Gutman designed the Sombor index (SO) based on a geometric approach in [4] and subsequently developed six SO-like vertex-degree-based (VDB) topological indices in conjunction with this approach in [5]. Within this framework, we propose five novel SO-like VDB topological indices based on geometry and calculus. We recall that two of the proposed indices are related to the elliptic Sombor index (ESO) and also highlight the relationship between two of the proposed indices and the second Gourava index (GO2). Afterward, we investigate the chemical usefulness of the four proposed indices and observe that two of them have better chemical applicability potential compared to the others. Moreover, we compare the obtained results with those obtained for ESO and GO2. Furthermore, we determine connected graphs and trees attaining the extremal values for the two proposed indices among all connected graphs and trees, respectively. Lastly, we present the relationships between the two proposed indices and SO from a mathematical perspective.