en>David Eppstein: copyedit

2013-11-21T08:31:09Z

copyedit

← Older revision		Revision as of 10:31, 21 November 2013
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	~~You start in a pair connected with a lovely island where your peaceful village~~ is in the ~~middle~~ of ~~beaches and woods till~~ the ~~enemies known as the BlackGuard led by Lieutenant Hammerman invades your destination~~. ~~After managing to guard against a little invasion force~~, ~~he proposes~~ to ~~avenge his loss while battle~~.~~<br><br>~~		{{Unreferenced\|date=November 2009}}
			In [[differential geometry]], especially the theory of space curves, the '''Darboux vector''' is the [[areal velocity]] [[vector (geometric)\|vector]] of the [[Frenet-Serret formulas\|Frenet frame]] of a space curve. It is named after [[Gaston Darboux]] who discovered it. It is also called '''angular momentum vector''', because it is directly proportional to [[angular momentum]].

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			:<math> \boldsymbol{\omega} = \tau \mathbf{T} + \kappa \mathbf{B} \qquad \qquad (1)</math>
			and it has the following [[symmetry\|symmetrical]] properties:
			:<math> \boldsymbol{\omega} \times \mathbf{T} = \mathbf{T'}, </math>

			:<math> \boldsymbol{\omega} \times \mathbf{N} = \mathbf{N'}, </math>

			:<math> \boldsymbol{\omega} \times \mathbf{B} = \mathbf{B'}, </math>
			which can be derived from Equation (1) by means of the [[Frenet-Serret formulas\|Frenet-Serret theorem]] (or vice versa).

			Let a rigid object move along a regular curve described parametrically by '''β'''(''t''). This object has its own intrinsic [[coordinate system]]. As the object moves along the curve, let its intrinsic coordinate system keep itself aligned with the curve's Frenet frame. As it does so, the object's motion will be described by two vectors: a translation vector, and a [[rotation vector]] '''ω''', which is an areal velocity vector: the Darboux vector.

			Note that this rotation is [[kinematics\|kinematic]], rather than physical, because usually when a rigid object moves freely in space its rotation is independent of its translation. The exception would be if the object's rotation is physically constrained to align itself with the object's translation, as is the case with the cart of a [[roller coaster]].

			Consider the rigid object moving smoothly along the regular curve. Once the translation is "factored out", the object is seen to rotate the same way as its Frenet frame. The total rotation of the Frenet frame is the combination of the rotations of each of the three Frenet vectors:
			:<math> \boldsymbol{\omega} = \boldsymbol{\omega}_\mathbf{T} + \boldsymbol{\omega}_\mathbf{N} + \boldsymbol{\omega}_\mathbf{B}. </math>

			Each Frenet vector moves about an "origin" which is the centre of the rigid object (pick some point within the object and call it its centre). The areal velocity of the tangent vector is:
			:<math> \boldsymbol{\omega}_\mathbf{T} = \lim_{\Delta t \rightarrow 0} {\mathbf{T}(t) \times \mathbf{T}(t + \Delta t) \over 2 \, \Delta t} </math>

			::<math> = {\mathbf{T}(t) \times \mathbf{T'}(t) \over 2}. </math>

			Likewise,
			:<math> \boldsymbol{\omega}_\mathbf{N} = {1 \over 2} \ \mathbf{N}(t) \times \mathbf{N'}(t), </math>

			:<math> \boldsymbol{\omega}_\mathbf{B} = {1 \over 2} \ \mathbf{B}(t) \times \mathbf{B'}(t). </math>

			Now apply the Frenet-Serret theorem to find the areal velocity components:
			:<math> \boldsymbol{\omega}_\mathbf{T} = {1\over 2} \mathbf{T} \times \mathbf{T'} = {1\over 2}\kappa \mathbf{T} \times \mathbf{N} = {1\over 2}\kappa \mathbf{B} </math>

			:<math> \boldsymbol{\omega}_\mathbf{N} = {1\over 2}\mathbf{N} \times \mathbf{N'} = {1\over 2}(-\kappa \mathbf{N} \times \mathbf{T} + \tau \mathbf{N} \times \mathbf{B}) = {1\over 2}(\kappa \mathbf{B} + \tau \mathbf{T}) </math>

			:<math> \boldsymbol{\omega}_\mathbf{B} = {1\over 2}\mathbf{B} \times \mathbf{B'} = -{1\over 2}\tau \mathbf{B} \times \mathbf{N} = {1\over 2}\tau \mathbf{T} </math>
			so that

			:<math> \boldsymbol{\omega} = {1\over 2}\kappa \mathbf{B} + {1\over 2}(\kappa \mathbf{B} + \tau \mathbf{T}) + {1\over 2}\tau \mathbf{T} = \kappa \mathbf{B} + \tau \mathbf{T}, </math>
			as claimed.

			The Darboux vector provides a concise way of interpreting [[curvature]] ''κ'' and [[Torsion of a curve\|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.

			{{DEFAULTSORT:Darboux Vector}}
			[[Category:Differential geometry]]
			[[Category:Vectors]]

en>Kilom691: ref corrected

2012-02-11T10:41:21Z

ref corrected

New page

You start in a pair connected with a lovely island where your peaceful village is in the middle of beaches and woods till the enemies known as the BlackGuard led by Lieutenant Hammerman invades your destination. After managing to guard against a little invasion force, he proposes to avenge his loss while battle. 

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K-minimum spanning tree - Revision history

en>David Eppstein: copyedit

en>Kilom691: ref corrected