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3.4.4.1b Krovak Oblique Conformal Conic

The normal case of the Lambert Conformal conic is for the axis of the cone to be coincident with the minor axis of the ellipsoid, that is the axis of the cone is normal to the ellipsoid at a pole. For the Oblique Conformal Conic the axis of the cone is normal to the ellipsoid at a defined location and its extension cuts the minor axis at a defined angle. This projection is used in the Czech Republic and Slovakia under the name "Krovak" projection. The projection method is similar to the Oblique Mercator (see section 3.4.4.5). The geographic co-ordinates on the ellipsoid are first reduced to conformal co-ordinates on the conformal Gaussian sphere. These spherical co-ordinates are then projected onto the oblique cone and transformed to grid co-ordinates. The pseudo standard parallel is the radius arc on the projected cone which is true to scale. A scale factor may be applied to this arc to increase the useful area of coverage.

The defining parameters for the oblique lambert conformal conic projection are:

jc	= latitude of center of the projection
lc	= longitude of center of the projection
ac	= azimuth (true) of the center line passing through the center of the projection = co-latitude of the cone axis at point of intersection with the ellipsoid.
j1	= latitude of pseudo standard parallel.
kc	= scale factor on the psuedo standard parallel
Ec	= False Easting of the center of the projection at the apex of the cone.
Nc	= False Northing of the center of the projection at the apex of the cone.

From these the following constants for the projection may be calculated:

To derive the projected Easing and Northing coordinates of a point with geographical coordinates (j,l) the forumlas for the oblique conic conformal are:

Easting:

Northing:

where

Note that the terms easting and northing here refer to the two map grid ordinates. Their actual geographic direction depends upon the azimuth of the center line.

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:

where

Example:

Oblique Lambert Conic Conformal

For Projected Coordinate System: Krovak

Note: Krovak projection uses Ferro as the prime meridian. This has a longitude with reference to Greenwich of 17 deg 40 min West. To apply the formulas the defining lingitudes must be corrected to the Greenwich meridian.

Parameters:

Projection constants:

B = 1.000597498

A = 6380703.61

g_o = 0.863239103

t_o = 1.003419164

n = 0.979924705

r_o = 1298039.005

Forward calculation for:

Gives

Then:

where Easting increases southwards and northing increases westwards.

Reverse calculation for the same Easting and Northing gives:

j1 =	0.876310601
j2 =	0.876312560
j3 =	0.876312566

Latitude	j =	0.876312566 rad	=	50°12'32.4416" N
Longitude	l =	0.595793792 rad	=	16°50'59.1790" E

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