<p>Two new conformal projections based on elliptic integrals of the first and second kinds are developed using the complex conformal latitude as the argument. The projection based on the elliptic integral of the first kind enables the entire Earth to be represented within a closed rectangular boundary and exhibits distortion characteristics comparable to those of the spherical transverse Mercator and ellipsoidal Thompson projections, indicating its potential suitability for global mapping. The projection based on the elliptic integral of the second kind, although unable to represent the entire ellipsoid within a finite region, closely approximates the Gauss–Krüger projection over extended wide zones. Within the standard 6° zone width on the GRS80 ellipsoid, the maximum differences in length distortion, meridional convergence, and projected coordinates are only <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$7.51\times 10^{-6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>7.51</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${0.11\,\mathrm{ ^{\prime \prime }}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.11</mn> <mspace width="0.166667em" /> <mmultiscripts> <mrow /> <mrow /> <mo>″</mo> </mmultiscripts> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${15.56\,\textrm{m}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>15.56</mn> <mspace width="0.166667em" /> <mtext>m</mtext> </mrow> </math></EquationSource> </InlineEquation>, respectively, all substantially smaller than the corresponding maximum projection distortions and also smaller than the differences between the Universal Transverse Mercator and Gauss–Krüger projections. According to the US National Map Accuracy Standards, these discrepancies are negligible for maps at publication scales no larger than 1 : 30, 500. The proposed projections may be classified as transverse elliptical cylindrical conformal mappings, indicating potential applications in both global mapping and large-scale geodetic mapping.</p>

Two new conformal projections based on elliptic integrals of the first and second kinds

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2026年7月9日

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Jia-Chun Guo

摘要

Two new conformal projections based on elliptic integrals of the first and second kinds are developed using the complex conformal latitude as the argument. The projection based on the elliptic integral of the first kind enables the entire Earth to be represented within a closed rectangular boundary and exhibits distortion characteristics comparable to those of the spherical transverse Mercator and ellipsoidal Thompson projections, indicating its potential suitability for global mapping. The projection based on the elliptic integral of the second kind, although unable to represent the entire ellipsoid within a finite region, closely approximates the Gauss–Krüger projection over extended wide zones. Within the standard 6° zone width on the GRS80 ellipsoid, the maximum differences in length distortion, meridional convergence, and projected coordinates are only $7.51\times 10^{-6}$ $7.51 \times 10^{- 6}$ , ${0.11\,\mathrm{ ^{\prime \prime }}}$ $0.11^{″}$ , and ${15.56\,\textrm{m}}$ $15.56 m$ , respectively, all substantially smaller than the corresponding maximum projection distortions and also smaller than the differences between the Universal Transverse Mercator and Gauss–Krüger projections. According to the US National Map Accuracy Standards, these discrepancies are negligible for maps at publication scales no larger than 1 : 30, 500. The proposed projections may be classified as transverse elliptical cylindrical conformal mappings, indicating potential applications in both global mapping and large-scale geodetic mapping.