Eye-hand coordination is geometrically complex. To compute the location of a visual target relative to the hand, the brain must consider every anatomical link in the chain from retinas to fingertips. Here we focus on the first three links, studying how the brain handles information about the angles of the two eyes and the head. It is known that people, even in darkness, reach more accurately when the eye looks toward the target, rather than right or left of it. We show that reaching is also impaired when the binocular fixation point is displaced from the target in depth: reaching becomes not just sloppy, but systematically inaccurate. Surprisingly, though, in normal Gaze-On-Target reaching we found no strong correlations between errors in aiming the eyes and hand onto the target site. We also asked people to reach when the head was not facing the target. When the eyes were on-target, people reached accurately, but when gaze was off-target, performance degraded. Taking all these findings together, we suggest that the brain's computational networks have learned the complex geometry of reaching for well-practiced tasks, but that the networks are poorly calibrated for less common tasks such as Gaze-Off-Target reaching.