## 3 HTP Tooth Profile Design Using IH Tooth Profile and Cycloidal Curve
### HTP Tooth Profile Design Theory
To determine the envelope of the external tooth addendum group in the HTP profile, the rack approximation method from the conventional IH tooth profile is applied, based on Equation (4), and expressed as Equation (26):  
\[ \begin{bmatrix}x_{ccg}(\theta)\\ y_{ccg}(\theta)\end{bmatrix} = \begin{bmatrix}x_{cc1}(\theta)\\ y_{cc1}(\theta)\end{bmatrix} + \begin{bmatrix}x_{l}\\ y_{l}\end{bmatrix} = \begin{bmatrix}-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p(\pi)+ p(\phi)-p(0)\\ -(1-\lambda)q(\theta)+q(0)+(1-\lambda)q(\pi)+q(\phi)-q(0)\end{bmatrix}  \tag{26}\]  
This method defines the external tooth addendum profile group by incorporating the motion trajectory of curve \(l\), and it is applied similarly to the IH tooth profile. The condition for envelope formation is that the Jacobian determinant equals zero, as also applied in the conventional IH profile. This condition is expressed as follows:  
\[\frac{\partial x_{fcg}}{\partial\phi}\frac{\partial y_{fcg}}{\partial\theta}- \frac{\partial x_{fcg}}{\partial\theta}\frac{\partial y_{fcg}}{\partial\phi}=0 \tag{27}\]  
Thus, the same envelope condition applies to the HTP tooth profile. In the HTP design, the envelope-forming partial differential condition retains the same form as in Equation (6):  
\[-(1-\lambda)\frac{\partial p(\phi)}{\partial\phi}\frac{\partial q(\theta)}{ \partial\theta}+(1-\lambda)\frac{\partial p(\theta)}{\partial\theta}\frac{ \partial q(\phi)}{\partial\phi}=0 \tag{28}\]  
This equation is applied with the same logic in the HTP tooth profile and is consistent with the basic envelope formation condition of the IH profile. To define the initial meshing position in the HTP design, Equation (7) is directly used and expressed as follows:  
\[ \begin{bmatrix}x_{fcg}(\phi=0)\\ y_{fcg}(\phi=0)\end{bmatrix} = \begin{bmatrix}\lambda p(\theta)+(1-\lambda)p(\pi) \\ \lambda q(\theta)+(1-\lambda)q(\pi)\end{bmatrix} = \begin{bmatrix}0.5 mn(\lambda( \theta-\kappa\sin\theta)+(1-\lambda)\pi)\\ \lambda q(\theta)+(1-\lambda)q(\pi)\end{bmatrix}  \tag{29}\]  
This condition clearly defines the initial point forming the envelope. In the HTP profile, \(\alpha\)-scaling is applied along the X-axis to allow for a more flexible design compared to the IH profile. This modification, building on Equation (20), expands \(\pi/2\) to \(\pi\) and incorporates X-axis deformation, as shown in Equation (30):\[ \begin{bmatrix}x_{cc}\\ y_{cc}\end{bmatrix} = \begin{bmatrix}\alpha\{-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p( \pi)\}+p(\phi)-\alpha p(0)\\ -(1-\lambda)q(\theta)+q(0)+(1-\lambda)q(\pi)+q(\phi)-q(0)\end{bmatrix}  \tag{30}\]  
This enhances design flexibility and enables broader shape control of the external tooth. The envelope condition is updated accordingly:  
\[-(1-\lambda)\frac{\partial p(\phi)}{\partial\phi}\frac{\partial q(\theta)}{ \partial\theta}+\alpha(1-\lambda)\frac{\partial p(\theta)}{\partial\theta} \frac{\partial q(\phi)}{\partial\phi}=0 \tag{31}\]  
Similarly, by using Equation (22), \(\beta\) scaling is applied in the Y-axis direction with an expansion of \(\pi/2\) to \(\pi\) to further widen the design scope:  
\[ \begin{bmatrix}x_{cc}\\ y_{cc}\end{bmatrix} = \begin{bmatrix}-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p( \pi)+p(\phi)-p(0)\\ \beta\{-(1-\lambda)q(\theta)+q(0)+(1-\lambda)q(\pi)\}+q(\phi)-\beta q(0) \end{bmatrix}  \tag{32}\]  
This provides additional design freedom and more precise adjustment of the external tooth addendum profile. The corresponding envelope condition is as follows:  
\[-\beta(1-\lambda)\frac{\partial p(\phi)}{\partial\phi}\frac{\partial q(\theta )}{\partial\theta}+(1-\lambda)\frac{\partial p(\theta)}{\partial\theta}\frac {\partial q(\phi)}{\partial\phi}=0 \tag{33}\]  
Up to this point, the HTP tooth profile design equations based on the IH tooth profile have been established. However, accurately describing the motion trajectory of the external tooth gear is difficult using only the rack approximation method applied in the IH tooth profile.