Quasi indicatrix approach for distortion visualization and analysis for map projections

Abstract

Tissot’s indicatrix or ellipse of distortion is a diagram that is the projection of an infinitesimal circle on the original surface. It is normally an ellipse of which elongation depends on the amount of distortion caused by map projection. It provides a medium for analyzing existing projections and developing new ones. The ellipse can be scaled and depicted on the map for visualization purposes. This paper presents an alternative approach, in which the projection of a finite small circle on the sphere is used. Its projection is normally an ellipse that can be very close to Tissot’s indicatrix, and is called quasi indicatrix, here. Its parameters can be derived from the forward projection equations without using partial derivatives. Therefore, it is a useful and practical approach from a programmer’s point of view. The quasi indicatrix approach is also numerically tested on Aitoff–Hammer projection with a set of points. The indicatrix parameters obtained by using this approach deviate 0.5% from the ground truths at most, being the average less than 0.2%.

Keywords:

• map projections
• Tissot’s indicatrix
• ellipse of distortion
• conic sections
• projection distortions

Acknowledgement

The author thanks Prof. Dr. Aydin Ustun from Kocaeli University, Turkey, and the anonymous reviewers for their valuable suggestions and contributions.

ORCID

Ibrahim Oztug Bildirici http://orcid.org/0000-0001-7717-586X