Discrete isogonal nets with similar parallelograms

High-quality surface designs are increasingly significant in industrial applications, such as architecture and product design, yet they pose challenges in balancing visual appeal and functional requirements. Isogonal nets (I-nets) stand out for their aesthetically pleasing patterns and engineering practicality. However, constructing such nets remains difficult due to their dependence on complex angle constraints or a narrow focus on orthogonal scenarios. We propose a novel representation and construction method for I-nets characterized by similar mid-edge subdivided parallelograms in the quad faces. This approach achieves a simple yet versatile representation that generalizes orthogonal nets and extends to the construction of isogonal 4-webs (I-webs). By focusing on constraining edge ratios, our method enables efficient integration into mesh optimization algorithms. We demonstrate the effectiveness of I-nets and I-webs in freeform shapes through conformal mapping and numerical optimization. Experiments on various surfaces validate our method, showcasing its potential for both theoretical advancements and practical applications.