# A Rapid and Inexpensive Method for Finding the Basic Parameters of Involute Helical Gears
## 2 Materials and Methods  
### Aspects Concerning the Geometry of Spur Gears with Involute Teeth  
Given that the geometric and kinematic evaluations of helical gears can ultimately be deduced from the analysis of a spur gear, we first consider the geometric and kinematic parameters of spur gears, before moving on to understanding those of a helical gear. From the above, it is plausible to leverage the fundamental law of gearing to determine a conjugate profile that corresponds to any given profile. From a manufacturing perspective, it is advantageous if both profiles are curves of the same type. Among all the planar curves, the involute of the circle is unique as its conjugate is of the same type. This curve is defined by a point attached to a mobile straight line that, without any slippage, rolls along an affixed circle, termed the base circle \(C_{b}\), as shown in Figure 1.  
In the initial position, the mobile straight line takes the position \(\Delta_{0}\). Three fixed points on this straight line are considered: \(A_{0}^{0}\) coincides to the point of tangency, \(A_{0}^{-}\) is inside the base circle, and \(A_{0}^{+}\). Allowing the straight line to roll without slipping over the base circle \(C_{b}\), it occupies a current position \(\Delta\) where the tangency with the base circle is in the point \(T\). The three points fixed to the generating straight line take the current positions \(A^{0}\), \(A^{-}\), and \(A^{+}\) respectively, describing during the motion of a normal involute, a prolate involute, and a curtate involute (Figure 1). The condition of pure rolling stipulates that the length of the line segment \(\overline{A_{0}T}\) should equal the arc length \(A_{0}^{0}T\). Based on this remark, the polar parametric equations of the normal involute can be obtained. \[ \begin{cases}\rho_{y}=OA_{0}=\frac{r_{b}}{cos\alpha y}\\ \theta_{y}=\angle A_{0}^{0}OA^{0}=tan\alpha_{y}-\alpha_{y}\end{cases}  \tag{3}\]  
The function that appears in the definition of the \(\theta_{y}\) angle is denoted \(inv\alpha\) and is defined as  
\[inv\alpha=tan\alpha-\alpha,\ [\alpha]rad \tag{4}\]  
Figure 1: The involute of the circle: generation and properties. and is frequently met in the theory of gearing. The \(\alpha_{y}\) parameter from Equation (3) is the pressure angle of the normal involute on the base circle \(C_{y}\) of radius \(r_{y}\). Within the theory of involute gears, we can identify a circle by specifying the pressure angle of the normal involute profile. This profile is created using a base circle with an already established centre and radius. Based on relations (3) and (4), the cartesian parametric equations of any of the three involutes can be obtained:  
\[ \begin{cases}x(\alpha)=\frac{r_{b}}{\alpha\alpha}sin(inv\alpha)+a\cdot sin(tan \alpha)\\ y(\alpha)=\frac{\alpha\alpha}{cos\alpha}cos(inv\alpha)+a\cdot cos(tan\alpha) \end{cases}  \tag{5}\]  
where for \(a<0\) the prolate involute is obtained, for \(a=0\) the normal involute is obtained, and for \(a>0\) the curate involute is obtained. Another observation with important consequences concerns the fact that the length of the segment \(TA_{0}\) is the curvature radius of the normal involute in the current point \(A^{0}\). \[\rho_{c}=length(TA_{0})=r_{b}tan\alpha \tag{6}\]  
As the tooth count of a gear wheel approaches infinity, it logically follows that the radius of the base circle expands towards infinity as well. Thus, a gear wheel evolves into a rack with an infinite number of teeth and the curvature radius of the rack profile also expands towards infinity. This implies that the profile of an involute gear rack takes the form of a straight line, as depicted in Figure 2. The base line of the rack \(\Delta_{0}\) is the line alongside which the thickness of the tooth \(s_{0}\) is equal to the space width \(e_{0}\).