The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the slope metric. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.
We show how to construct new Finsler metrics, in two and three dimensions, whose indicatrices are pedal curves or pedal surfaces of some other curves or surfaces. These Finsler metrics are generalizations of the famous slope metric, also called Matsumoto metric.
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to certain well-know surface of revolution.
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities for the derivative of the associated function on the Cartesian coordinate and the polar coordinate. In particular, we apply this result to a certain well-known surface of revolution.
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