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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.


[No one ignorant of geometry admitted]
Entrance of Plato's Academy (founded 387 BC)

Related articles on this site:

Related Links (Outside this Site)

Gearing  by  John J. Coy, Dennis P. Townsend, Erwin V. Zaretsky  (1985).
Old Gear  by Dr. James B. Calvert  (University of Denver)
Gear Types  by  Stephen Canfield   |   Gear (Wikipedia)
Gear Design and Engineering  at  Engineers Edge   |   Tribology ABC
Introduction to Mechanisms   ( Ch. 7 )   by Yi Zhang, S. Finger, S. Behrens
Spur Gears  by  Douglas Wright   |   List of Gear Standards
Engrenages & Routes et roues (in French) with animations by Alain Esculier.
Designing Cycloidal Gears  by  Hugh Sparks   |   Cycloidal Gear Generator
How Gears Work  by  Karim Nice   |   Harmonic Drive Gearing
How Calculating Machines Worked   ( The Museum of  HP Calculators )
Oriental Motor   |   Gearing Solutions   |   DR Gears   |   Meshing with Gears
Tungsten and molybdenum fabrication   by  Diamond Ground Products.

Bibliography :

Analytical Mechanics of Gears   by  Earle Buckingham (1887-1978)
McGraw-Hill, 1949.  Dover Publications, 1963 & 1988  (ISBN 0-486-65712-4).

Wheel and Pinion Cutting in Horology   by  J. Malcolm Wild, FBHI (b. 19??)
The Crowood Press Ltd, 2001.  Hardcover reprint 2012, 253 pp.  (ISBN 0-978-1-86126-245-5).  Appendix includes extracts from relevant Swiss standards  (NIHS)  20-01, 20-02, 20-10, 20-25  and British standard  BS 978-2 (1952).

Machinery's Handbook25th edition  (1996)   by  Erik Oberg 1881-1951,  Franklin D. Jones 1879-1967,  Holbrook L. Horton 1907-2001,  Henri H. Ryffel 1920-2012.
Editors:   Robert E. Green  and  Christopher J. McCauley.
Industrial Press Inc., New-York 1996   (ISBN 0-8311-2575-6).
(First published in  1914.  The  29th edition was released on January 2, 2012.)

Nanjing Plusplus Software Co., Ltd.
Gears,  textbooks & training:  Salem Company,  Woodstown, NJ 08098
Videos:  Non-circular gears and planetary gear   |   Super Oval Flowmeter
How to make wooden gears  by  Matthias Wandel.
Wooden Gear Template Generator (video demo)  by  Matthias Wandel.
Cutting Wood Gear & Clock Wheel Teeth  by  Ronald Walters.
These Gears Really Work?  by  Dr. Clayton Boyer.
Around the Corner:  How a differential gear works  (Jam Handy, 1937).
Making Gears    by  TheMetalCutter.



gear  is a toothed wheel, rigidly attached to an axis of rotation  (axle).  It meshes with other gears to transmit rotary motions to other axles.

The following geometrical study is mostly concerned with the exact shape of ideal gears.  In this context, we may use the word  pinion  to denote a  single-tooth  gear which may lack the axial symmetry of gears with several teeth.  This usage is more restrictive than the ordinary meaning of the word, which is part of the following mechanical jargon:

  • Addendum  For a straight spur gear, this is the maximum height of a tooth above the pitch circle.
  • Addendum Curve  The part of the profile that's above the pitch circle.
  • Annular Gear  A gear whose teeth are cut on the  inside  of a ring.
  • Axle  The axis around which a gear revolves.  Also called  shaft.
  • Backlash  The amplitude of the back-and-forth motion allowed in one gear when a meshing gear is held in place.  (This is normally measured in module units, along the pitch circle).
  • Base Circle  The circle of which an involute tooth profile is an  involute.
  • Bevel Gear  A conical gear, used to connect intersecting shafts.
  • Cage Gear  Another name for a  lantern gear  (which see).
  • Cam  A smooth solid imparting a specific motion to a so-called follower in contact with it (often spring-loaded).  A disk cam, is a rotating cylinder whose lateral surface drives a flat follower, whereas the active surface of a cylinder cam is actually helical...  A cam in straight motion is called a translation cam.
  • Center Distance  The distance between the two axles  (parallel or not).
  • Centrode  The motion of a rigid plane in a fixed plane, is either a pure translation or a rotation about a so-called  instantaneous center of rotation  (which has zero speed in either plane)  whose path is dubbed  centrode.  The centrode in one plane rolls without slipping on the centrode in the other.
  • Circular Pitch  (Also called tooth space.)  The curvilinear distance between the centers of two adjacent teeth, as measured along the pitch circle.  In module units, the cicular pitch is always equal to  p.
  • Clearance  The amount by which the dedendum of a gear exceeds the addendum of another gear when both mesh.
  • Cog  Another name for the tooth of a gear  (especially for wooden gears).
  • Crown Gear  A wheel with [straight or helical] teeth on its flat side.
  • Dedendum  For a straight spur gear, this is the maximum depth of [the fillet of] a tooth, below the pitch circle.
  • Dedendum Curve  The part of the profile that's below the pitch circle.
  • Elliptic gears  The family of compatible gears described below which are syntrepent to an ellipse rotating about one of its foci  (which can be viewed as the basic single-tooth pinion belonging that family).
  • Elliptical gears  Meshing gears in the general shape of  syntrepent ellipses  possibly endowed with small matching teeth on their respective circumferences to prevent slipping.  Opposed to  circular gears.
  • External gears  are regular gears, as opposed to internal or annular ones.
  • Face of a Gear  See  flank.
  • Fillet  The deep part of the teeth  (near the  dedendum)  which is never in contact with meshing gear.  (As opposed to the kinematically relevant flank.)
  • Flank  The surface of the gear which comes in contact with meshing gears.  We consider flank and face to be synonymous.  However, some authors reserve the word flank for the part of active surface which is inside the pitch surface and call face the part outside of the pitch surface.
  • Gear Ratio  The ratio of the (average) angular velocity of the input gear to the (average) angular velocity of the output gear.  Unless the input and output axles are perpendicular, the notion of a signed gear ratio  (or  algebraic gear ratio  can usefully be introduced.
  • Gear Train  Any system of two or more meshing gears.
  • Gender  Except in the special case of  genderless  families  (of which elliptic gears are an example)  the tooth profiles in a family of compatible spur gear come in two distinct genders.  Only teeth of opposite genders can mesh  externally.
  • Helical Gear  A gear whose flank spirals around the shaft.
  • Herringbone Gear :  Two helical gears of opposite handedness, side by side on the same shaft  (to cancel the axial thrust of a  single  helical gear).
  • Hypoid gears  connect two shafts that do  not  intersect.  (The term is a contraction of "hyperboloid", which is the pitch surface for such a gearing.)
  • Internal Gear :  See annular gear.
  • Isotrepent  The qualifier applying to a curve which is syntrepent with itself with respect to one of its points.  Examples include ellipses and logaritihmic spirals  (French:  courbe isotrépente).
  • Lantern Gear :   Lantern Gear 
 (Cage Gear) Also called:  cage gear.  A wooden gear  (at right)  consisting of two disks connected by rods that serve as gearing teeth.  Omitting one of the supporting disks turns this into a  pinwheel gear,  which can also be used as a  crown gear  (usually in a  mitter gear  arrangement).  The term  pinwheel  may also denote a rudimentary gear obtained by attaching rods  radially  to a solid disk.
  • Leaf  The tooth of a gear, in clockmaking parlance.
  • Miter Gear  A conical gear transmitting rotation between two shafts intersecting at a right angle  (the most common type of  bevel  gear).
  • Module  A unit of length equal to the diameter of the pitch circle divided by the number of teeth.  It's used to describe the tooth profile in general terms.  In module units, the circular pitch of a gear is always equal to  p.
  • Pinion  A small gear with few teeth  (possibly, a  single  tooth).  When discussing a pair of meshing gears, the smaller one is called the pinion whereas the larger one is the wheel  (or the rack, for an infinite radius).
  • Pinwheel Gear  See  lantern gear.
  • Pitch Circle  Loosely speaking, what the cross section of a straight spur gear would become without its teeth  (see pitch surface, below, for more precision).
  • Pitch Point  Let K be the point of contact of two planar gears as they mesh.  Their common normal through K intersects the line of the two rotation centers at a point P called the pitch point.  (P = K if and only if there's no slipping).
  • Pitch Surface  The pitch surfaces of two meshing gears are the abstract surfaces attached to each of them which roll without slipping on each other in a uniform rotation equal to the  angular  average of the actual motion.  A pitch surface is always a ruled surface of revolution, namely:  a plane for a crown gear, a cylinder for a spur gear, a cone for a bevel gear, an hyperboloid for an hypoid gear.
  • Profile  The shape of a gear's tooth.  The planar curve corresponding to its cross-section in the case of a straight spur gear.
  • Rack  A toothed bar, which may be viewed as a gear of infinite radius.
  • Shaft  The axis around which a gear revolves.  Also called  axle.
  • Spur Gear  A cylindrical wheel, with teeth cut across its circumference.
  • Straight Gear :  Spur gear or conical gear whose teeth are cut along straight lines parallel to the shaft or intersecting it.  Opposed to  helical gear.
  • Syntrepent Curves  Planar curves which roll on each other without slipping as they rotate about two centers.  (Miquel 1838.  French:  courbes syntrépentes).
  • Tooth Profile  Shape of a tooth  (the same shape is repeated for all teeth).
  • Tooth Space  See  circular pitch.
  • Worm Gear  An endless screw driving an helical gear perpendicular to it.

(2012-11-21)   Gear Ratio   ( signed  or  unsigned )
The ratio of the driver's rotation to the  output  rotation.

The  gear ratio  of any gear train is defined as the ratio of the (average) angular velocity of the input gear to the (average) angular velocity of the output gear.  Thus, if the driving gear rotates  5  times faster than the (final) driven gear, the gear ratio is  5.

In the automotive realm, "overdrive" denotes a gear ratio less than 1  (French:  surmultipliée ).  A gear ratio of 1 is called "direct drive".

That ratio is often taken only as a positive quantity involving the  magnitudes  of the rotation rates, irrespective of their directions.

However, if the input and output axles are  not  perpendicular  (in particular, when they are parallel)  the directions of their rotations can be compared unambiguously.  The gear ratio can then usefully be given a sign  (the same sign as the dot product of the relevant rotation vectors).

For a simple train of two spur gears, the  algebraic gear ratio  so defined is negative is the two gears are meshing externally and positive when they are meshing internally  (one of the gears is  annular  in the latter case).

The gear ratio is zero if the driver is a  rack  in rectilinear motion,  unless the output gear is itself also a rack  (in which case the gear ratio is undefined).  If the output is a rack and the driver isn't, the gear ratio is  infinite.

(2005-12-11)   Gears which Roll without Slipping 
"Perfect" straight spur gears roll against each other without slipping.

When two rigid planar curves roll against each other without slipping, the point of contact has zero velocity with respect to  either  curve.

The planar cross-sections of two straight spur gears rotate respectively around two points O and O'.  If these curves roll against each other in the above sense, the velocity of the point of contact M is perpendicular to  both  OM and O' M.  This implies that M is on the line OO'  joining the two centers of rotation.

Some slipping is thus necessarily involved in gear pairs (see involute gears)  which hold the rotational velocity ratio strictly constant.  (Otherwise, the point of contact would maintain constant distances from both centers of rotation, because such distances would have a fixed sum and a fixed ratio...)

The polar coordinates of the point of contact (M) in the systems bound to either curve obey the following differential equation.  The distance  a  between the centers or rotation is r+r' for external gearing, and  | r-r' |  for internal gearing  (where one of the gears is an annular gear).  Polar coordinates for 
 two planar gears

r dq  +  r' dq'   =   0

Genders :

If two curves mesh with a third, they'll mesh  internally  with each other.  Two  genders  are thus defined so that profiles of the same gender mesh internally with each other.  Curves of opposite genders mesh  externally.

If one curve meshes  externally  with itself  (as shown next in the case of an ellipse)  then all curves that mesh with it do so both internally  and  externally, thus forming a  genderless  family of compatible gears.

 Congruent ellipses symmetrical 
 with respect to a common tangent. (2005-12-11)   Ellipses are Isotrepent Curves 
Syntrepent  planar curves roll on each other without slipping as they rotate around two fixed centers.  A curve syntrepent to a copy of itself [with respect to matching centers] is said to be  isotrepent.
An ellipse is isotrepent about its focus.

Both terms (French: courbes syntrépentes, courbe isotrépente)  were coined by the French mathematician  Auguste Miquel  (1816-1851)  in 1838.

Ellipses are isotrepent because congruent ellipses may roll on each other without slipping, as they rotate around their respective foci. In such a motion, the two ellipses are symmetrical about their tangent of contact, as illustrated above.

In this symmetrical configuration, the line joining two "opposite" foci goes through the point of contact.  This may be proved using the fact that an ellipse reflects any ray from a focus back to the other focus.  (HINT:  Draw the four lines going from the contact point to each focus,  then deduce collinearity from angular relations.)

This gearing does not allow one pinion to drive the other in practice, since it pushes against the other for only half of each cycle.  Instead, the same motion can be reproduced in a gear-free mechanism, by tying the two moving foci with a rigid rod...  This tranfers rotary motion from one shaft to the other in a 1:1 ratio.

Unfortunately, that simple mechanism retains a dead point when the 4 foci are aligned.  In the absence of a flywheel, the direction of rotation can indeed reverse itself from this dead position  (both shafts may rotate in the same direction if the bar tying the moving foci remains parallel to the line joining the fixed foci).

Elliptical Gear Applet

(2005-12-10)   From Ellipse Pinion to Sinewave Rack   (Michon, 1975)
Elliptic spur gears  roll on each other as they rotate  (no sliding).

This family of gears involves only pure roll  (no sliding or slipping)  at the expense of Euler's conjugate action  (which would make the driven gear rotate at a uniform rate if the driver does).  These gears are thus more suited for unlubricated clockwork than high-power lubricated machinery.

I devised this as a teenage student  (in 1974 or 1975)  mostly to test my calculus proficiency.  I was convinced that the same idea must have occurred to many people and left it at that.  It seems that nobody ever bothered to publish it, though.  The simplicitity of the final result could be expressed and/or justified purely in geometrical terms, but I'll derive it here using the same differential approach as my younger self:

If a focus is used as origin, the polar coordinates   (r,j)   of an ellipse of eccentricity  e  and parameter  p  obey the equation:

r   =   p / (1 + e cos j )

This polar equation applies to any conic section.  The scaling parameter  (p)  specifies the size of the curve and the eccentricity  (e)  speciies its shape:  e=0  for a circle,  e<1  for an ellipse,  e=1 for a parabola,  e>1  for an hyperbola.  Here, we assume  0<e<1.

Thus, the polar coordinates   (r,q)   of a planar curve which rolls without slipping on that ellipse, while rotating around a center orbiting at distance  A  from the origin, obey the following differential equation:

(r-A) dq  =  r dj
So:   dq     =     - dj / ( A/r - 1 )     =     - p dj / ( A + e A cos j - p )

Introducing the variable   t  =  tg(j/2)   we have   dj  =  2 dt / (1+t2 )
and   cos j  =  (1-t2 ) / (1+t2 ).  Therefore:

dq   =     - 2p dt  
A + e A (1-t2 ) - p

 Come back later, we're
 still working on this one...

Introducing  n  such that   n2 p2  =  (A-p)2 - (Ae)2,  this boils down to:

Polar equation of an elliptic gear of order  n :
r   =     n2 p
[ n2 (1-e2) + e2 ] ½  +  e cos(nq)

Closed contours are obtained when  n  is an  integer  (which is what we normally want for an actual gear, except in the rare situations when the gear will never execute a complete turn while meshing with another gear).

For given values of the parameter  (p)  and eccentricity  (e)  elliptic gears form a  genderless  family:  Every curve meshes with any other, either externally or internally  (for different values of  n  in the latter case).

Sinusoidal rack meshing with elliptic gears :

For  very large  values of  n,  the gear's median radius is nearly equal to:

R   =   n p / ( 1-e2 ) ½

The limit of such a gear is best described as a  straight rack  whose cartesian equation is obtained, as  n  tends to infinity, via the substitutions:

x   =   R q  
y   =   r - R

This yields, neglecting relative errors proportional to  1/n2  or less,

n q   =   n x/R   =   x  ( 1-e2 ) ½ / p

The relation   y = r-R   then gives the cartesian equation of a  sine wave :

y   =     - p e     cos [    ( 1-e2 ) ½   x ]
Vinculum Vinculum
1-e2 p

 (definition of a and b) Let's express this in terms of the traditional notations  a, b and c  for, respectively, the  major radiusminor radius  and  focal radius  of the matching  ellipse:

Sinusoidal Rack   (n = ¥)
y   =   - a e  cos ( x / b )   =   - c  cos ( x / b )

The unessential appearances of a  negative  sign and a  cosine  (rather than a  sine )  come from choosing the origin of  x  at a point where  y  is smallest.

To summarize, the ellipse and the sinewave so described can roll without slipping on each other as one of the foci of the ellipse remains at a fixed distance from the axis of the sinewave.

This fact implies that the  perimeter of the ellipse  is equal to the length of one full arch of a sinewave of  wavelength   2p b   and amplitude   c = a e.  (The distance between the two foci is  2c.)

Family of compatible elliptic gears :

We may define the  nominal radius  Rn  of an elliptic gear of order  n  as the half-sum of its smallest and largest radius  (i.e., the distances from the axle to the root of a tooth and to the tip of a tooth).

Rn   =   a  [ n2 (1-e2 ) + e2 ] ½   =   [ n2 b2 + c2 ] ½   =   [ a2 + (n2-1) b2 ] ½

Nominal radius of an elliptic gear of order  n :
Rn   =   n b   Ö   1  +   e2  
n2 (1-e2 )

In particular, for the basic ellipse  (n = 1)  we have:   R1  =  a  =  p / (1-e)

The  axle  of an elliptic gear of order  1  is at a focus of its elliptical contour.  This isn't the center of symmetry of that  single-tooth  gear.

If the axles of two compatible elliptic gears  (i.e., same  p  and same  e )  are separated by a distance  A  equal to the sum  (resp. the difference)  of their nominal radii, those gears can roll externally  (resp. internally)  on each other  [without any sliding]  as they rotate about their respective shafts.

(2012-11-21)   Pitch Radius   vs.  Nominal  (Median)  Radius 
The pitch radius of a gear  does  depend on what it meshes with.

We'll use elliptic gears to quantify the distinction, which is often butchered.

The  median radius  or  nominal radius  Rn  is an  intrinsic  mesurement of an  n-tooth  gear.  You can measure it on a given gear without knowing anything about the rest of the mechanism.

On the other hand, the  pitch radius  of a gear depends on what it actually meshes with.  If an  n-tooth gear  of median radius  Rn  meshes  externally  with an  m-tooth gear  of median radius  Rm ,  their pitch radii are:

R'n   =   ( Rn + Rm )  n / (n+m)
R'm   =   ( Rn + Rm )  m / (n+m)

The sum   R'n + R'm  =  Rn + Rm   is the distance between the two axles.

For gears meshing  internally,  that distance is the difference between the radius of the larger gear  (the annular one)  and that of the smaller one:

R'n   =   ( Rn - Rm )  n / (n-m)
R'm   =   ( Rn - Rm )  m / (n-m)

This is to say that the previous formulas remain true if we make the convention that an  annular  gear has a negative number of teeth and a negative radius.  With our previous expression of the nominal radius of an elliptic gear as an  odd  function of its order, we may simply view annular gears as gears of negative order.

In any  genderless family of gears,  if   m = n ,  then   R'n = Rn .  That is also the case when  m  is infinite  (an n-tooth gear meshing with a rack)  as is readily seen by envisioning the rack profile moving forward between two gears with the same number of teeth meshing externally...

As an  m-tooth  driver meshes with an  n-tooth  wheel, the quantity  R'n - Rn  (for a constant value of  n)  can be viewed as a function of the  gear ratio  x.

We have  |x| = n/m.  Recall that the gear ratio  x  is negative for external meshing, positive for internal meshing and zero if the driver is a rack).  x  goes from  -n  to  n  (x = 1  being the dubious case of a  frozen  gear meshing internally with itself).

In the case of  elliptic gears, this function is:

 Come back later, we're
 still working on this one...

(2012-11-21)   Envelope of an orbiting gear
How to detect flawed meshing, the way Euler could have done it...

 Come back later, we're
 still working on this one...

So far, we've been considering gears only as pairs of smooth mathematical contours that keep sharing a common tangent as they rotate about two fixed centers of rotation.

Such contours can .../...

Consider a plane where one of the two gears is fixed and the other orbits around .../...

The successive contours of the moving gear form a parametrized family of curves whose  envelope  consists of two parts:

  • The  trivial part  is simply the contour of the fixed gear.
  • The  nontrivial part  consist of other locally extreme positions of the moving contour.

The contour of the moving gear will never intersect the contour of the fixed gear if and only if those trivial and nontrivial parts of the envelope never  cross  each other.

They are allowed to be tangent to each other at certain points.  (This happens with zero-tolerance designs, either when there's no backlash or when the tips of a gear touch the roots of the other.)

 Come back later, we're
 still working on this one...

(2005-12-25)   Split Elliptic Gearing 
One-way gearing featuring rolling without slipping.

With the elliptic gears described above, one gear can drive the other only half of the time.  By retaining only the active half-tooth, we obtain an asymmetrical design in which one gear pushes against the other all the time, in a predetermined direction of rotation.

 Come back later, we're
 still working on this one...

(2012-11-17)   Traditional Watchmaker Gears 
The acting surfaces of horological  wheels  are radial  planes.

Fig. 77
A correct depth...

for a pinion with 8 teeth meshing with a wheel of 32 teeth.

From the book:

With the Watchmaker
at the Bench

(1943)   by
Donald de Carle
   Watchmaker Gears

In traditional clockwork, the protruding gear teeth are called  leaves  (French:  ailes, meaning  wings). The  wheels  (i.e., the large driven gears with many leaves)  have  flat  contact surfaces.  The  pinions  have  ogival  profiles  (so-called)  matching such planar contact surfaces.

No specialized tools are required for machining the wheels but the ogival shapes of pinion leaves require horological  pinion cutters.  As far as I know, only two manufacturers are still supplying those nowadays  (see footnotes).  Those tools aren't cheap in either case, but you can easily obtain a single size from  P. P. Thornton (UK) whereas Bergeon-Tecnoli (Switzerland) sells only  expensive  complete sets.

In horology, the gears are not at all expected to rotate at a constant  instantaneous  rate.  Therefore, there's absolutely no reason to invoke Euler's  conjugate tooth action  to preserve the constancy of rotational speed from one gear to the next.  As conjugate action is not required, neither is the  involute gearing  based on it  (which is virtually mandatory for lubricated high-speed machinery).

Horological mechanisms must work without any lubrication.  Their gear teeth could thus be designed to roll on each other's contact surfaces without any slipping or sliding  (which would be impossible to achieve with rotating gears obeying Euler's conjugate action law). 

 Come back later, we're
 still working on this one...

Designing Cycloidal Gears  (rack cutting)  by  Hugh Sparks  (2012)
Cycloidal Gear Builder  by  Dr. Rainer Hessmer  (2012)
Gears for instruments and clockwork mechanisms.  Cycloidal type gears.  Double circular arc type gears :
British Standard :   BS 978-2:1952  Addendum  1:1959
Swiss standards :   NHS 26702   |   NIHS 20.01, 20.02, 20.25  (formerly NHS 56702 and 56703)
P. P. Thornton Watch & Clock Wheel and Pinion Cutters   |   Bergeon-Tecnoli Gear Cutters

(2012-11-24)   La Hire's theorem   (1694)
A two-cusped hypocycloid is a straight line.

The simplest result in the theory of rolling curves:  If a circle rolls without slipping inside a fixed circle twice as big, then any point on it remains on a straight line  (others point attached to the moving circle describe ellipses).

Using modern nomenclature:  An  hypocycloid  of ratio 2 is a straight line.  An  hypotrochoid  of ratio 2 is an ellipse.

Proclus Diadochus  (AD 411-485)   |   Nasir al-Din al-Tusi  (1201-1274)
Nicolaus Copernicus  (1473-1543)   |   Philippe de la Hire  (1640-1718)
Tusi couple   |   Trammel of Archimedes  (ellipsograph)
Secrets of the Nothing-Grinder (12:52)  by  Burkard Polster  (Mathologer, 2018-12-07).

 Albrecht Duerer 
 1471-1528 (2005-12-30)   Cycloidal Gearing
Epicycloidal addendum curve.  Hypocycloidal dedendum curve.

This is to say that any acting part of the tooth profile  outside  the pitch circle is an arc of an  epicycloid,  whereas any acting part of the tooth profile  inside  the pitch circle is an arc of an  hypocycloid,  whereas

Both genders of cycloids are mathematically generated by two congruent circles that roll with slipping on the pitch circle.  Switching genders at the pitch circle  (where the tangents of both cycloids are purely radial)  is just a  practical  necessity.  Otherwise tooth profiles would feature points with infinite curvature, pointing either outward  (hypocycloid, yang)  or inward  (epicycloid, yin).

In practical gears, at most half an arch of either gender of cycloid can be used  (whichever of the two gears is acting as the driver can only "push" the other; it cannot "pull" it).

The cycloidal shapes were first described by  Albrecht Dürer  around 1525.  The idea to combine the two genders of cycloid into  genderless  gears is attributed to the French mathematician  Philippe de la Hire  (c. 1694).

 Come back later, we're
 still working on this one...

Albrecht Dürer  (1471-1528)   |   Philippe de la Hire  (1640-1718)   |   Cycloidal Gear

 Leonhard Euler 
 1707-1783 (2012-11-17)   Law of conjugate action   (Euler, c. 1754)
Gears featuring a steady rotational speed ratio.

As shown above, if two rotating curves are engaged in pure roll on each other  (without any sliding)  then their point of contact is on the straight line joining their fixed centers of rotation.  Also, the rate of rotation of either curve varies inversely as the distance from that point of contact to the center of rotation.

Therefore, the ratio of the rates of rotation of two such gears  cannot  be constant  (except when both are circles, in which case the point of contact does remain at a fixed distance from either center of rotation).  However, if the curves are allowed to slide tangentially to each other, some profiles can maintain a constant rate of rotation of both gears at all times...

 Come back later, we're
 still working on this one...

Conjugate tooth action  by  Douglas Wright   |   Leonhard Euler (1707-1783)  by  Walter Gautschi

 Leonhard Euler 
 1707-1783 (2005-12-26)   Involute Tooth Profile   (Leonhard Euler, 1754)
Involute tooth profiles provide constant rotational speed ratios.

The  rack  conjugate to an involute spur gear has  straight  flanks.

 Come back later, we're
 still working on this one...

Involute (or "evolvent")   |   Involute gear   |   Gear tooth generation  (rack cutting)  by  Douglas Wright

(2005-12-25)   Harmonic Drive  (patented by C. Walton Musser in 1955)
A "wave generator" rolls against a  flexspline  inside a  circular spline.

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 still working on this one...

Harmonic drive   |   Clarence Walton Musser (1909-1998)

(2012-11-25)   Circular Arc Helical Gears  (Ernest Wildhaber, 1923)

The Wildhaber-Novikov gears feature a large contact area between the convex and concave mating teeth.

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 still working on this one...

US Patent 1601750 "Helical Gearing" (filed Nov. 2, 1923) issued Oct. 5, 1926  to  Ernest Wildhaber (1892-1979).
USSR Patent 109750 "Helical Gearing" (1956) issued to M.L. Novikov.
Wildhaver-Novikov gear geometry  by  Stepan V. Lunin  (2001)  |  Gallery

(2012-11-25)  Double Circular Arc (DCA) Gear Technology   (1965)
Chinese standards were established in 1981.

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 still working on this one...

Chinese standards  (1981) for helical double circular arc (DCA) gear.

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