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Geometric model for motion of curves specified by acceleration - Research Proposal Example

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This research contributes to present knowledge by applying the existing equations to modern scenarios to determine whether the existing data remains relevant and provide new details pertinent to current trends and examines the primary curvature properties developed during the evolution of curves. …
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Geometric model for motion of curves specified by acceleration
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Geometric model for motion of curves specified by acceleration

The intention of this study a geometric model generally that deals with the kinematics of a one dimensional manifold in a higher dimensional space. The model is specified by acceleration fields which are local or global functions of the intrinsic quantities of the manifold. This research intends to examine the evolution of one dimensional manifold embedded in the Euclidean space as it evolves under a stochastic flow of diffeomorphisms. Within the manifold, motion depends on the intrinsic invariants immersed in the space. During the course of this research, we will obtain the system of differential equations that governs the motion of the curve, keeping in mind that the processes driving the stochastic flows are chosen to be the most common class of Gaussian processes with stationary increments in time, which is the family of fractional Brownian motions with Hurst parameter. A family of random mappings is called a stochastic (Brownian) flow and is formulated as follows:
“ɸst, 0 ͟< s ͟< t < ∞, Rn into itself such that:
ɸst, for each s ͟< t is a diffeomorphisms of Rn into itself.
ɸut ͦ ɸsu = ɸst, for all s ͟< u ͟< ∞.
ɸtt is the identity map on Rn for all t.
ɸs1t1, ɸs2t2, …, ɸsntn are independent if s1 ͟< t1 ͟< s2 ͟< t2 ͟< … ͟< sn ͟< tn. ”
Using some applications to give geometric meanings to each solution to the governing system of (Partial Differential Equations) PDE,s corresponding to the model length and local time investigated, this profile will also demonstrate how the geometric problem can be transformed to a fully nonlinear parabolic system of equations for the curvature, the position, and orientation.

This research will also examine the primary curvature properties developed during the evolution of curves. Another facet of the study will explore the evolution of derive time equations using the Frenet frame. Further derive time equations will be determined regarding the intrinsic quantities satisfied by curves. The investigation will also propose a model using the solution of the evolution equation for the curvature and torsion and the Fundamental theorem for space curves to ... Read More
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