Page 410 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

Notes
dY  dS . Sin   G . d 
 
 on the generating globe
dX  dS . cos   E . d  



dy  ds . Sin '  g . d  
 on the projection

dx  ds . cos '  e . d  
1
d  dS .cos
E
1
d  dS .sin
G
1
 dy  g . dS .sin
G

1
and : dx  e . dS .cos
E

dx 2 2 2 ; dy 2 2 2
E/e  dS . cos  G/g  dS . sin 

dx 2 dy 2 2 2 )dS 2
E/e  G/g  (sin q  cos 

dx 2 dy 2 dS 2
E/e  G/g 

If dS = 1 then the elementary circle on the globe has a radius of 1 (remember that capital letters
denote elements on the generating globe, and small letters elements on the projection.)

dx 2 dy 2 1
E/e  G/g 

This is an equation of an ellipse.

Analysis of Deformation Characteristics using Tissot’s Indicatrix

If we call the semi-major and semi-minor axes of the ellipse a, and b, then these are the directions
of maximum and minimum distortion i.e. the principal directions. a and b are also thus called
the principal scale factor

x 2 y 2 1
b  a 
For convenience we will consider the plane x and y axes to be in the principal directions.

Length Distortion
 = ds cos ’ on the plane
x
 = dS cos  = 1 on globe
X
(There is no distortion on the globe)

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