Page 412 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

If 2 = 0 then no angular distortion occurs and the projection is called conformal. The property Notes
of a conformal projection is that a = b and Tissot’s Indicatrix is a circle with equal scale distortion
in all directions. This is consistent with the previously derived conditions for conformality,


namely that     . and '  2 . The area is not preserved and the projected circle increases in

size as one moves away from the line of zero distortion.
Areal Distortion (s): Second Theorem of Appolonius.
This is found by dividing the projected area by the area of the circle on the globe (radius =1)

 ab
   ab
 R 2
When looking at equal area projections earlier. It was found that:

 =   sin ’, thus   sin ’ = ab.
   
This is called the Second Theorem of Appolonius. When ab = 1 then the projection is equal-area
or equivalent.

Notes Conformality and equivalence are exclusive: ab = 1 and a = b cannot occur at
the same time.

31.5 Summary

Geodesics play a very important role in surface theory and in dynamics.

One of the main reasons why geodesics are so important is that they generalize to curved
surfaces the notion of “shortest path” between two points in the plane.

More precisely, given a surface X, given any two points p = X(u , v ) and q = X(u , v ) on X,
0
0
1
1
let us look at all the regular curves C on X defined on some open interval I such that
p = C(t ) and q = C(t ) for some t , t  I.
0
1
0
1
It can be shown that in order for such a curve C to minimize the length l (pq) of the curve
C
segment from p to q, we must have k (t) = 0 along [t , t ], where k (t) is the geodesic
g
1
0
g
curvature at X(u(t), v(t)).
Given a surface X :   E , let  : I  E be a regular curve on X parameterized by arc
3
3

length. For any two points p = (t ) and q = (t ) on , assume that the length l (pq) of the
1
0

curve segment from p to q is minimal among all regular curves on X passing through p
and q. Then,  is a geodesic.
At this point, in order to go further into the theory of surfaces, in particular closed surfaces,
it is necessary to introduce differentiable manifolds and more topological tools.
Nevertheless, we can’t resist to state one of the “gems” of the differential geometry of
surfaces, the local Gauss-Bonnet theorem.
The local Gauss-Bonnet theorem deals with regions on a surface homeomorphic to a
closed disk, whose boundary is a closed piecewise regular curve  without self-intersection.
Such a curve has a finite number of points where the tangent has a discontinuity.
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