Spherical objects and vanish points are often used for camera calibration. An occluding contour of a sphere is projected to a conic in the perspective image, and using a moving active camera, the trajectory of a vanishing point in the perspective images is also a conic when the camera is rotated about a fixed 3D axis whereas the translation of the camera is arbitrary. In fact, the problems of camera calibration using conics from spheres or vanishing points can be described by same mathematic representations. Two linear approaches to the problems are proposed in this paper: one based on the geometric interpretation of the relation between image conics and the image of the absolute conic, and the other using the special structure of the problems in algebra. Only three such conics are needed for the two linear approaches, and the minimum number for previous nonlinear optimization methods is also three. All five intrinsic parameters are recovered linearly without making assumptions, such as, zero-skew or unitary aspect ratio which are often used in previous methods. The two linear algorithms have been tested in extensive experiments with respect to noise sensitivity and also made comparisons with recent calibration techniques.