Abstract
The cutter systems of hypoid gear cutting machines contain groups of inside and outside blades. In these cutter systems, the side cutting edges of the blades machine the convex and concave gear teeth while rotating about the cutter rotation axis. The side cutting edges lay on the rake face formed through the blade, rake, and relief angles; hence, the normal crosssection of the cutter swept surface forms hyperboloid gear teeth. Using the accurate geometry of the cutter system, a relationship between the pressure and spiral angles of the gear tooth and the parameters of the cutter system is developed for the FORMAT machining of a hypoid gear. A new parameterization of the gear tooth surfaces is introduced to determine these angles for the accurate gear tooth by the accurate cutter system. A numerical example with different cutter systems and blade parameters is presented, demonstrating the effects of rake and relief angles over the pressure and spiral angles on mean point projections and gear tooth surface. Finally, the change in pressure and spiral angles with respect to the rake and relief angles are plotted, and the results are analyzed. Finally, it is concluded that the pressure and spiral angles are changed up to a few seconds of a degree in the operating area of the tooth with the change in the back and side rake angles. The side relief angle exhibited little or no effect over the geometry of the gear tooth.
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Abbreviations
 k:

k is a generic variable and can be replaced by i or o for the inside or outside blade, or convex or concave sides of the gear tooth
 \(A_{1}\) :

Distance from the root cone apex to the pitch cone apex for \(M\)
 \(a_{\text{og}}\) :

Gear tooth addendum at the heel
 \(b_{\text{og}}\) :

Gear tooth dedendum at the heel
 \(a_{\text{G}}\) :

Gear tooth addendum in the mean crosssection
 \(b_{\text{G}}\) :

Gear tooth dedendum in the mean crosssection
 \(b_{\text{P}}\) :

Pinion tooth dedendum in the mean crosssection
 \(c\) :

Clearance between the gear and pinion teeth in the mean crosssection
 ECE:

Equivalent cutting edge
 \(F\) :

Face width of the gear tooth
 \(F_{\text{r}}\) :

Projection of face width along the root cone generatrix
 \(h_{\text{M}}\) :

Height of mean point \(M\) at the center of the tooth land
 \(h_{\text{P}}\) :

Height of the point \(P\) along the normal at the middle of the tooth land
 \(L_{\text{p}}\) :

Distance between pitch cone apex and point \(M\)
 \(L_{\text{r}}\) :

Distance between root cone apex and middle of the tooth land
 \(l_{\text{r}}\) :

Distance between root cone apex and point \(P\)
 \(M\) :

Mean point
 \(M_{\text{i}} ,M_{\text{o}} \varvec{ }\) :

Mean point projections on the gear profile or ECE curves
 \(P\) :

Parametric point on gear tooth
 \(P_{\text{i}} ,P_{\text{o}}\) :

Projections of point \(P\) at the ECE
 \(\varvec{P}_{\text{kb}}\) :

ECE curves in \(O_{\text{b}}\) coordinate system
 \(P_{\text{w}}\) :

Point width of the cutter
 \(\varvec{R}_{\text{kb}}\) :

ECE curves rotated about \(X_{\text{cg}}\) axis in \(O_{\text{b}}\) coordinate system
 \(R_{\text{cg}}\) :

Average radius of the cutter
 \(\varvec{S}_{\text{cek}}\) :

Vector along the side cutting edge of the blade
 \(\varvec{s}_{\text{cek}}\) :

Unit vector along the side cutting edge of the blade
 \({\mathbf{s}}\) :

Normal vector at the middle of gear tooth land
 \(\varvec{s}_{\text{g}}\) :

Symmetric axis of the gear tooth flanks
 \(\varvec{s}_{\text{e}}\) :

Symmetric axis of the tangents at projections \(M_{\text{i}} ,M_{\text{o}}\) on ECE
 \(\varvec{t}_{\text{k}}\) :

Unit tangent vectors \(\varvec{t}_{\text{i}}\) and \(\varvec{t}_{\text{o}}\) at the ECE curves at points \(P_{\text{i}} ,P_{\text{o}}\)
 \(\varvec{n}_{\text{k}}\) :

Unit normal vectors \(\varvec{n}_{\text{i}}\) and \(\varvec{n}_{\text{o}}\) at the ECE curve at points \(P_{\text{i}} ,P_{\text{o}}\)
 \(\varvec{n}_{\text{p}}\) :

Unit normal to the plane Ω formed by the curves \(\varvec{R}_{\text{k}}\)
 \(u_{\text{k}}\) :

Lengthwise parameters of the inside \((u_{\text{i}} )\varvec{ }\) and outside cutting edges \((u_{\text{o}} )\)
 \(u_{\text{p}}\) :

Lengthwise parameter of the gear tooth
 \(v_{\text{k}}\) :

Parameters of the inside \((v_{\text{i}} )\varvec{ }\) and outside ECE curves \((v_{\text{o}} )\)
 \(v_{\text{kM}}\) :

Parameter of the ECE curve at the points \(M_{\text{i}} ,M_{\text{o}}\), i.e., \(v_{\text{iM}} \varvec{ }\) and \(v_{\text{oM}}\)
 \(\varvec{x}_{\text{b}}\) :

Unit vector along the negative direction of \(X_{\text{b}}\) axis
 \(\varvec{x}_{\text{p}}\) :

Unit vector along the gear rotation axis
 \(\varvec{x}_{\text{k}}\) :

Unit vector perpendicular to the vectors \(\varvec{x}_{\text{p}} \varvec{ }\) and \(\varvec{y}_{\text{k}}\)
 \(\varvec{y}_{\text{k}}\) :

Unit vector along the pitch cone generatrix towards the points \(P_{\text{i}} ,P_{\text{o}}\)
 \(\varvec{z}_{\text{k}}\) :

Unit vector perpendicular to the vectors \(\varvec{x}_{\text{k}} \varvec{ }\) and \(\varvec{y}_{\text{k}}\)
 \(\alpha_{{{\text{e}},{\text{k}}}}\) :

ECE angles \(\alpha_{{{\text{e}},{\text{i}}}}\) and \(\alpha_{{{\text{e}},{\text{o}}}}\) formed by tangents \(\varvec{t}_{\text{i}}\) and \(\varvec{t}_{\text{o}}\) with the \(\varvec{x}_{\text{b}}\)
 \(\alpha_{{{\text{g}},{\text{k}}}}\) :

Gear flank angles \(\alpha_{{{\text{g}},{\text{i}}}}\) and \(\alpha_{{{\text{g}},{\text{o}}}}\) of convex and concave gear tooth
 \(\alpha_{{{\text{o}},{\text{k}}}}\) :

Back rake angles \(\alpha_{{{\text{o}},{\text{i}}}}\) and \(\alpha_{{{\text{o}},{\text{o}}}}\) of inside and outside blades
 \(\alpha_{{{\text{f}},{\text{k}}}}\) :

Side rake angles \(\alpha_{{{\text{f}},{\text{i}}}}\) and \(\alpha_{{{\text{f}},{\text{o}}}}\) of inside and outside blades
 \(\alpha_{{{\text{b}},{\text{k}}}}\) :

Blade angles \(\alpha_{{{\text{b}},{\text{i}}}}\) and \(\alpha_{{{\text{b}},{\text{o}}}}\) of inside and outside blades
 \(\gamma_{{{\text{f}},{\text{k}}}}\) :

Side rake angles \(\gamma_{{{\text{f}},{\text{i}}}}\) and \(\gamma_{{{\text{f}},{\text{o}}}}\) of inside and outside blades
 \(\beta_{\text{k}}\) :

Spiral angle at points \(P_{\text{i}} ,P_{\text{o}}\)
 \(\beta_{\text{r}}\) :

Spiral angle on the root cone tangent plane
 \(\theta_{\text{cg}}\) :

Angular parameter of the cutter swept surface
 \(\theta_{\text{P}}\) :

Angular parameter of the gear tooth
 \(\varGamma\) :

Pitch angle of the gear
 \(\varGamma_{\text{o}}\) :

Face angle of the gear
 \(\varGamma_{\text{R}}\) :

Root angle of the gear
 \(\delta_{\text{G}}\) :

Dedendum angle of the gear
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Rababah, M., Wasif, M. & Iqbal, S.A. Parametric relationship between hypoid gear teeth and accurate facemilling cutter. Adv. Manuf. 8, 537–555 (2020). https://doi.org/10.1007/s4043601900286x
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Keywords
 Hypoid gear
 Pressure angle
 Spiral angle
 Accurate blade
 Cutter head