Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 650

condition. There is a beautiful geometric representation of the Hesse scalar h 2 : If we introduce
the first and second differential invariance of the clothoid, namely

= gdx 2 | : d x | d x (23.179)

=+ d 2 x f 2 = hd x 2 || : − d f 2 | d x

d x 2 |

(23.180)

We are led to the celebrated representation of the curvature scalar (Figs. 23.38 and 23.39 )

hg − 1 κ = − (23.181)

Fig. 23.37. Minimal distance mapping of a point close to the clothoid C ,point X “ outside ”and“ inside ”the
clothoid, clothoidal height h c

Fig. 23.38. Height h> 0 of a point outside the clothoid