<P> The relations between y (φ) and properties of the projection, such as the transformation of angles and the variation in scale, follow from the geometry of corresponding small elements on the globe and map . The figure below shows a point P at latitude φ and longitude λ on the globe and a nearby point Q at latitude φ + δφ and longitude λ + δλ . The vertical lines PK and MQ are arcs of meridians of length Rδφ . The horizontal lines PM and KQ are arcs of parallels of length R (cos φ) δλ . The corresponding points on the projection define a rectangle of width δx and height δy . </P> <P> For small elements, the angle PKQ is approximately a right angle and therefore </P> <Dl> <Dd> tan ⁡ α ≈ R cos ⁡ φ δ λ R δ φ, tan ⁡ β = δ x δ y, (\ displaystyle \ tan \ alpha \ approx (\ frac (R \ cos \ varphi \, \ delta \ lambda) (R \, \ delta \ varphi)), \ qquad \ qquad \ tan \ beta = (\ frac (\ delta x) (\ delta y)),) </Dd> </Dl> <Dd> tan ⁡ α ≈ R cos ⁡ φ δ λ R δ φ, tan ⁡ β = δ x δ y, (\ displaystyle \ tan \ alpha \ approx (\ frac (R \ cos \ varphi \, \ delta \ lambda) (R \, \ delta \ varphi)), \ qquad \ qquad \ tan \ beta = (\ frac (\ delta x) (\ delta y)),) </Dd>

Mercator projection maps are used to determine great-circle routes