## 3 HTP Tooth Profile Design Using IH Tooth Profile and Cycloidal Curve
### HTP Tooth Profile Design Theory
In the IH model, gear movement is approximated using a linear rack, but in reality, the external tooth gear follows a nonlinear curved path rather than a simple linear motion. To better reflect this behavior, the epitrochoid curve equation was introduced and incorporated into the HTP tooth profile design. In this design, the gear meshing trajectory is defined using the epitrochoid curve to more precisely capture the movement of the external tooth gear. An epitrochoid curve represents the path traced by a point on a small circle as it rolls along the outside of a fixed larger circle. It is mathematically expressed as follows:  
\[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}(R-r)\cos\theta+d\cdot\cos\left(\frac{R-r}{r} \theta\right)\\ (R-r)\sin\theta-d\cdot\sin\left(\frac{R-r}{r}\theta\right)\end{bmatrix}  \tag{34}\]  
where \(R\) is the radius of the fixed circle drawn by the center of the external tooth gear, \(r\) is the actual rotation radius of the external tooth gear (the radius of the small circle), \(d\) is the distance from a specific point (tooth profile generating point) on the external tooth gear, \(\theta\) is the angle by which the external tooth gear rotates, and \(\frac{(R+r)}{r}\) represents the ratio indicating how many times the small circle rotates while making one revolution around the fixed circle. Equation (34) shows the trajectory drawn by a specific point on the gear as the external tooth gear rotates along the fixed circle. This is essential for analyzing the exact movement path of the external tooth gear, and through this, the meshing between the external tooth gear and the internal tooth gear can be more precisely designed. In the HTP tooth profile design, in order to apply the parameters used in general gear design, Equation (34) is transformed using the module, m, and the gear ratio, \(z\). \[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}m(z-1)\cos\theta+\kappa m\cdot\cos((z-1)\theta) \\ m(z-1)\sin\theta-\kappa m\cdot\sin((z-1)\theta)\end{bmatrix}  \tag{35}\]Through this, the epitrochoid curve was transformed into a form that can be used in the gear meshing system. In other words, it was transformed into a mathematical expression that reflects the physical characteristics of the gear. Equation (36) is a form where the coordinate system has been transformed to be suitable for gear design. That is, it reflects the relative movement trajectory based on the center of the gear:  
\[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}m(z-1)\sin\theta-\kappa m\cdot\sin((z-1)\theta) \\ m(z-1)\cos\theta+\kappa m\cdot\cos((z-1)\theta)-(R-r)\end{bmatrix}  \tag{36}\]  
Equation (37) simplifies the parameters by substituting the new variable, \(t=(z-1)\theta\). Through this, the mathematical expression of the gear meshing can be further simplified, and it enables precise calculation of the movement trajectory of the external tooth gear in the HTP tooth profile:  
\[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}m(z-1)\sin(\frac{t}{z-1})-\kappa m\cdot\sin(t) \\ m(z-1)\cos(\frac{t}{z-1})+\kappa m\cdot\cos(t)-m(z-1)\end{bmatrix}  \tag{37}\]  
By applying the small-angle approximation, the complex trigonometric terms were converted into a simple polynomial form and simplified into Equations (38) and (39). In particular, it can provide a more accurate approximation in the region where \(\theta\) is small. \[ \begin{gathered}\cdot\sin(\frac{t}{z-1})\approx\left(\frac{t}{z-1} \right),\cos(\frac{t}{z-1})\approx 1-\frac{1}{2}\big{(}\frac{t}{z-1}\big{)}^{2} \\ \begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}m(z-1)\big{(}\frac{t}{z-1}\big{)}-\kappa m\cdot \sin(t)\\ m(z-1)\Big{(}1-\frac{1}{2}\big{(}\frac{t}{z-1}\big{)}^{2}\Big{)}+\kappa m\cdot \cos(t)-m(z-1)\end{bmatrix}\end{gathered}  \tag{38}\]  
\[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}m\cdot t-\kappa m\cdot\sin(t)\\ \kappa m\cdot\cos(t)-m\Big{(}\frac{1}{2}\frac{t^{2}}{(z-1)}\Big{)}\end{bmatrix}  \tag{39}\]  
Finally, the tooth profile equation of the external tooth gear based on the epitrochoid curve is organized as follows:  
\[