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The Frenet–Serret frame on a curve is the simplest example of a moving frame. In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space .
9 KINEMATICS OF MOVING FRAMES 71 This result is often written in terms of body-referenced velocity v: ∂ r v = ω × r + + vo, ∂t where vo is the body-referenced velocity of the origin. The total velocity of the particle is equal to the velocity of the reference frame origin, plus a component due to rotation of this frame.
2. Equivariant Moving Frames. We begin by describing the general equivariant moving frame construction. Let Gbe an r-dimensional Lie group acting smoothly on an m-dimensional manifold M. Definition 2.1. A moving frame is a smooth, G-equivariant map ρ:M→ G. There are two principal types of equivariance: ρ(g·z) = ˆ g·ρ(z) left moving frame
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geometry in terms of moving frames. Our indices shall have the range i;j;k;:::= 1;2 and A;B;C;:::= 1;2;3 and we follow the convention that sums are over repeated indices. Let SˆR3 be a surface and let the dot product of R3 be given h;i. Let a local chart for Sbe given by the map X: U!Swhere UˆR2 is an open set.
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Learn about the new, algorithmic theory of moving frames, a powerful tool for studying geometric properties and invariants under transformation groups. The paper surveys applications in geometry, computer vision, invariant theory, and numerical analysis.
Apr 23, 2021 · 9.1: Introduction to Moving-Frame Kinematics. Page ID. Franz S. Hover & Michael S. Triantafyllou. Massachusetts Institute of Technology via MIT OpenCourseWare. An understanding of inertial guidance systems for navigation of vehicles and robots requires some background in kinematics.
We discuss kinematics, focusing on transformation of vectors through change of reference frame, as a background to understanding inertial guidance systems for vehicles and robots.
