  We report a systematic study on the directional sensitivity of a direct dark
matter detector that detects the polar angle of a recoiling nucleus. A weakly
interacting massive particle (WIMP)-mass independent method is used to obtain
the sensitivity of a general detector in an isothermal galactic dark matter
halo. By using two-dimensional distributions of energy and polar angle, a
detector without head-tail information with 6.3 times the statistics is found
to achieve the same performance level as a full three-dimensional tracking dark
matter detector. Optimum operation orientations are obtained for various
experimental configurations, with detectors that are space- or Earth-fixed,
have head-tail capability or not, and use energy information or not.
Earth-fixed detectors are found to have best sensitivity when the polar axis is
oriented at a 45 degree angle from the Earth's pole. With background
contamination that mimics the WIMP signal's energy distribution, the
performance is found to decrease at a rate less than the decrease of signal
purity. The WIMP-mass dependence of the performance of a detector with various
energy thresholds that uses gaseous xenon as target material is reported. We
find that with a $5\times 10^{-46} \mathrm{cm}^2$ spin-independent WIMP-nucleon
cross-section and a 30 GeV WIMP, a $770$ kg$\cdot$year's exposure with a polar
detector of 10 keV threshold can make a three sigma discovery of directional
WIMPs in the isothermal galactic dark matter halo. For a columnar recombination
detector, experimental considerations are discussed.
