Paracatadioptric camera calibration based on properties of polar line of infinity point with respect to circle and line

Two linear calibration methods based on space-line projection properties for a paracatadioptric camera are presented. Considering the central catadioptric system, a straight line is projected into a circle on the viewing spherical surface for the first projection. The tangent lines in a group at antipode point pairs with respect to the circle are parallel, with the infinity point being the intersection point; therefore, the infinity line can be obtained from two groups of antipode point pairs. Further, the direction of the polar line of an infinity point with respect to the circle is orthogonal to the direction of its infinity point. Hence, on the imaging plane, images of the circular points or orthogonal vanishing points are used to determine the intrinsic parameters. On the basis of the properties of the antipodal point pairs and a least-squares fitting, a corresponding optimization algorithm for line image fitting is proposed. Experimental results demonstrate the robustness of the two calibration methods, that is, for images of the circular points and orthogonal vanishing points.

Pencils Of Polarities In Projective Space

1. Introduction. A polarity in complex projective space of two dimensions (S2) is completely determined by a self-polar triangle ABC, and a pair of corresponding elements: a point P and its polar line p. We denote the polarity by (ABC) (Pp). We follow Coxeter (2) in denning a pencil of polarities as the ∞1 polarities (ABC) (Pp) where A, B, C, P are fixed while p varies in a pencil of lines.

On the Three-Cusped Hypocycloid

I have had a recent opportunity to recall an early article (1884) which I wrote on the three-cusped hypocycloid. My starting point was the property that the asymptotes of any pencil of equilateral hyperbolas envelop such a hypocycloid. I proved this analytically in the aforesaid article ; perhaps there is some interest in finding geometrical reasons for it. Principles on pencils of conies are well known. According to these principles : (1) The polars of any point a with respect to the various conies of the pencil are concurrent at one and the same point a, which we shall call the corresponding point of a. (2) If a describes a straight line D, then a. describes a certain conic C. (3) This conic C is also the locus of the poles of D with respect to the conies of the pencil, a consequence being: (4) If m, a point of C, is the pole of D with respect to one of the conies H of the pencil and a a point of D with the corresponding point α, then the polar line of a with respect to H is mα.

On the centre of spherical curvature of a curve

The position of the centre S of spherical curvature at a point P of a given curve C may be found in the following manner, regarding S as the limiting position of the centre of a sphere through four adjacent points P, P1, P2, P3 on the curve, as these points tend to coincidence at P. The centre of a sphere through P and P1 lies on the plane which is the perpendicular bisector of the chord PP1 and so on. Thus the centre of spherical curvature is the limiting position of the intersection of three normal planes at adjacent points. Let s be the arc-length of the curve C, r the position vector of the point P, and t, n, b unit vectors in the directions of the tangent, principal normal and binormal at P. Then if s is the position vector of the current point on the normal plane at P, the equation of this plane isSince r and t are functions of s, the limiting position of the line of intersection of the normal planes at P and an adjacent point (i.e. the polar line) is determined by (1) and the equation obtained by differentiating this with respect to s, viz.