# CircleVis: A Visualization Tool for Circular Labeling Arrangements and Overlap Removal
## 3 Proposed Approach
### Angular Arrangement:  
As previously stated, \(A1\) is frequently used due to the large amount of space available for drawing label characters while handling a large number of label elements. Furthermore,  
Figure 1: Study workflow and contribution. Figure 2: Label problem formulation in the context of circle layout. the arc size has no bearing on the label. Figure 3 shows an example of a set of labels arranged for each arc by rotating \(q\) from the circle center. In the following cases, \(q\) is updated to \(q^{\prime}\):  
* If \(\frac{\pi}{2}\leq q_{i}\leq\pi\), then \(C_{v}=C_{v}+l_{i}\); the \(p_{i}\) is calculated in association with the new \(C_{v}\), and \(q^{\prime}=-(\pi-q)\);
* If \(\pi<q<2*\pi\), then \(q^{\prime}=2*\pi-q\). A2: Path Arrangement:  
Labels in path arrangement follow the arc's orientation. Each arc has a border length (\(L\)) that corresponds to the related arc length. However, arc lengths can be too short to draw labels with fixed sizes. Thus, users cannot see label without further interactions. In Figure 4, for example, \"Madagasc\" has a limited drawing space. For \(A2\), \(p_{i}\) is calculated using the same equation as \(A1\). Labels in this case are paths that follows the arc orientation. However, they are placed in the middle of \(p_{i}\), and if \(l_{Arci}\) is the length of associated arc, each label length must be smaller or equal to arc length \(l_{i}<=l_{Arci}\). If \(l_{i}>l_{Arci}\), the label will be updated by plotting just the available length from the beginning of the label, which will be \(l_{Arci}\). Figure 4: Path arrangement example. Figure 3: Angle label arrangement. A3: Linear Arrangement  
For drawing labels around circles without orientation, many approaches have used 1D layout. This arrangement is simple to read, even without turning the head. It employs two linear arrangements. The first (\(A3_{L}\)) involves drawing a rectangle around the real circle \(C_{r}\) and then placing labels on the left, right, top, and bottom of the rectangle, as shown in Figure 5. The number of areas available to draw labels in this first layout is limited. If we choose the left and right sides, for example, we can use the space in the top and bottom. As a result, this arrangement is not widely used in the InfoVis community (see Section 2.1), but it may be found in website programming ([[http://bl.ocks.org/dbuezas/9306799](http://bl.ocks.org/dbuezas/9306799)]([http://bl.ocks.org/dbuezas/9306799](http://bl.ocks.org/dbuezas/9306799)) (last accessed on 1 December 2021)). The second arrangement (\(A3_{Ln}\)) is illustrated in Figure 5. For this approach, there is no rectangle around the circle and each label is drawn according to the virtual circle and arc center. Furthermore, the label anchor is related to each arc position, such as: the start point in the left, the finish point in the right, the midpoint point on the top or bottom. This arrangement is also used in the literature; however, there is a problem with overlap between the label pairs. For \(A3\) arrangement, there are three modifications of position:  
* If \((0\leq q_{i}\leq\frac{\pi}{2})\cup(\frac{3\pi}{2}\leq q_{i}\leq 2\pi)\), then \(p_{i}\) is the same;
* If \(q_{i}=\frac{\pi}{2}\) or \(q_{i}=\frac{3\pi}{2}\), then \(p_{i}=p_{i}+\frac{l_{i}}{2}\);
* If \((\frac{\pi}{2}\leq q_{i}\leq\frac{3\pi}{2})\), then \(p_{i}=p_{i}+l_{i}\). A4: Proposed Enhanced Linear Arrangement  
Classical linear approaches place the center of labels according to a virtual circle with fixed radius, as shown in Figure 6. As a result, all labels with varying bounding box (rectangle encompassing the label) sizes will be placed in accordance with the fixed radius. As shown in Figure 6A, the radius of the virtual circle is assigned \"long labels\" and \"short labels\" (with the point drawn). We can see that the \"long label\" requires more space to display the label.