Characterization of Rectifying Curves by Their Involutes and Evolutes

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Direction Curves Associated with Darboux Vectors Fields and Their Characterizations

In this paper, we consider the Darboux frame of a curve α lying on an arbitrary regular surface and we use its unit osculator Darboux vector D ¯ o , unit rectifying Darboux vector D ¯ r , and unit normal Darboux vector D ¯ n to define some direction curves such as D ¯ o -direction curve, D ¯ r -direction curve, and D ¯ n -direction curve, respectively. We prove some relationships between α and these associated curves. Especially, the necessary and sufficient conditions for each direction curve to be a general helix, a spherical curve, and a curve with constant torsion are found. In addition to this, we have seen the cases where the Darboux invariants δ o , δ r , and δ n are, respectively, zero. Finally, we enrich our study by giving some examples.

Gauss Map Based Curved Origami Discretization

This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.

A distance on the equivalence classes of spherical curves generated by deformations of type RI

In this paper, we introduce a distance [Formula: see text] on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text] satisfy a certain technical condition, then [Formula: see text] is obtained from [Formula: see text] by a single weak RIII only. In Theorem 3, we show that if [Formula: see text] and [Formula: see text] satisfy other conditions, then [Formula: see text] is ambient isotopic to a spherical curve that is obtained from [Formula: see text] by a sequence of a particular local deformations, which realizes [Formula: see text].

A Novel Mechanism Model of Actual Precision Transmission Devices

A mechanism approach is presented in this paper to deal with machining errors and model the accuracy of a precision transmission device in connection with kinematic geometry. The 3D motion of a rotor with six DOFs is perfectly represented by a redundant mechanism [1]. Positions and orientations of two rotors are determined by solving the vector equations of the redundant mechanisms at different instants. The geometric properties of loci traced by the characteristic points and lines of the rotors are analyzed. The invariants of the discrete line-trajectories, the image spherical curve and striction curve, are introduced into the accuracy evaluation for the precision transmission device. The rotary table of a machine tool is used as an example to test the proposed model. The results show that the kinematic geometry is advantageous in modeling effects of errors in multiple body mechanical systems.

A study on motion of a robot end-effector using the curvature theory of dual unit hyperbolic spherical curves

In this paper we study the motion of a robot end-effector by using the curvature theory of a dual unit hyperbolic spherical curve which corresponds to a timelike ruled surface with timelike ruling generated by a line fixed in the end-effector. In this way, the linear and angular differential properties of the motion of a robot end-effector such as velocities and accelerations which are important information in robot trajectory planning are determined. Moreover, the motion of a robot end-effector which moves on the surface of a right circular hyperboloid of one sheet is examined as a practical example.