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Deformable Models and Image Segmentation
Published in Ayman El-Baz, Jasjit S. Suri, Level Set Method in Medical Imaging Segmentation, 2019
Ahmed ElTanboly, Ali Mahmoud, Ahmed Shalaby, Magdi El-Azab, Mohammed Ghazal, Robert Keynton, Ayman El-Baz, Jasjit S. Suri
Geodesic Curvature is another form of curve representation. Given a parametric curve C(S) embedded into the surface S, this curve will have the following acceleration vector: Css=kgT^+knN^
Particle on a Curve and on a Surface
Published in John G. Papastavridis, Tensor Calculus and Analytical Dynamics, 2018
Contravariant components of normalized geodesic curvature, of C at P, KG=K(G)yEγ=K(G)G[⇒K(G)=(MaβK(G)K(G)β)½: normalized geodesic curvature]
Differential geometry
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2018
The coefficients are the normal curvature and the geodesic curvature, respectively. Taking the inner product of the last equation with the b¯ vector and exploiting the perpendicularity conditions present, we obtain b¯⋅k¯=κg.
Three-dimensional fixed-trajectory approaches to the minimum-lap time of road vehicles
Published in Vehicle System Dynamics, 2022
The angles are obtained by integration of the relative torsion , normal curvature , and geodesic curvature , which are the angular rate s (rad/m) of the Darboux frame expressed in the Darboux frame itself. The resulting relationships are obtained from the well-known Frenet-Serret formula where the prime means derivative with respect to s. In practice, the angular rates are estimated from the measured coordinates of the road borders, see e.g. [2,11,18,31]. The position of the origin of the Darboux frame, expressed in the absolute frame, is obtained by integration of the tangent vector , i.e. the first column of (1)
Mesh segmentation via geodesic curvature flow
Published in Computer-Aided Design and Applications, 2018
Zhiyu Sun, Ramy Harik, Stephen Baek
The geodesic curvature flow (GCF) is a geometric flow, or informally, a continuous evolution of a curve that minimizes the arc length of a curve. Given a closed self-avoiding rectifiable curve lying on a differential -manifold embedded in (), the energy functional of the GCF is defined as follows: where is an infinitesimal segment defined on the curve for the integration. Here, we restrict our curve to be rectifiable in order to make sure that it is integrable. A curve on a manifold is said to be rectifiable if and only if the length of every geodesic polygon formed by vertices , can be bounded from above by the length of the curve for some parameterization , and under the induced metric of . This consequently means that the curve is a function with bounded variations, and thus integrable.
A three-dimensional free-trajectory quasi-steady-state optimal-control method for minimum-lap-time of race vehicles
Published in Vehicle System Dynamics, 2022
The road centreline is defined using its curvature κ and torsion τ, which are given as a function of the travelled distance along the road, i.e. the curvilinear coordinate s. The road plane is created by adding width w and twist ν to the road centreline. A moving trihedron, called Darboux frame, moves with the vehicle along the centreline and its x−y plane represents the road tangent plane. The orientation of such frame is expressed using the attitude-slope-banking convention. The related angles, θ-μ-ϕ, are obtained by integration of and ν or alternatively by integration of the relative torsion , normal curvature and geodesic curvature , which are the angular rate of the Darboux frame, expressed along the axes of the Darboux frame itself. The resulting relationships are obtained from the well-known Frenet–Serret formula where the prime means derivative with respect to s. The position of the origin of the Darboux frame, expressed in the absolute ground frame, is obtained from where is the tangent to the road centreline, which is the first column of the rotation matrix associated with the Darboux frame.