## 3 HTP Tooth Profile Design Using IH Tooth Profile and Cycloidal Curve
### HTP Tooth Profile Design Theory
By setting point C as the center of rotation, the similarity curve is rotated by \(180^{\circ}\) to obtain the second curve. Up to this step, the concept used in the IH tooth profile is applied as it is when \(\lambda\) is given. By multiplying only the X-coordinates of the second curve by \(\alpha\), the third curve corresponding to Equation (18) can be defined. This curve becomes an important feature that serves as the new external addendum tooth profile. \(\theta\) is a new parameter representing the tooth profile shape, and it is used separately from \(\phi\) in the IH tooth profile. By adjusting the values of \(\alpha\) and \(\beta\), the designer can modify the tooth thickness, radius of curvature, and contact distribution as intended. \[ \begin{bmatrix}x_{fc}\\ y_{fc}\end{bmatrix} = \begin{bmatrix}\alpha\big{\{}-(1-\lambda)p(\theta)+p(0)+(1- \lambda)p\big{(}\frac{\pi}{2}\big{)}\big{\}}\\ -(1-\lambda)q(\theta)+q(0)+(1-\lambda)q\big{(}\frac{\pi}{2}\big{)}\big{\}} \end{bmatrix} ,\ 0\leq\theta\leq\frac{\pi}{2} \tag{18}\]  
By scaling only the Y-coordinate using factor \(\beta\), the fourth curve is defined as the external tooth addendum profile in Equation (19):  
\[ \begin{bmatrix}x_{fc}\\ y_{fc}\end{bmatrix} = \begin{bmatrix}-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p \big{(}\frac{\pi}{2}\big{)}\\ \beta\big{\{}-(1-\lambda)q(\theta)+q(0)+(1-\lambda)q\big{(}\frac{\pi}{2}\big{)} \big{\}}\end{bmatrix} ,\ 0\leq\theta\leq\frac{\pi}{2} \tag{19}\]  
By applying the rack approximation from Equation (18), the general form of the external tooth addendum profile group is given by Equation (20), with its envelope condition expressed in Equation (21):  
\[ \begin{bmatrix}x_{fcg}\\ y_{fcg}\end{bmatrix} = \begin{bmatrix}\alpha\big{\{}-(1-\lambda)p(\theta)+p(0)+( 1-\lambda)p\big{(}\frac{\pi}{2}\big{)}\big{\}}+p(\phi)-\alpha p(0)\\ -(1-\lambda)q(\theta)+q(0)+(1-\lambda)q\big{(}\frac{\pi}{2}\big{)}+q(\phi)-q(0) \end{bmatrix}  \tag{20}\]  
\[-(1-\lambda)\frac{\partial p(\phi)}{\partial\phi}\frac{\partial q(\theta)}{ \partial\theta}+\alpha(1-\lambda)\frac{\partial p(\theta)}{\partial\theta} \frac{\partial q(\phi)}{\partial\phi}=0 \tag{21}\]  
Since obtaining a simple analytical solution may be difficult, the relationship between \(\varphi\) and \(\theta\) is typically calculated numerically. Substituting the numerical result into Equation (20) allows for the internal tooth addendum tooth profile to be derived. Similarly, by applying Equation (19), the external tooth addendum tooth profile group can be generalized as shown in Equation (22), with the corresponding envelope condition expressed in Equation (23). By numerically calculating the relationship between \(\phi\) and \(\theta\) and substituting it into Equation (23), the internal tooth addendum profile is obtained. \[ \begin{bmatrix}x_{fcg}\\ y_{fcg}\end{bmatrix} = \begin{bmatrix}-(1-\lambda)p(\theta)+p(0)+(1-\lambda)p\big{(} \frac{\pi}{2}\big{)}+p(\phi)-p(0)\\ \beta\big{\{}-(1-\lambda)q(\theta)+q(0)+(1-\lambda)q\big{(}\frac{\pi}{2}\big{)} \big{\}}+q(\phi)-\beta q(0)\end{bmatrix}  \tag{22}\]  
\[-\beta(1-\lambda)\frac{\partial p(\phi)}{\partial\phi}\frac{\partial q(\theta )}{\partial\theta}+(1-\lambda)\frac{\partial p(\theta)}{\partial\theta}\frac {\partial q(\phi)}{\partial\phi}=0 \tag{23}\]  
The tangential polar coordinates are given by Equation (24) for when the external gear tooth is defined as an elliptical shape instead of a simple circle. If the internal tooth meshing is approximated using rack meshing, then curve \(l\) is expressed in the form of Equation (25) [26]:  
\[p=r_{n}+\kappa nm\cos(2\psi),\;0\leq\psi\leq 2\pi \tag{24}\]  
\[ \begin{bmatrix}x_{l}\\ y_{l}\end{bmatrix} = \begin{bmatrix}0.5 mn(2\phi-\kappa\sin(2\phi))\\ \kappa nm\cos(2\phi)\end{bmatrix} ,\;0\leq\phi\leq\frac{\pi}{2} \tag{25}\]  
Using the above concepts, the HTP tooth profile design equations can be mathematically defined.