolling on the xed plane the frame A v w n shall b e moving

A A A

along in suchaway that A is the running p ointofcontact b etween k and This

d d

implies that A k is always coincident with the corresp onding p oint A k Therefore

the elements of the moving frame get the following co ordinates with resp ect to the cartesian

co ordinate system U I II III attached to

A I t II t v cos sin

A A A

w sin cos n

A A

Let x y z denote the co ordinates of any p oint P attached to the Oloid The required

representation of the motion consists of a matrix equation whichallows to compute the in

stantaneous co ordinates I II III ofpoint P with resp ect to the xed plane in dep endence

H Dirnbock H Stachel The Development of the Oloid

of the motion parameter t In order to obtain this equation we rstly compute the co ordinates

ofP with resp ect to the moving frame A v w n Though P is attached to

A A A

these co ordinates are dep endenton tFrom and we get

c c s c s

s x

C B

B C C B B C

C B

y c

A A A

s c c c c

Secondly according to the motion of the moving frame with resp ect to see Fig

reads

I I t cos sin

B C B C B C B C

II II t sin cos

A A A A

III

Nowwe eliminate from these two matrix equations with orthogonal matrices

After substituting and we obtain by straightforward calculation

Theorem Based on the cartesian co ordinate systems x y z in the mo ving space and

I II III in the xed space the rolling of the Oloid on the tangent plane can b e represented

cs c c

q p

arccos

B C

c c

B C

I x

B C

c c

B B C C B C

B II C a y where

A A

B C

c c

B C

III z

B C

p p

c c c cs cs

q q q

C B

c c c c c

C B

c c s c c c

C B

c c c

C B

p p

C B

c s

q q q

c c c

Here c and s stand for cos t and sin t resp ectively while the motion parameter t ob eys

The rst vector on the right side of this matrix equation represents the path k of the

Oloids center O under this rolling motion In particular the third co ordinate of this vector

gives the oriented distance

between O and the tangent plane for each t Due to the intro duced moving frame this

distance r equals the dot pro duct n AO One can verify that for each t the velo cityvector A

H Dirnbock H Stachel The Development of the Oloid d l een and traces w tact b et of con l e while the curv Figure The Oloid and the inscrib ed ellipsoid are rolling on

H Dirnbock H Stachel The Development of the Oloid

d d

of the center curve k is p erp endicular to the axis A B of the instantaneous rotation see

orthogonal view k of k in Fig or Fig

The rolling of the Oloid is truly staggering It is related to the Turbula motion see or

and the references there which is used for shaking liquids It turns out that the Turbula

motion is inverse to the motion of the moving frame in

b b

The circles k and k can also b e seen as singular surfaces k k of class The

A B A B

co ordinates u u u u of their tangent planes

u u x u y u z

match the tangential equations

b b

k u u u u u k u u u u u

A B

Then due to a standard theorem of Pro jective Geometry the torse is not only tangentto

k and k but to all surfaces of nd class included in the range which is spanned by k and

A B A

k Among these surfaces there is an ellipsoid of revolution ob eying the equation

b b b

x y z or k k u u u u

A B

and cf p The curve l of contact with fo cal p oints M M and semiaxes

A B

between and the torse is lo cated on cylinders which are the images of k and k

A B

resp ectively in the p olarity with resp ect to Therefore this curve has the representations

z y or l x y

c s c

x y z

c c c

Together with the Oloid also the inscrib ed ellipsoid is rolling on In the xed plane the

p oint of contact with the rolling ellipsoid traces a curve l The parameter representation

p p p

c c

arccos ln III II l I

c c

of this isometric image of l is obtained by transforming the co ordinates of l given in

negative sign under the matrix equation of Theorem

d d d

k and l Fig shows not only the xed tangentplane with the develop ed curves k

A B

in true shap e In this gure also an orthogonal view of the Oloid with the inscrib ed ellipsoid

and the curve l of tangency is displayed

In general the instantaneous motion is a helical motion However when a torse is rolling on a plane the

helical parameter must vanish cf p or

In this pap er a very particular planesymmetric sixbar lo op is studied which is also displayed in

Figure In each p osition of this lo op and for eachtwo opp osite links there is a plane of symmetry

It turns out that relatively to these planes are tangent to a torse of typ e The Turbula motion is the

motion of relativeto whenin the generator of the torse is kept xed

In the cases treated in k and k are ellipses but is a sphere This implies that the center O of

A B

gravity has a constant distance to during the rolling motion

H Dirnbock H Stachel The Development of the Oloid

Surface Area and Volume of the Oloid

As the development of a torse is lo cally an isometry the computation of the area of can

b e carried out either in the space or after the developmentinto the plane We prefer the

latter and use a formula given in p eq The area swept out by the line segment

AB under a planar motion for t t t can b e computed according to

S v v AB dt

A B

where v v are the velo cityvectors of the endp oints For vectors in R the norm in this

A B

formula can b e cancelled which gives rise to an even oriented area

In the co ordinate system I II of we obtain due to and

p p

dII dI

A A

AB v cose sin e

A AB AB

dt dt

p p

dI de de dII

A AB AB

v sine cos e

B AB AB

dt dt dt dt

and according to and

p p

dII de de dS dI

A AB AB A

sin e cos e sin

AB AB

dt dt dt dt dt

Eq and the derivation of lead to

cos t

dS cost

cos t

cos t cos t

which nally results in

cos t dS

cos t cos t

After integration we obtain up to a constant k

cost cos t

S t arcsin arcsin k

cos t

and for the complete torse

S S S S S

Theorem The surface area of the Oloid equals that of the unit sphere

The computation of the Oloids volume starts from Each surface elementof is

the base of a volume element forming a pyramid with ap ex O Its altitude r has already b een

computed in as it equals the distance b etween O and the corresp onding tangent plane

Thus weobtain

r cos t

dV dt dS

cos t cost

Anumerical integration gives

V V V

H Dirnbock H Stachel The Development of the Oloid

Acknowledgements

The authors express their gratitude to Christine Nowak Klagenfurt for her successful help

in solving the integrals Sincere thanks also for p ermanentcontributions given by Jakob

Kofler Klagenfurt The authors are indebted to Gerhard Hainscho Wolfsb erg for

fruitful discussions and to Sabine Wildberger Klagenfurt for manufacturing a mo del

Finally the authors thank Manfred Husty Leob en for useful comments and Gunter Weiss

d d

Dresden for p ointing their attention to the singularities of k and k

A B

References

JE Baker The Single Screw Reciprocal to the General PlaneSymmetric SixScrew

LinkageJGG

W Blaschke HR Muller Ebene KinematikVerlag von R Oldenbourg Munchen

O Bottema B Roth Theoretical Kinematics NorthHolland Publ Comp Ams

terdam

H Brauner Lehrbuch der Konstruktiven Ge ometrie SpringerVerlag Wien

C Engelhardt C Ucke ZweiScheibenRol le r MNU Math Naturw Unterr

M Husty A Karger H Sachs W Steinhilper Kinematik und Robotik

Springer Berlin

S Kunze H Stachel Uber ein sechsgliedriges raumliches Getriebe Elem Math

P Schatz Deutsches Reichspatent Nr in der allgemeinen Getrieb e

klasse

P Schatz Rhythmusforschung und TechnikVerlag Freies Geistesleb en Stuttgart

K Strubecker Dierentialgeometrie I Au Sammlung Goschen Bd a

Walter de Gruyter Co Berlin

K Strubecker Dierentialgeometrie II Au Sammlung Goschen Bd

a Walter de Gruyter Co Berlin

TJ Willmore AnIntroduction to Dierential Geometry Oxford UniversityPress

gelmaigen sechsgliedrigen Wurfelkette Pro W Wunderlich Umwendung einer re

ceedings IFToMM Symp osium Mostar May

Received July nal form Novemb er