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.
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