  Only one exoplanet has so far been mapped in both longitude and latitude, but
the James Webb Space Telescope should provide mapping-quality data for dozens
of exoplanets. The thermal phase mapping problem has previously been solved
analytically, with orthogonal maps---spherical harmonics---yielding orthogonal
lightcurves---sinusoids. The eclipse mapping problem, let alone combined
phase+eclipse mapping, does not lend itself to such a neat solution. Previous
efforts have either adopted spherical harmonics, or various ad hoc map
parameterizations, none of which produce orthogonal lightcurves. We use
principal component analysis to construct orthogonal "eigencurves," which we
then use to fit published 8 micron observations of the hot Jupiter HD 189733b.
This approach has a few advantages over previously used techniques: 1) the
lightcurves can be pre-computed, accelerating the fitting process, 2) the
eigencurves are orthogonal to each other, reducing parameter correlations, and
3) the eigencurves are model-independent and are ranked in order of
sensitivity. One notable result of our analysis is that eclipse-only mapping of
HD 189733b is far more sensitive to the central concentration of dayside flux
than to the eastward offset of that hotspot. Mapping can, in principle, suffer
from degeneracies between spatial patterns and orbital parameters. Previous
mapping efforts using these data have either assumed a circular orbit and
precise inclination, or have been pessimistic about the prospects of eclipse
mapping in the face of uncertain orbital parameters. We show that for HD
189733b the combined photometry and radial velocity are sufficiently precise to
retire this concern. Lastly, we present the first map of brightness
temperature, and we quantify the amplitude and longitude offset of the dayside
hotspot.
