## 1 Introduction
[PERSON] conducted a geometric analysis of the meshing characteristics between the flexspline and circular spline, demonstrating that continuous contact during meshing can be achieved by designing the tooth profile based on the flexspline's deformation trajectory. This approach improves both efficiency and fatigue life [13; 14]. [PERSON] proposed a new tooth profile to overcome conventional limitations, improving fatigue strength and torque transmission. The design maintains high stiffness even under low torque, thus enhancing durability and extending gear life [15]. [PERSON] and [PERSON] analyzed the kinematic and geometric characteristics of tooth meshing, treating the flexspline as a rigid body. They defined the tooth profile based on gear mechanism theory, assuming constant gear spacing at the neutral line where common normals intersect at the pitch point [16]. [PERSON] addressed the kinematic limitations of non-involute profiles in existing harmonic drives and developed a new wave generator using an involute profile. This approach eliminated tooth interference and enabled precise gear motion [17]. [PERSON], et al. applied a double-arc tooth profile to the flexspline and circular spline, and, through finite element analysis, confirmed superior durability and meshing characteristics compared to conventional profiles [18]. [PERSON] and [PERSON] validated a new geometric tooth profile model via Ansys Workbench simulations, using explicit dynamic analysis to predict elastic and nonlinear behavior. The design's accuracy was confirmed through comparison with theoretical and experimental results [19]. [PERSON], et al. analyzed flexspline deformation based on ring theory and introduced a tooth profile coordinate adjustment method. This approach prevents undercutting and improves meshing reliability [20]. [PERSON] et al. designed a new tooth profile combining hypocycloidal and epicycloidal curves, enabling more simultaneous tooth engagement and offering a simpler mathematical formulation. Their design was validated through finite element analysis and prototype testing [21]. [PERSON] et al. introduced a high-contact-ratio two-dimensional tooth profile based on central engagement. By leveraging the kinematic relationship between the neutral layer and pitch circle of the flexspline and circular spline, they enabled continuous engagement and zero backlash. The design was verified through mathematical analysis and kinematic simulations [22]. In this paper, the tooth profiles used in conventional harmonic reducers have structural limitations such as meshing discontinuity, restrictions on the radius of curvature of the tooth base, and distortion of the contact trajectory, especially when the number of teeth is small. These problems limit the design freedom of the reducer and make it difficult to secure contact stability and durability under precision driving conditions. To solve these problems, this paper proposes a new tooth profile design equation, the IH (Involute Harmonic) tooth profiles and the HTPs (Hybrid Tooth Profiles), using the cycloid curve to overcome the structural limitations of the conventional harmonic tooth profile, which is difficult to design under small-tooth-number conditions, and to enable tooth design without restrictions on the number of teeth. Theoretical equations for the HTP design were derived, and the x and y trajectories for each profile were plotted using key parameters to illustrate differences from previously studied profiles. The proposed equation is a generalized design formula that allows tooth profiles to be generated without constraints on the number of teeth. Additionally, a Python-based HTP tooth profile design program was developed, enabling real-time visualization of the tooth meshing state through simulation. To evaluate the field applicability of the proposed profile, an R21_z3 single-stage high-reduction-ratio reducer was designed, and its strength and meshing condition were verified through static analysis in Ansys.