A near Earth asteroid, 2012 DA14, will pass within 35,000 kilometers of the Earth on Feb 15, 2013 at 19:25 UT according to JPL's ephemeris. If the direction of the asteroid as it flies by the Earth is plotted in rectangular coordinates one gets an odd curve indicating that it will spend most of its time near the poles.
The path appears to be a little simpler if it is plotted on a celestial globe and is very nearly a great circle. The swing from the Celestial South Pole to the Celestial North Pole takes place at the time of closest approach. The positions are given for one hour intervals. The center of the Celestial Sphere is 12h RA. The relatively large south to north motion is due the the fact that the asteroid is in an orbit similar to that of the Earth's and travels along with it.
The path appears to wrap around the Earth making an angle greater than 180°. The asteroid will be deflected by its proximity to the Earth but as it moves farther away the angle relative to the Ecliptic will also decrease since the relative distance above orbital plane changes at a slower rate. The rate of change for the distance from the Earth is fairly linear and appears to be "V" shaped over the 20 days included in the plot. The bottom of the "V" is rounded at the time of minimum distance. The apparent magnitude will be greatest near close approach rising to 7.48.
Supplemental (Jan 4): The JPL ephemeris was set to generate geocentric positions which are relative to the Earth's center and so the minimum distance from the Earth's surface will be about 29,000 km or 4.5 Earth radii. For comparison the average distance to the Moon from the center of the Earth is about 60 Earth radii.