The Virbhadra-Ellis lens equation in astronomy and mathematics relates to the angular positions of an unlensed source ( β ) {\displaystyle \left(\beta \right)} , the image ( θ ) {\displaystyle \left(\theta \right)} , the Einstein bending angle of light ( α ^ ) {\displaystyle ({\hat {\alpha }})} , and the angular diameter lens-source ( D d s ) {\displaystyle \left(D_{ds}\right)} and observer-source ( D s ) {\displaystyle \left(D_{s}\right)} distances.
tan β = tan θ − D d s D s [ tan θ + tan ( α ^ − θ ) ] {\displaystyle \tan \beta =\tan \theta -{\frac {D_{ds}}{D_{s}}}\left[\tan \theta +\tan \left({\hat {\alpha }}-\theta \right)\right]} . This approximate lens equation is useful for studying the gravitational lens in gravitational fields of any strength when the angular source position is small.
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