## 3 HTP Tooth Profile Design Using IH Tooth Profile and Cycloidal Curve
### HTP Tooth Profile Design Theory  
The HTP tooth profile is a novel gear tooth design method that enhances meshing characteristics by incorporating the IH tooth profile and the cycloidal curve. In conventional gear systems, achieving a high contact ratio and continuous engagement has been a key objective. The IH tooth profile was introduced to address this, improving continuous engagement by increasing the contact ratio compared to the traditional involute profile. However, the IH profile has limitations in enabling optimal rolling contact for low-tooth-count gears due to its fixed-envelope-based design. To overcome this, the HTP tooth profile builds upon the fundamental concept of the IH profile while integrating a modified cycloidal curve based on the epitrochoid to optimize contact characteristics. The HTP tooth profile maintains the envelope-based structure of the conventional IH tooth profile, but it adopts a hybrid design by applying a cycloidal curve based on the epitrochoid to more precisely reproduce the actual contact trajectory in the external tooth addendum region. The IH profile simplifies gear motion into a linear trajectory through rack approximation, providing a high contact ratio. However, the actual motion of external gears is rotational based on a fixed center, resulting in nonlinear changes in the position of the actual contact point and the tangential velocity vector. To account for this nonlinearity, this study substitutes an epitrochoid curve in a specific section of the external addendum region of the IH profile. In this process, scaling factors \(\alpha\) and \(\beta\) are applied in the X and Y directions, respectively, to finely adjust the geometric shape and meshing characteristics. The overall tooth profile is designed to maintain the envelope condition of the IH profile while being continuously connected with the epitrochoid trajectory. The cycloidal curve (based on the epitrochoid) was adopted on the following theoretical grounds. First, this curve has characteristics that induce pure rolling contact, which can minimize friction and wear. Second, as shown in Figure 4, when scaling correction is applied, it exhibits a shape almost identical to the ideal meshing trajectory, and compared to the existing IH and harmonic tooth profiles, it provides a more precise approximation. Furthermore, Figures 6-8 confirm that the HTP profile maintains conformity with the ideal trajectory even under low-tooth-count gear conditions, demonstrating that this design choice offers higher structural accuracy and consistent design compared to conventional profiles. Through this, the proposed method offers the potential to optimize rolling contact in traditional gear design, thereby reducing friction, enhancing durability, and achieving high transmission efficiency. It is particularly expected to be a suitable design approach for high-efficiency mechanical systems such as precision reducers and robotic gear systems. The HTP tooth profile design equations are derived from Equations (1)-(7) of the conventional IH tooth profile. The IH tooth profile defines a rack-shaped curve formed geometrically by specifying the two endpoints (points A and B) and the midpoint (point C). At this point, \(\lambda\) is used as a displacement coefficient and serves as a parameter to adjust the overall position and shape of the tooth profile curve. By applying this coefficient, the shapes of the addendum and flank regions are specified, and the contact conditions are controlled so that the meshing is not interrupted. Furthermore, by additionally introducing scaling factors such as \(\alpha\) for the X-coordinate and \(\beta\) for the Y-coordinate, the root width, curvature radius, and tooth thickness can be designed more flexibly. To newly construct the external addendum, the following transformation is performed. Taking point B as the center of similarity, the entire curve \(l\) is scaled down by a factor of \((1-\lambda)\), and this is referred to as the similarity curve.