## The image of the absolute conic

• Points on the plane at infinity ( ), which may be written as are mapped to the image plane by a general camera as

• Thus is the planar homography between and the image plane. Note that the mapping is independent of the position (translation) of the camera and depends only on the orientation. (An explanation as to why the images of stars stay fixed on the retinae as we translate?)
• Since the absolute conic ( ) is on , we can compute its image as

Proof. Note that under a point homography which maps to , a conic is mapped to . Hence on maps to

• Like , is an imaginary point conic with no real points. It cannot really be observed in an image. It is really an useful mathematical device.
• depends only on the internal parameters of the camera and is independent of the cameras position or orientation.
• It follows from above that the angle between two rays is given by the simple equation

• The above expression is independent of the choice of the projective coordinate system on the image. To see this consider any 2D projective transformation . The points are transformed to , and transforms (as any image conic) to . Hence the expression for is unchanged.
• We may define the dual image of the absolute conic as

• Once (equivalently ) is identified in an image is uniquely determined; since a symmetric matrix can be uniquely decomposed into an upper triangular matrix and its transpose ( ) by Cholesky decomposition.
• An arbitrary plane intersects in a line, and this line intersects in two points (imaginary) which are circular points of . The image of the circular points line on at the points at which the vanishing lines of the plane intersects .
Subhashis Banerjee 2008-01-20