Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

Generalization of the pedal concept in bidimensional spaces. Application to the limaçon of Pascal

The concept of a pedal curve is used in geometry as a generation method for a multitude of curves. The definition of a pedal curve is linked to the concept of minimal distance. However, an interesting distinction can be made for ℝ2. In this space, the pedal curve of another curve C is defined as the locus of the foot of the perpendicular from the pedal point P to the tangent to the curve. This allows the generalization of the definition of the pedal curve for any given angle that is not 90º.In this paper, we use the generalization of the pedal curve to describe a different method to generate a limaçon of Pascal, which can be seen as a singular case of the locus generation method and is not well described in the literature. Some additional properties that can be deduced from these definitions are also described.

Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space

For the spherical unit speed nonlightlike curve in pseudo-hyperbolic space and de Sitter space [Formula: see text] and a given point P, we can define naturally the pedal curve of [Formula: see text] relative to the pedal point P. When the pseudo-sphere dual curve germs are nonsingular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the pseudo-sphere dual curve germs are nonsingular. Furthermore, we obtain the extension results in dualities, which has wide influence on the open and closed string field theory and string dynamics in physics, and can be used to better solve the dynamics of trajectory particle condensation process.

Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse

The Negative Pedal Curve (NPC) of the Ellipse with respect to a boundary point M is a 3-cusp closed-curve which is the affine image of the Steiner Deltoid. Over all M the family has invariant area and displays an array of interesting properties.

Galileo’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola (16.03.2019)

Galileo’s Parabola describing the projectile motion passed through hands of all scholars of the classical mechanics. Therefore, it seems to be impossible to bring to this topic anything new. In our approach we will observe the Galileo’s Parabola from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola. For the description of events on this Galileo’s Parabola (this conic section parabola was discovered by Menaechmus) we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the Galileo’s empty focus that plays an important function, too. We will study properties of this MAG Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. For the visible Galileo’s Parabola in the Aristotelian World, there might be hidden curves in the Plato’s Realm behind the mechanism of that Parabola. The analysis of these curves could reveal to us hidden properties describing properties of that projectile motion. The parabolic path of the projectile motion can be described by six expressions of projectile speeds. In the Dürer-Simon’s Parabola we have determined tangential and normal accelerations with resulting acceleration g = 9.81 msec-2 directing towards to Galileo’s empty focus for the projectile moving to the vertex of that Parabola. When the projectile moves away from the vertex the resulting acceleration g = 9.81 msec-2 directs to the center of the Earth (the second focus of Galileo’s Parabola in the “infinity”). We have extracted some additional properties of Galileo’s Parabola. E.g., the Newtonian school correctly used the expression for “kinetic energy E = ½ mv2 for parabolic orbits and paths, while the Leibnizian school correctly used the expression for “vis viva” E = mv2 for hyperbolic orbits and paths. If we will insert the “vis viva” expression into the Soldner’s formula (1801) (e.g., Fengyi Huang in 2017), then we will get the right experimental value for the deflection of light on hyperbolic orbits. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Newton’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Castillon’s Cardioid, and Ptolemy’s Circle (Hodograph) (09.02.2019)

Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia…” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician). We will study properties of this PAN Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. In the Plato’s Realm some curves carrying hidden information might be waiting for our research. One such curve - the evolute of parabola - discovered Newton behind his famous gravitational law. We have used the Castillon’s cardioid as the curve describing the tangent velocity of objects on the parabolic orbit. In the PAN Parabola we have newly used six parameters introduced by Gottfried Wilhelm Leibniz - abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal. We have obtained formulae both for the tangent and normal velocities for objects on the parabolic orbit. We have also obtained the moment of tangent momentum and the moment of normal momentum. Both moments are constant on the whole parabolic orbit and that is why we should not observe the precession of parabolic orbit. We have discovered the Ptolemy’s Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting objects. In this case the Ptolemy’s Circle plays a role of the hodograph rotating on the parabolic orbit without sliding. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Newton’s Hyperbola Observed from Newton’s Evolute (1687), Gudermann’s Circle (1833), the Auxiliary Circle (Pedal Curve and Inversion Curve), the Lemniscate of Bernoulli (1694) (Pedal Curve and Inversion Curve) (09.01.2019)

Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets orbiting around the Sun. After the intensive study of those conic sections during the last four hundred years it is believed that this topic is practically closed and the 21st Century cannot bring anything new to this subject. Can we add to those visible orbits from the Aristotelian World some curves from the Plato’s Realm that might bring to us new information about those conic sections? Isaac Newton in 1687 discovered one such curve - the evolute of the hyperbola - behind his famous gravitation law. In our model we have been working with Newton’s Hyperbola in a more complex way. We have found that the interplay of the empty focus M (= Menaechmus - the discoverer of hyperbola), the center of the hyperbola A (= Apollonius of Perga - the Great Geometer), and the occupied focus N (= Isaac Newton - the Great Mathematician) together form the MAN Hyperbola with several interesting hidden properties of those hyperbolic paths. We have found that the auxiliary circle of the MAN Hyperbola could be used as a new hodograph and we will get the tangent velocity of planets around the Sun and their moment of tangent momentum. We can use the lemniscate of Bernoulli as the pedal curve of that hyperbola and we will get the normal velocities of those orbiting planets and their moment of normal momentum. The first derivation of this moment of normal momentum will reveal the torque of that hyperbola and we can estimate the precession of hyperbolic paths and to test this model for the case of the flyby anomalies. The auxiliary circle might be used as the inversion curve of that hyperbola and the Lemniscate of Bernoulli could help us to describe the Kepler’s Equation (KE) for the hyperbolic paths. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Kepler´s Ellipse Observed from Newton´s Evolute (1687), Horrebow´s Circle (1717), Hamilton´s Pedal Curve (1847), and Two Contrapedal Curves (28.10.2018)

Johannes Kepler discovered the very elegant elliptical path of planets with the Sun in one focus of that ellipse in 1605. Kepler inspired generations of researchers to study properties hidden in those elliptical paths. The visible elliptical paths belong to the Aristotelian World. On the other side there are invisible mathematical objects in the Plato´s Realm that might describe the mechanism behind those elliptical paths. One such curve belonging to the Plato´s Realm discovered Isaac Newton in 1687 - the locus of radii of curvature of that ellipse (the evolute of the ellipse). Are there more curves in the Plato´s Realm that could reveal to us additional information about Kepler´s ellipse? W.R. Hamilton in 1847 discovered the hodograph of the Kepler´s ellipse using the pedal curve with pedal points in both foci (the auxiliary circle of that ellipse). This hodograph depicts the moment of the tangent momentum of orbiting planets. Inspired by the hodograph model we propose newly to use two contrapedal curves of the Kepler´s ellipse with contrapedal points in both the Kepler´s occupied and Ptolemy´s empty foci. Observers travelling along those contrapedal curves might bring new valuable experimental data about the orbital angular velocity of planets and a new version of the Kepler´s area law. Based on these contrapedal curves we have defined the moment of the normal momentum. The first derivation of the moment of the normal momentum reveals the torque of the ellipse. This torque of ellipse should contribute to the precession of the Kepler´s ellipse. In the Library of forgotten works of Old Masters we have re-discovered the Horrebow´s circle (1717) and the Colwell´s anomaly H (1993) that might serve as an intermediate step in the solving of the Kepler´s Equation (KE). Have we found the Arriadne´s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?