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Topic: Frenet Frame

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	Differential geometry of curves - Wikipedia, the free encyclopedia
		The differential geometric properties of a curve (length, frenet frame and generalized curvature) are invariant under reparametrization and therefore properties of the equivalence class.The equivalence classes are called C
		A Frenet frame is a moving reference frame of n orthonormal vectors e
		The Frenet frame and the generalized curvatures are invariant under reparametrization and therefore differential geometric properties of the curve.
	en.wikipedia.org /wiki/Frenet_frame (1118 words)

	Darboux vector - Wikipedia, the free encyclopedia
		In differential geometry, especially the theory of space curves, the Darboux vector is the areal velocity vector of the Frenet frame of a space curve.
		Each Frenet vector moves about an "origin" which is the center of the rigid object (pick some point within the object and call it its center).
		The Darboux vector provides a concise way of interpreting curvature κ and torsion τ geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector.
	en.wikipedia.org /wiki/Darboux_vector (380 words)

	diff geo images page
		For a reference, a second copy of the Frenet frame is shown with its 'center' at the origin of 3-space.
		For this longer thinner frame we take the initial conditions to be the same as those for torus knot Frenet frame, at a point on the torus knot where curvature is small and torsion is large (right after the first 'hilltop' is crossed by the torus knot Frenet frame).
		This graphic shows the first frame of an animated view of the Theorema Egregium of Gauss, that the (Gauss) curvature K of a surface is invariant under isometry.
	www.math.uiowa.edu /~wseaman/DGImage53100.htm (3122 words)

	Computation of the Frenet frame of the plumbline
		, the torsion of the plumbline, are defined at the origin of the local Frenet frame by the expressions in Eqs.
		The coordinates computed refer to the Frenet frame the origin of which is actually a running point along the plumbline.
		Therefore X,Y and Z have to be transformed to the x, y, and z coordinate system of the density model to obtain the coordinates of the extrapolated point of the plumbline being investigated.
	www.ggki.hu /new/gravity/node5.html (213 words)

	Untitled Document (Site not responding. Last check: 2007-10-24)
		The main tools that we use to describe the geometry of tire tracks are the Frenet frames of the tire tracks and the unsigned curvature of each tire track.
		The Frenet frame of a plane curve is the pair of unit vectors
		The importance of the Frenet frame is that the geometry of the curve is given by how the Frenet frame changes.
	www.rose-hulman.edu /~finn/research/bicycle/tracks_3.htm (296 words)

	Space Curves, Frenet Frames, and Torsion
		The last two arguments determine the number of frames plotted and the length of the normal and binormal vectors.
		Use curveframeplot to plot the trefoil with its Frenet frame.
		Use ezsurf to show the twisting of the Frenet frame of the trefoil around the curve.
	www.math.umd.edu /~jmr/241/curves2.htm (862 words)

	Torsion and the Frenet Frame (Site not responding. Last check: 2007-10-24)
		Thus, B is a unit vector normal to the plane spanned by T and N at time t.
		The 3 vectors T, N, and B taken together are called either the TNB frame or the Frenet Frame of the curve.
		Since both v and a are in the plane spanned by T and N, the binormal vector B is also the unit vector in the direction of v×a.
	math.etsu.edu /MultiCalc/Chap1/Chap1-8/part4.htm (221 words)

	FieldFrenet (Site not responding. Last check: 2007-10-24)
		The Field Frenet module calculates the Frenet frame of a field along a particular indices using central differences.
		The Field Frenet input port is a field of any type that has regular topology.
		The user must supply one input values via the module GUI, the axis for the Frenet Frame and the component of the Frenet Frame to output.
	software.sci.utah.edu /future/src/Dataflow/XML/FieldFrenet.html (85 words)

	[No title]
		(left) The Frenet Frame is a (tangent, normal) coordinate frame that is adapted to the local structure of each point along a curve; and (right) the osculating circle is that circle with the largest contact with the curve among all circles tangent at that point.
		Of course, for the full system the complete Frenet 2-frames are used to infer the Frenet 3-frame attached to the space curve.
		Depth scale is shown at right (units: meters).}}{9}} \newlabel{fig:stereo-result}{{7}{9}} \@writefile{toc}{\contentsline {section}{\numberline {5}Covariant Derivatives, Oriented Textures, and Color}{9}} \newlabel{connection_equations}{{1.4}{10}} \newlabel{connection_equations_1}{{1.5}{10}} \newlabel{Kt_and_Kn}{{1.6}{10}} \newlabel{connection_equations_2}{{1.7}{10}} \newlabel{frenet_equation}{{1.8}{10}} \citation{ohad-zucker:03} \newlabel{figure:the-connection-equation}{{5}{11}} \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Displacement (transport) of a Frenet frame within a vector field or an oriented texture amounts to rotation, but differs for different displacements.
	www.cs.yale.edu /homes/zucker/paragios.aux (893 words)

	Frenet Frame Display Package (Site not responding. Last check: 2007-10-24)
		# Typical usage: T([f(var),g(var),h(var)],var); T := proc(gamma,t) local T_len,T_before; T_before := vector(3,tang(gamma,t)); T_len := sqrt(norm(T_before,2)); evalm((1/T_len) * T_before) end: subs2 := proc(a,b) subs(b,a) end: # Return a Frenet frame and curvature at a parameter value t=t0.
		Pi,2); # Return a graph of num_steps Frenet frames, equally spaced along the curve gamma.
		# Each frame is drawn with T in red, N in blue, and B in green.
	mathlab.cit.cornell.edu /local_maple/mvc/Math222/lib/frame6 (178 words)

	[No title] (Site not responding. Last check: 2007-10-24)
		on the Frenet frame which seems to be something like what I'm
		(By "Frenet frame", in case that's not a common name, I mean the
		frame which parameterizes by curvature and torsion in 3 dimensions).
	mathforum.org /kb/plaintext.jspa?messageID=110692 (147 words)

	Geometric interpretation of the Frenet-Serret frame description of circular orbits in stationary axisymmetric spacetimes (Site not responding. Last check: 2007-10-24)
		Geometric interpretation of the Frenet-Serret frame description of circular orbits in stationary axisymmetric spacetimes
		A geometrical interpretation is given for the Frenet-Serret structure along constant speed circular orbits in orthogonally transitive stationary axisymmetric spacetimes.
		This gives a simple visualization of the acceleration of these orbits and of the Fermi-Walker angular velocity of the usual symmetry-adapted frame vectors along them and provides an elegant description of the various observers characterized by critical values of the variables parametrizing the acceleration.
	stacks.iop.org /0264-9381/16/1333 (273 words)

	Citations: Geometric continuity - Gregory (ResearchIndex) (Site not responding. Last check: 2007-10-24)
		Analogous to the osculating plane, the linear space spanned by the first d Gamma 1 curvature vectors is called an osculating linear space.
		Definition 9 (Frenet frame) In IR d, the Frenet frame is defined as the first d curvature vectors (t 1 (w) t d (w) Remark 5 The Frenet....
		By contrast, continuity of the kth geometric invariant, also called kth order Fr enet frame continuity [30] 37] and abbreviated F k, does not require that the entry (or, more generally, any subdiagonal entry) depend on other entries in the connection matrix.
	citeseer.ist.psu.edu /context/209261/0 (729 words)

	Differential geometry of curves - Gurupedia
		For a given parametrized curve c(t) the natural parametrization is unique up to shift of parameter.
		If c is interpreted as the path of a particle then the tangent vector can be visualized as path the particle takes when free from outer force.
		The solution is the set of frenet vectors describing the curve specificied by the generalized curvature functions χ;
	www.gurupedia.com /d/di/differential_geometry_of_curves.htm (1012 words)

	Frenet frame of a 3D curve - Maple Application Center - Maplesoft
		Frenet frame of a 3D curve - Maple Application Center - Maplesoft
		In this worksheet we will see how Maple and the vec_calc package can be used to analyse a parametrized curve.
		Examples include the winding line on a torus and the frenet frame of a curve.
	www.maplesoft.com /applications/app_center_view.aspx?AID=727 (64 words)

	► » Frenet frame question (Site not responding. Last check: 2007-10-24)
		Let c is a regular curve and C is unit speed curve by reparametrization of c.
		then, " Frenet frame {t,n,b} of c " = " Frenet frame {T,N,B} of C "
		If this is true, then how to prove it ?
	www.science-chat.org /Frenet-frame-question-6890450.html (157 words)

	Citations: Shape preserving interpolation using Frenet frame continuous curves of order - Kong, Ong (ResearchIndex) (Site not responding. Last check: 2007-10-24)
		Citations: Shape preserving interpolation using Frenet frame continuous curves of order - Kong, Ong (ResearchIndex)
		Kong and B. Ong, Shape preserving interpolation using Frenet frame continuous curves of order 3, to appear.
		Finally we mention the papers [2,3] which give local G schemes using a piecewise polynomial of degree six, also allowing optimisation of a fairness measure.
	citeseer.ist.psu.edu /context/2272651/0 (197 words)

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