1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

In this paper, we discuss a special property of conics on the projective plane and answer questions in enumerative algebraic geometry such as ”How many points determine a conic?” and ”How many conics do we expect to pass through m points and tangent to n lines?”

In this paper we prove that a point set in PG(2, q) meeting every line in 0, 1 or r points and having a unique tangent at each of its points is either an oval or a unital. This answers a question of Blokhuis and Szőnyi [1].

This paper considers and solves the problem of estimating camera pose given a pair of point-tangent correspondences between the 3D scene and the projected image. The problem arises when considering curve geometry as the basis of forming correspondences, computation of structure and calibration, which in its simplest form is a point augmented with the curve tangent. We show that while the standa...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...

We consider a string of length l with ends fixed, and rest state coinciding with x-axis. The string is plucked into oscillation. Let u (x, t) be the position of the string at time t. Assumptions: 1. Small oscillations, i.e. the displacement u (x, t) is small compared to the length l. (a) Points move vertically. In general, we don't know that points on the string move vertically. By assuming the...

This paper considers and solves the problem of estimating camera pose given a pair of point-tangent correspondences between the 3D scene and the projected image. The problem arises when considering curve geometry as the basis of forming correspondences, computation of structure and calibration, which in its simplest form is a point augmented with the curve tangent. We show that while the standa...