ELERP: An ellipsoidal linear interpolation for ellipsoidal graph visualization

The Digital Earth (DE) is a digital twin of the real earth, whose shape is approximated by a 3D reference ellipsoid. Abstract relationship among objects in the world can be represented by graph visualization on the DE. However, because the diversity of curvature on ellipsoid, it is generally difficult to construct accurate and concise curvature-fitting edges for 3D graph visualization. Therefore, a curvature-fitting ellipsoidal graph visualization algorithm (CFE) is proposed to address the gap. The core work is a novel ellipsoidal linear interpolation (ELERP) with closed form. ELERP is derived by the combination of spherical linear interpolation (SLERP) and composite transformation W. W accomplishes the scaled transformation from general ellipsoid into unit sphere and implements the rotation from 3D inclined plane into 2D XY-plane. In W, Householder matrix H is from the Hadamard product generated from the geometric composite information of ellipsoid and plane. H has concise computation stemmed from the excellent properties that its inverse and transpose are itself. Lastly, using ELERP and Bezier interpolation to construct a new beret-style and curvature-fitting edge. The experimental results verify the validity of CFE. Compared with classical complex geodesic algorithm, CFE is faster and more concise. Generally, CFE provides an ellipsoidal graph visualization, which will be applied on ellipsoid-based celestial bodies in the vast universe, such as, earth, moon and Mars.