Metric-Induced Anisotropic Rotations with Gielis-Based Geometry

In this work, we present a new geometric framework that integrates quaternion-based rotation and translation operators with a generalized inner product and vector product framework defined on Gielis-type superquadrics. By incorporating the multiplicative shape factor ρ(ϕ), we construct a family of rotation matrices and quaternion mappings adapted to the elastic and non-Euclidean behavior of biological growth surfaces. This framework enables smooth, direction-dependent deformations and provides a unified representation for curvature-induced growth, differential thickening, and torsional motions observed in plants. The proposed model provides a mathematically tractable and biologically interpretable tool in many applications in plants, animals and biomolecules. Potential applications include computational botany, growth-based animation, and the design of biologically inspired structures.