Research on the pedals trajectory on cone surface of inside point to element line

Abstract:

This paper studies the trajectory equations of pedals of an inside arbitrary point to all element lines of a cone for solving the problem of cone curve in actual engineering. Based on the feature of a cone and the theory of differential geometry, when the vector from an inside arbitrary point to an element line is perpendicular to the vector of that element line, their scalar product has to be zero, thus the parameter expression of the pedals is derived. After calculated and simplified, the parameter equation of track points shows an intersection track line of the cone and an offset sphere. And the expression in the Descartes coordinate is given. Furthermore, a conclusion is obtained that if there defines a sphere by a diameter from the summit to an arbitrary point in the cone, then lines from the given point to the intersection of that sphere and cone are perpendicular to the intersected conical element lines. Meanwhile, the projection property of the trajectory is studied, and the frontal projection is a parabola, and the horizontal projection is a quadratic curve.