in x and y. The degree of the curve is the degree of the polynomial f(x, y). A parametric curve α(t) = (p(t), q(t)), where p(t) and q(t) are polynomials in t, is an algebraic curve. This is because we can eliminate t from the two polynomial equations x = p(t) and y = q(t) using the resultant technique, and obtain a polynomial equation, called their resultant, in x and y. The resultant has a deg...

The rotation-minimizing frame is the main research object for a spatial curve. Although mathematical description not complicated, it easy to directly make an exact minimizing-rotation Euler-Rodrigues frame. condition non-normalized of Pythagorean-Hodograph curve become given in this article, which ordinary differential equation with rational form, analytical solution that does always exist. To ...

The shot noise of a tunneling current passing through a molecule-motor can sustain a one-way rotation when populating the molecular excited states by tunneling inelastic excitations. We demonstrate that a ratchet-like ground state rotation potential energy curve is not necessary for the rotation to occur. A relative shift in energy difference between the maxima of this ground state and the mini...

Rotation Transformation is the backbone of visualization of 3D objects. Rotation transformations allow viewing objects from different angles. Rotations about the principal axes are straightforward whereas the rotation about an arbitrary axis is complex. In mobile environment, rotations is computed continuously as the objects move and is thus a costly operation. We give intuitive, constructive, ...

A binary chain code for representing 3D curves is presented. 3D curves are digitalized and represented as 3D discrete curves. 3D discrete curves are composed of constant straight-line segments using only orthogonal directions. The proposed binary notation is based on the orthogonal direction change chain code. Thus, using the proposed binary chain code it is possible to represent any 3D curve b...