Metric-Induced Rotations with Gielis-Based Geometry

In this work, we present a generalized geometric framework that extends quaternion-based rotation and translation operators within a generalized inner product and vector product setting defined on Gielis-type superquadrics. By incorporating the multiplicative shape factor ρ(ϕ), we formulate rotation matrices and quaternion mappings adapted to the elastic and non-Euclidean behavior of biological growth surfaces. The proposed framework extends classical Euclidean constructions to the generalized metric setting, enabling smooth, direction-dependent deformations and providing a unified description of curvature-induced growth, differential thickening, and torsional motions observed in plants. The resulting formulation offers a mathematically consistent and biologically interpretable tool with potential applications in computational botany, growth-based animation, and the design of biologically inspired structures.