## 2 Methodology
### Geometry  
Multiple mathematical shapes exist with all different reflection characteristics: parabola, ellipse, etc. The reflective characteristic of a parabola is confirmed by multiple studies [10; 11; 12; 13] and states that incident parallel rays are reflected to the focus point of the parabolic reflector. Consequently, rays originating in the focal point are always reflected parallel to the symmetry axis of the parabola. These reflected rays automatically create a wide heated region. We intend to concentrate the radiation in a small point for every section of the reflector in order to minimise the heated region; therefore, a parabola can not be used as a reflector shape for this purpose. Another interesting mathematical shape is an ellipse because of its reflective property and could offer a solution for our application. This reflective property is better known as the focal property of an ellipse [14]. We could mount the tungsten halogen lamp in F1 of the ellipse to focus the radiation in F2. Thanks to this approach, all radiation originating in the halogen lamp will be reflected to the heated area on the specimen (F2). Figure 2 visualises the different properties of a parabola and an ellipse. According to the focal property of an ellipse, it is theoretically possible to focus the electromagnetic radiation on a small point as long as the elliptical shape remains closed. As can be seen in Figure 2, cutting a part of the elliptical reflector results in rays not being reflected towards the second focal point. For our application, it is not possible to keep the elliptical reflector closed in order to be able to heat the object. The radiation has to leave the reflector in order to induce a temperature increase of the sample. Figure 2: The reflecting characteristics of a parabola and an ellipse are visualised. All parallel rays entering a parabolic reflector will be focussed on the focal point. Each ray leaving one focal point of the ellipse will be reflected to the other focal point of the ellipse. One can see that cutting the ellipse on the red line will result in rays not being reflected by the elliptical shape towards the second focal point. Removing a part of the elliptical reflector leads to a wider radiation pattern as a result of the rays not being reflected towards the focus point. Using simple mathematics, the percentage of direct heating rays reaching the object can be calculated. In Figure 3, a simplified representation of the halogen tube lamp and the direct heated region of the object can be seen. The desired focus distance of the heat source is represented by \(\overline{OF}\). The width of the opening in the reflector shape is called \(\overline{E_{2}E}\) and \(\overline{H_{2}H}\) describes the wanted illuminated region. In the right-angled triangle \(\triangle\)OFE (the angle between \(\overline{OF}\) and \(\overline{FE}\) equals 90 degrees), the length of \(\overline{OE}\) can be found using Pythagoras Theorem:  
\[\overline{OE}^{2}=\overline{OF}^{2}+\overline{FE}^{2} \tag{1}\]  
The value for \(\alpha\) can be found using:  
\[\alpha=sin^{-1}(\overline{\overline{OF}}\over\overline{OE}) \tag{2}\]  
The width of the total heated region equals \(\overline{FE}\) and the wanted illuminated region is presented by \(\overline{FH}\). The length of the undesired heated region can easily be found by subtracting \(\overline{FH}\) of \(\overline{FE}\).