This paper presents four alternative ways of initializing camera parameters using essentially the same calibration tools (orthogonal wands) as nowadays popular 3D kinematic systems do. However, the key idea presented here is to sweep the volume with an orthogonal pair or triad of wands instead of a single one. The proposed methods exploit the orthogonality of the used wands and set up familiar linear constraints on certain entities of projective geometry. Extracted initial camera parameters values are closer to the refined ones, which should generally assure faster and safer convergence during the refinement procedure. Even without refinement, sometimes not necessary, reconstruction results using our initial sets are better than using commonly obtained initial values. Besides, the entire calibration procedure is shortened since the usual two calibration phases become one.