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A gear base circle is the fixed circle from which an involute tooth flank is generated. Imagining a taut string unwinding from that circle, the free end traces the involute. It is a geometric construction for the tooth shape, distinct from the gear reference circle used for sizing and the operating pitch circle describing a particular meshing pair.^[ref]

For base radius $r_b$ and unwinding parameter $u\ge 0$, one orientation of the involute isSquaring and adding gives $r=\sqrt{x^2+y^2}=r_b\sqrt{1+u^2}$. The local pressure-angle relation $\tan \alpha =u$ therefore gives $r_b=r\cos \alpha$. At the gear reference circle, this becomesFor reference diameter $d=60\mathrm{mm}$ and reference gear pressure angle $20{}^{\circ }$, the base diameter is $56.382\mathrm{mm}$, approximately. A change in mounting separation does not change this base diameter or the cut tooth profile; it changes which operating geometry describes the mesh.

In a compatible external pair, a common tangent to the two gear base circles is the gear line of action. Their fixed geometry gives $r_{b1}+r_{b2}=a\cos {\alpha }_w$, explaining why the operating pressure angle changes when $a$ changes. Repeating a tooth profile every $2\pi /N$ also gives gear base pitch $p_b=\pi d_b/N$ along the base circumference. Finally, an ideal involute generated this way lies on or outside its gear base circle; a real tooth's lower connecting portion can have a different shape. The gear base circle is not a physical boundary of the gear material.

Gear base pitch is the spacing of successive involute tooth profiles referred to the gear base circle. For base diameter $d_b$, tooth count $N$, gear module $m$ and reference gear pressure angle $\alpha$,It also gives the spacing between successive corresponding tooth contacts measured along the gear line of action. The gear circular pitch uses the reference circle instead, giving $p=\pi m$ and $p_b=p\cos \alpha$.^[ref]

For $m=2\mathrm{mm}$ and $\alpha =20{}^{\circ }$, $p=6.2832\mathrm{mm}$ but $p_b\approx 5.9043\mathrm{mm}$. A second gear with the same module but $25{}^{\circ }$ reference angle has $p_b\approx 5.6945\mathrm{mm}$. Matching reference tooth spacing alone therefore does not make these the same specified mating geometry.

Changing the mounting separation of a compatible involute pair does not change its cut profiles, base circles or gear base pitch. In operating-circle notation, $p_w=2\pi r_w/N$ changes with mounted radius, whilestays fixed. This links the mounted operating pressure angle to an invariant tooth-profile spacing.

For a tooth rotating at angular speed $\omega$, base-circle tangential speed is $r_b|\omega |$. Dividing by gear base pitch gives the gear mesh frequency,A 30-tooth gear at $600$ revolutions per minute gives $f_m=300\mathrm{Hz}$. Finally, the gear contact ratio divides usable gear path of contact length by this base pitch, so profile spacing and available contact length jointly determine overlap between successive tooth engagements.

Gear base-roll speed $u_b$ is the rate at which base-circle rolling advances an ideal involute contact along the gear line of action. For a fixed-axis external pair at constant speed,The equality follows from the tooth-count speed ratio and compatible base pitches. It describes the rate of change of contact position along a geometric line, rather than the gear sliding velocity between material points on the flanks. Base circles and their common tangent define this geometry.^[ref]

For standard module2 gears with20 and40 teeth at20degree reference angle, base radii are $r_{b1}=20\cos 20{}^{\circ }\mathrm{mm}$ and $r_{b2}=40\cos 20{}^{\circ }\mathrm{mm}$. Signed angular velocities ${\omega }_1=50rad/s$ and ${\omega }_2=-25rad/s$ give the same $u_b\approx 0.93969m/s$. The first gear's reference-circle tangential speed is1m/s, so base-roll and pitch-circle speeds are distinct.

In one tooth-entry interval, the base circle advances one gear base pitch:With $f_m\approx 159.155\mathrm{Hz}$, the example advances5.90426mm per entry. An active contact length9.65457mm therefore gives gear contact duration $L/u_b\approx 10.2742\mathrm{ms}$.

For compatible involute teeth, changing center distance changes the operating pressure angle and operating pitch circles, while leaving the manufactured base circles unchanged. At the same shaft speeds, base-roll speed is therefore unchanged even though the usable contact length may change. This separates a kinematic rate from the active interval needed to calculate contact time. If speed varies, contact displacement follows $\int u_b(t)dt$; a constant-speed quotient is then only an approximation.

Gear contact ratio measures how much successive tooth engagements overlap. For an external spur gear pair, its transverse value iswhere $L$ is usable gear path of contact length along the gear line of action and $p_b$ is gear base pitch. One tooth pair remains engaged while contact advances through length $L$; successive pairs begin after an advance of $p_b$. Thus ${ϵ}_{\alpha }$ is also the ideal mean number of tooth pairs simultaneously engaged.^[ref]

For a tip-limited, noninterfering external involute pair with tip radii $r_{a1},r_{a2}$, base radii $r_{b1},r_{b2}$, axis separation $a$ and operating pressure angle ${\alpha }_w$, right-triangle geometry givesFor an original example, use module $2\mathrm{mm}$, tooth counts 20 and 40, reference angle $20{}^{\circ }$ and standard separation $60\mathrm{mm}$. With tip radii $22$ and $42\mathrm{mm}$, the formula gives $L\approx 9.6546\mathrm{mm}$ and $p_b\approx 5.9043\mathrm{mm}$, hence ${ϵ}_{\alpha }\approx 1.6352$.

In the ideal periodic timing model, two tooth pairs are engaged for about $63.52\%$ of a tooth-passage period and one pair for about $36.48\%$. This does not mean that force is divided equally between engaged pairs: tooth flexibility and contact geometry affect load sharing. A value below one would leave gaps in ideal engagement, while a value exactly one leaves no overlap margin for tolerances and deformation.

The plot shows contact timing, rather than physical tooth shapes or force sharing. For helical gears, the transverse measure must additionally be distinguished from gear overlap ratio caused by the inclination of teeth across the gear face. Finally, real usable contact can be shortened by modified or unavailable flank portions; the tip-circle formula is conditional on its stated geometric assumptions.

The gear line of action is the common normal along which ideal tooth contact transmits force. For meshing involute profiles in a transverse plane, it is a straight common tangent to the two gear base circles. The angle it makes with the common tangent to the operating pitch circles is the operating pressure angle.^[ref]

Its intersection with the line of gear centers is the gear pitch point. The actual contact point moves along only a finite portion, the gear path of contact, bounded by usable tooth profiles. Thus a diagram's infinite straight line is a geometric construction, not a claim that teeth can contact everywhere on it.

The geometry explains constant ratio. Let the tooth-normal force magnitude be $F_n$ and the base radii $r_{b1},r_{b2}$. The perpendicular distance from each gear center to the gear line of action is its base radius, so the ideal transmitted moments have magnitudesThese moment arms do not change as the contact point advances along that line. The no-penetration contact condition correspondingly gives $r_{b1}{\omega }_1+r_{b2}{\omega }_2=0$ for an external pair. Compatible profiles have $r_{b2}/r_{b1}=N_2/N_1$, recovering the tooth-count gear ratio.

For example, reference radii $20$ and $40\mathrm{mm}$ at reference gear pressure angle $20{}^{\circ }$ give base radii approximately $18.794$ and $37.588\mathrm{mm}$. A $100\mathrm{N}$ normal force produces moment magnitudes $1.8794$ and $3.7588\mathrm{N}\mathrm{m}$. Their ratio stays two as the point of contact moves.

Ideal normal transmission does not eliminate gear tooth sliding. Relative surface slip generally changes along the gear path of contact and vanishes at the gear pitch point for the ideal external pair. Actual friction associated with that slip adds a tangential contact force; its resultant need not lie exactly on the geometric gear line of action.

The gear path of approach is the part of the active gear path of contact followed before the contact reaches the gear pitch point. For a conventional external involute pair in the stated driving direction, contact starts where the driven gear's tip circle intersects the usable gear line of action, and then moves toward the pitch point. This is a straight-line distance along the common normal, not an arc around a pitch circle.^[ref]

Let gear1 drive gear2. Write $r_{a2}$ for gear2's tip radius, $r_{b2}$ for its base radius, $r_{w2}$ for its operating radius and ${\alpha }_w$ for operating pressure angle. A right triangle from the base-circle tangency gives tip-to-tangency distance $\sqrt{r_{a2}^2-r_{b2}^2}$. Subtracting the pitch-point-to-tangency distance givesThis formula assumes the full tip-limited interval is usable without interference or profile truncation.

For an original standard module2,20/40-tooth pair at60mm center distance and20degree reference angle, gear2 has tip radius42mm, operating radius40mm and base radius $40\cos 20{}^{\circ }\mathrm{mm}$. Consequently $L_a\approx 5.0586\mathrm{mm}$. If gear1 rotates at $50rad/s$, contact advances along the line at $u_b=|{\omega }_1|r_{b1}=0.93969m/s$, giving an approach contribution to gear contact duration of about $5.383\mathrm{ms}$. This contact-location speed differs from gear sliding velocity, which tends to zero on arriving at the pitch point.

The total active length adds gear path of recess. Reversing the followed engagement direction exchanges which segment is called approach and which is recess. If altered usable flanks exclude the nominal pitch point, an assumed positive tip-limited formula must not replace an inspection of the actual active interval.

The gear path of contact is the locus of actual tooth-contact points during an engagement. For a conventional external involute pair, its active part is a finite straight segment of the gear line of action. Its endpoints are set by the usable tooth flanks, rather than by the infinite geometric line alone. The part before the gear pitch point is the gear path of approach; the part after it is the gear path of recess in the stated direction of transmission.^[ref]

For a standard noninterfering external pair driven by gear1, let $r_{ai}$ be tip radii, $r_{bi}$ base radii, $r_{wi}$ operating radii and ${\alpha }_w$ the operating pressure angle. When tip circles bound the full usable contact,Adding gives $L=L_{\mathrm{app}}+L_{\mathrm{rec}}$. The two lengths need not be equal even when the gears use the same module and reference angle.

For the original module2,20/40-tooth example at20degree reference angle and60mm standard separation, tip radii22 and42mm give $L_{\mathrm{app}}\approx 5.0586\mathrm{mm}$ and $L_{\mathrm{rec}}\approx 4.5960\mathrm{mm}$. Their sum is $9.6546\mathrm{mm}$; dividing by gear base pitch $5.9043\mathrm{mm}$ gives transverse gear contact ratio about1.6352.

These are distances along the line of contact, not arc lengths traveled around a pitch circle. A shortened or altered flank can reduce the active interval, and interference can invalidate the full tip-limited construction. The labels approach and recess also depend on which direction is being followed. A practical contact inspection therefore needs the profile geometry and transmission direction, not just a sketch of two nominal pitch circles.

The gear path of recess is the part of the active gear path of contact followed after passing the gear pitch point, in the stated driving direction. For a standard external involute pair driven by gear1, contact ends at the driving gear's tip circle when that tip bounds the usable profile. The recess length lies on the gear line of action; it is not an arc of the gear pitch circle.^[ref]

Use driving-gear tip radius $r_{a1}$, base radius $r_{b1}$, operating radius $r_{w1}$ and operating pressure angle ${\alpha }_w$. The base-circle tangent triangle givesFor a standard module2,20/40-tooth pair at60mm separation and20degree reference angle, $r_{a1}=22\mathrm{mm}$, $r_{w1}=20\mathrm{mm}$ and $r_{b1}=20\cos 20{}^{\circ }\mathrm{mm}$ yield $L_r\approx 4.5960\mathrm{mm}$. The corresponding gear path of approach is about5.0586mm, so the pitch point does not bisect the active contact interval.

At driving angular velocity $50rad/s$, the contact-location speed is $u_b=|{\omega }_1|r_{b1}\approx 0.93969m/s$. Recess therefore lasts about4.891ms. Adding approach gives gear contact duration $T\approx 10.274\mathrm{ms}$. Tooth-passage frequency is $20(50)/(2\pi )\approx 159.155\mathrm{Hz}$, so $T{f}_m\approx 1.6352$, agreeing with the transverse gear contact ratio calculated as active length divided by gear base pitch.

A shorter usable driving flank reduces recess and can change gear sliding-friction loss, because sliding grows with distance from the pitch point. The tip-circle formula alone cannot describe profile modifications or an interfering contact. The geometry and direction must be specified before applying the approach/recess labels.

The gear pressure angle specifies the direction of tooth-normal contact relative to tangential motion. For an external spur gear pair with involute tooth flanks, its operating value is the angle between the gear line of action and the common tangent to the operating pitch circles. The gear line of action is the common normal to the contacting flanks and a common tangent to their gear base circles. Reference gear pressure angle $\alpha$ and operating pressure angle ${\alpha }_w$ coincide at the corresponding reference mounting, but need not coincide at another axis separation.^[ref]

Neglecting friction, let the tooth-normal force magnitude be $F_n$. Its components areThe gear tangential force $F_t$ transmits torque, $\tau =F_t{r}_w$, at operating radius $r_w$. The gear separating force $F_r$ pushes the gear centers apart in the external mesh and loads their supports. Increasing gear pressure angle increases this separating load for the same transmitted tangential load.

For an original example, a spur gear transmits $\tau =3\mathrm{N}\mathrm{m}$ at $r_w=30\mathrm{mm}$. Then $F_t=100\mathrm{N}$. At ${\alpha }_w=20{}^{\circ }$,If the angle were $25{}^{\circ }$ at the same $F_t$, the separating load would rise to approximately $46.63\mathrm{N}$. This comparison concerns load direction, not gear efficiency: an ideal normal force can transmit power without dissipating it. Actual frictional contact adds another force component and can alter both losses and support loads.

Gear separating force is the component of an external gear mesh load along the line joining the gear centers, tending to push those centers apart. For a frictionless external spur gear pair,where $F_n$ is tooth-normal force, $F_t$ is gear tangential force and ${\alpha }_w$ is operating pressure angle. This component acts on the shaft supports even when the mesh is ideal and lossless.^[ref]

For example, take operating radius $r_w=20\mathrm{mm}$ and transmitted torque $4\mathrm{N}\mathrm{m}$. Then $F_t=200\mathrm{N}$. At ${\alpha }_w=20{}^{\circ }$, $F_r\approx 72.79\mathrm{N}$; at $25{}^{\circ }$ and the same tangential load, $F_r\approx 93.26\mathrm{N}$. The larger separating load is a change in load direction, not by itself an efficiency loss.

If the gear centers are ideally fixed, this radial component produces no work through a center-separation displacement, since that displacement is zero. Flexible supports can nevertheless deflect under it and store elastic potential energy. The resulting change in axis separation can change operating pressure angle, backlash and tooth contact, so the assumed geometry and support stiffness interact.

A support-load experiment can apply a known torque slowly to a mounted external pair and measure the reaction along the line of centers while maintaining the stated axis separation. Agreement with $F_r/F_t=\tan {\alpha }_w$ tests the force resolution under low-friction assumptions. Additional reactions, tooth friction and mounting preload must be accounted for before identifying a measured support force with this mesh component. Internal meshes and nonparallel-axis gearing require their own direction conventions; the external separating-force picture must not be transferred unchanged.

The operating pitch circles of two meshing gears are the imaginary circles that reproduce their motion by rolling without slipping. They are properties of the working pair and its mounting, whereas a gear reference circle specifies one gear's nominal geometry. Working pitch circle and intermeshing pitch circle are alternative names.^[ref]

Consider an external pair of spur gears on fixed parallel axes, with axis separation $a$ and tooth counts $N_1,N_2$. Their operating radii $r_{w1},r_{w2}$ must satisfy both contact geometry and the tooth-count gear ratio:Solving givesThus the operating radii scale with the actual separation, while the gear ratio stays fixed by tooth counts. These are kinematic equations, conditional on the teeth actually maintaining compatible contact.

For an original example, choose gear module $m=2\mathrm{mm}$ and tooth counts 20 and 40. Reference radii are $20\mathrm{mm}$ and $40\mathrm{mm}$; their standard separation is $60\mathrm{mm}$. If a compatible mounting uses $a=61\mathrm{mm}$ instead, operating radii become $20.333\dots \mathrm{mm}$ and $40.666\dots \mathrm{mm}$. At input angular velocity $30rad/s$ the output has $-15rad/s$, and both operating tangential speed magnitudes are $0.61m/s$. Using the reference radii in that speed calculation would incorrectly give $0.60m/s$.

For tooth flanks formed as involutes, the fixed gear base circles permit constant-ratio motion over a range of separations. If their radii are $r_{b1},r_{b2}$, the common-normal geometry giveswhere ${\alpha }_w$ is the operating pressure angle. Increasing separation therefore increases that angle, while leaving each gear reference circle and gear base circle unchanged. This geometric possibility does not guarantee adequate tooth contact, clearance or strength at an arbitrary separation: the physical tooth extents and mounting must still be checked.^[ref]

The operating pressure angle ${\alpha }_w$ is the gear pressure angle of an actual meshing pair at its mounted axis separation. For external spur gears with involute profiles, it is the angle between the gear line of action and the common tangent to the operating pitch circles. Changing the separation changes this angle even though the individual gear base circles remain fixed.^[ref]

Let the two base radii be $r_{b1},r_{b2}$ and the mounted axis separation be $a$. The right-triangle geometry of the gear line of action givesIf $a_0=r_1+r_2$ is the reference separation and both reference gear pressure angles equal $\alpha$, then $r_{bi}=r_i\cos \alpha$, henceThe gear reference circles and gear ratio remain unchanged by this mounting adjustment; the operating pitch circles expand or contract to fit $a$.

For example, reference radii $20\mathrm{mm}$ and $40\mathrm{mm}$ with $\alpha =20{}^{\circ }$ give $a_0=60\mathrm{mm}$. Mounting a compatible pair at $a=61\mathrm{mm}$ givesAt an imposed gear tangential force $100\mathrm{N}$, the associated gear separating force is about $41.30\mathrm{N}$, compared with $36.40\mathrm{N}$ at the reference angle. Holding torque instead of tangential load would additionally require using the changed operating radius.

The angle formula is only a geometric condition. Usable tooth flanks must still reach the contact region and maintain an adequate gear contact ratio; excessive separation can destroy useful contact. Likewise, the existence of a real inverse cosine does not establish that a proposed mounting has suitable backlash or mechanical strength.^[ref]