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?

Geomorphology Characterization of Ica Basin and Its Influence on the Dynamic Response of Soils for Urban Seismic Hazards in Ica, Peru

We evaluated the influence of the geomorphology of Peru’s Ica Basin on the dynamic response of soils of the city of Ica. We applied five geophysical methods: spectral ratio (H/V), frequency-wavenumber (F-K), multichannel analysis of surface waves (MASW), multichannel analysis of microtremor (MAM), and Gravimetric Analysis. Our results indicate that the soils respond to two frequency ranges:F0(0.4–0.8 Hz) andF1(1.0–3.0 Hz). TheF-K, which considers circular arrays, shows two tendencies with a jump between 1.0 and 2.0 Hz. MASW and MAM contribute to frequencies greater than 2.0 Hz. The inversion curve indicates the presence of three layers of 4, 16, and 60 m with velocities of 180, 250, and 400 m/s. The Bouguer anomalies vary between −17.72 and −24.32 mGal and with the spectral analysis we identified two deposits, of 60 m and 150 m of thickness. Likewise, the relationship between the velocities of 400 and 900 m/s, with the frequency = 1.5 Hz, allows us to determine the thickness for the layers of 60 (slightly alluvial to moderately compact) and 150 m (soil-rock interface). These results suggest that the morphology of the Ica Basin plays an important role in the dynamic behavior of the soils to low frequency.