We present an algorithm for creating contiguous cartograms using meshes. We use numerical optimization to minimize cartographic error and distortion by transforming the mesh vertices. The vertices can either be optimized in the plane or optimized on the unit sphere and subsequently projected to the plane. We also present a hybrid "best of both worlds" method, where the vertices are optimized on the sphere while anticipating the distortion caused by the final projection to the plane. We show a significant improvement in the preservation of region shapes compared to existing automated methods. Outside the realm of cartograms, we apply this hybrid technique to optimized map projections, creating the Liquid Earth projection.

翻译：本文提出了一种利用网格生成连续等值区域图的算法。我们通过数值优化变换网格顶点，以最小化制图误差与形变。顶点优化可在平面内直接进行，也可在单位球面上优化后投影至平面。此外，我们提出一种融合两者优势的混合方法：在球面优化顶点时，预先考虑最终平面投影将产生的形变。实验表明，相较于现有自动化方法，本方法在区域形状保持方面有显著提升。在等值区域图领域之外，我们将此混合技术应用于优化地图投影，创建了"液态地球投影"。