Coordinate Transformation Rules Research Articles

Electrostatic potential of a uniformly charged triangle in barycentric coordinates

We compute the electrostatic potential of a uniformly charged triangle. Barycentric coordinates are employed to express the field point, the parametrization of the surface integral, and the gradient operator. The resultant analytic expression for the electrostatic potential is expressed in terms of the side lengths of the triangle, the altitude of the field point from the plane in which the triangle is placed, and the barycentric coordinates of the field point relative to the triangle. Our results are in good agreement with available numerical results. The asymptotic behavior of the analytic expression is investigated in special limits that satisfy known values. The resultant analytic expressions for the asymptotic regions are useful in improving the numerical convergence at boundaries. As an application, we provide a strategy to compute the electrostatic potential of a uniformly charged polygon. The electrostatic potential of a uniformly charged rectangle is considered as a simple example that agrees with a previous result. Appendices provide a complete set of integral tables that are necessary to evaluate the double integral over the barycentric coordinates, an explicit parametrization of the gradient operator in the barycentric coordinates, and useful coordinate-transformation rules between the barycentric and Cartesian coordinates.

Coordinate transformation of three-dimensional image in integral imaging system illuminated by a point light source

The point light source is an important kind of directional light source. By using directional illumination, the performances of the integral imaging system will be enhanced. In this paper, the coordinate transformation characteristics of integral imaging systems illuminated by a point light source have been explored in detail. By using a dynamic point light source such as a moving light emitting diode to illuminate the imaging system, the position and size of the rendered three-dimensional image can be transformed in real time without updating the elemental images, and the depth of field of the imaging system is doubled. The coordinates of the three-dimensional image are changed with the distance between the point light source and the micro-lens array. The optical mechanism of the three-dimensional dynamic imaging phenomenon is analyzed. The coordinate transformation rules of the three-dimensional image are deduced and verified by experiments. The phenomenon may find potential applications in interactively three-dimensional display or augmented reality system.

The correct and unusual coordinate transformation rules for electromagnetic quadrupoles.

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.

Fractional differential forms

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case. Coordinate transformation rules are also computed. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. The metric for the fractional form spaces is given, based on the coordinate transformation rules. All results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one.