Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 632

( s ) as well as normal vector v ( s )ofthe clothoid can
be conveniently described by solving the initial value problem α = κ (s) = s /a 2 ,α 0 = α ( s 0),
solved by
For the proof we have to find the general solution of the homogeneous equation

The circular motion of the tangent vector μ

α =0anda

α = s/α 2 . First the general solution of the

particular solution of the inhomogeneous equation

homogeneous equation is

(s) = α 0 + s 0 (s) = α 0 + α 0 ( s

α − s 0 ) or α

a 2 ( s

− s 0 ) (23.133)

Second a particular solution of the inhomogeneous equation α = s/α 2 is based upon the integral
α (s) = s
s 0

a 2 s

κ ( s ) ds = 1

1

2 a 2 ( s − s 0 ) 2

sds = (23.134)

s 0

The superposition of the general solution of the homogeneous equation and the particular solution
of the inhomogeneous equation leads directly to the result of Corollary 23.2
The clothoid C

2 has finally to be constructed from its
curvature k ( s )= x | v ( s ) indeed a problem of global differential geometry. Since in the first
step we have already characterized the circular motion of its tangent vector as well as its normal
vector, in the second step we shall concentrate on its embedding function x ( s ). The tangent vector
x ( s ) at the points enjoys a particular form in the ambient space

2

⊂ R

2 isometrically embedded in R

R

2 ,namely

x (s) = e 1 x (s) + e 2 y (s) = e 1 cos α ( s )+ e 2 sin α ( s ) (23.135)

2 appear. They can be thought as
conformal coordinates of type Gauss-Krueger or UTM with respect to an International Reference
Ellipsoid, e.g. WGS80. In comparing the left and right representation of the tangent vector x ( s )
we are led to the system of differential equations of first order which govern the computation of ( x ,
y ) coordinates from the orientation parameter α of the tangent map

Here, for the first time Cartesian coordinates ( x , y ) covering R

1 ,namely

μ ( s ) ∈ T M → S s

x = cosα ( s ) ,y =sin α ( s ) (23.136)

Series expansion of the Fresnel integrals

If we integrate directly these differential equations, in particular with respect to α ( s )repre-
senting via Corollary 23.2 the clothoid C , we are led to the famous Fresnel integrals. Such an
approach is not suited here since first Fresnel integrals are only tabulated and second they are
too inflexible to account for those parts of the clothoid which are by purpose interrupted by cir-

cular and stretch lines. Instead a local representation is searched for which can be easily adjusted
to curve sections of type circle and/or straight line. Our result of integration ( x ,y ), respectively,
is collected in

C

Lemma 23.1 (local representation of the clothoid C ).

Given the system of ordinary differential equations of first order

x = cosα ( s ) ,y = sinα ( s ) ,

(23.137)

then the solution of its initial value problem ( x 0 ,y 0 )=( x 0 ,y 0 )=( x ( s 0 ) ,y ( s 0 )) is