In this paper, we propose a topology optimization (TO) framework where the design is parameterized by a set of convex polygons. Extending feature mapping methods in TO, the representation allows for direct extraction of the geometry. In addition, the method allows one to impose geometric constraints such as feature size control directly on the polygons that are otherwise difficult to impose in density or level set based approaches. The use of polygons provides for more more varied shapes than simpler primitives like bars, plates, or circles. The polygons are defined as the feasible set of a collection of halfspaces. Varying the halfspace's parameters allows for us to obtain diverse configurations of the polygons. Furthermore, the halfspaces are differentiably mapped onto a background mesh to allow for analysis and gradient driven optimization. The proposed framework is illustrated through numerous examples of 2D structural compliance minimization TO. Some of the key limitations and future research are also summarized.

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