Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others - Map Projections: Cartographic Information Systems - page 645

refers to the 2 2 Hesse matrix of second derivatives with respect to ( x, y )atthepoint( x, y )to

be positive-definite. Indeed the unit matrix I 2

×

R 2 is always positive-definite. ∈

End of Example.

1 by

Finally we collect the results of the minimal distance mapping with respect to L

Corollary 23.7 (minimal distance mapping of a point close to the straight line).

2 be a point close to the straight line 1 . The point ( x, y ) 1 is at minimal

Let ( X,Y ) ∈ R L ∈ L

x 2

1 if

Euclidean distance X − to ( X,Y ) ∈ L

( x, y ) ∈ X − x

2 =min | ( X,Y ) ∈ R 2 , ( x, y ) ∈ L 1

(23.162)

and

x
y

= + cos 2 α 0 X

Y

+ sinα 0 cosα 0

+ sin 2 α 0 + sinα 0 cosα 0

+ + sin 2 α 0

x 0
y 0

(23.163)

− sin α 0 cos α 0

+ cos 2 α 0 − sin α 0 cos α 0

solves the optimization problem.

End of Corollary.

Examples 23.3 and 23.4 have prepared us to construct a minimal distance mapping
d ( x, X )=

2 to an unknown point x

as illustrated by
Fig. 23.35 .Itwillturnoutthatwehavetoconsidertwocases,namely case onex ∈ C,h c > 0 , where
h c describes the clothoidal height as the length of the vector X − x ; Always a minimal distance
mapping exists. In contrast, for case two,h c < 0 , a minimal distance mapping only exists if the
given point X inside the clothoid is located in between the clothoid and its evolute. The evolute
is the set of points of type curvature centres. Of course, in practice this condition is most often
met. At the beginning we present to you the result by means of Lemma 23.3. It will turn out that
the minimal distance mapping

X − x = min of a given point X ∈ R ∈ C

is based upon a universal equation of
polynomial type which can either be solved by standard software or by a Newton type iteration
process outline in Corollary 23.6 . Next to the end we present the elaborate proof of Lemma 23.3.
And finally we illustrate our results by a numerical example.

R

2

( X,Y ) → ( x, y ) ∈ C

The minimal distance mapping of close to the clothoid

With the experience of the minimal distance mapping of points close to the straight line or
to the circle we are prepared to understand Lemma 23.3 which governs the minimal distance
mapping of the points close to the clothoid . In particular, we take advantage of the polynomial
representation of the clothoid. The nonlinear normal equations are solved by Newton iteration as
outlined in Corollary 23.8 .Example 23.4 is a detailed exercise of a point close to the clothoid,
namely in polygon approximation of the clothoid.
To find the unique solution of the normal equations generated by the minimal distance mapping

( X,Y ) → ( x, y ) ∈ C and which are of polynomial type ( 23.165 ) is an elaborate procedure.
Since we know already the closed-form solution of the minimal distance mapping R 2

R

2

( X,Y ) →