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Projection

Dodecahedron face and its reflections — polar (ρ, γ)
Sphere — spherical (θ, φ)
max realised cusp2.03°sag area, res 40.273%
A5's choice
sampling…

Jacobian at the cursor

J =1.0127-0.09390.00000.9875
γ̂ρ̂
on the face
J
→
θ̂φ̂
on the sphere
J = Mrotate · Mshear · Msquash · Mscale
Mrotate =cos α−sin αsin αcos αα = 0.00°Mshear =1s01s = -0.095Msquash =q001/qq = 1.0127Mscale =k00kk = 1.0000
DSEA Projection
i

A5 uses the DSEA projection to map the planar coordinates from the dodecahedron face onto the sphere.

The Jacobian describes how this mapping behaves locally, and can be decomposed into more intuitive rotation, scale, shear and squash matrices.

While there are a number of other similar projections, DSEA was chosen as it produces minimal sag (deviation from great circles) in cell edges, compared to ISEA and RTSEA. It has minimal cusps (2°) and while ISEA is better here both are practically imperceptible.

The remaining three Parallel Small Circle projections have similar metrics to DSEA but no known closed-form inverse, so are more computationally intensive.

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