Difference between revisions of "Curvature"
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| − | '''Curvature''' is a a number associated with every point on a [[smooth curve]]s that describes "how curvy" the curve is at that point. In particular, the "least curvy" curve is a [[line]], and fittingly lines have zero curvature. For a [[circle]] of [[radius]] <math>r</math>, the curvature at every point is <math>\frac{1}{r}</math>. Intuitively, this grows smaller as <math>r</math> grows larger because one must turn much more sharply to follow the path of a circle of small radius than to follow the path of a circle with large radius. | + | '''Curvature''' is a a number associated with every point on a [[smooth]] [[curve]]s that describes "how curvy" the curve is at that point. In particular, the "least curvy" curve is a [[line]], and fittingly lines have zero curvature. For a [[circle]] of [[radius]] <math>r</math>, the curvature at every point is <math>\frac{1}{r}</math>. Intuitively, this grows smaller as <math>r</math> grows larger because one must turn much more sharply to follow the path of a circle of small radius than to follow the path of a circle with large radius. |
Given a twice-[[differentiable]] [[function]] <math>f(x)</math>, the curvature of the [[graph]] <math>y = f(x)</math> of the function at the point <math>(x, f(x))</math> is given by the formula | Given a twice-[[differentiable]] [[function]] <math>f(x)</math>, the curvature of the [[graph]] <math>y = f(x)</math> of the function at the point <math>(x, f(x))</math> is given by the formula | ||
<cmath>\kappa(x) = \dfrac{f''(x)}{(f'(x)^2+1)^{3/2}}.</cmath> | <cmath>\kappa(x) = \dfrac{f''(x)}{(f'(x)^2+1)^{3/2}}.</cmath> | ||
| − | For a curve given in [[parametric]] | + | For a curve given in [[parametric form]] by the pair <math>(x(t), y(t))</math>, the curvature at a point is |
<cmath>\kappa(t) = \?.</cmath> | <cmath>\kappa(t) = \?.</cmath> | ||
This expression is invariant under positive-velocity reparametrizations, that is the curvature is a property of the curve and not the way in which you traverse it. | This expression is invariant under positive-velocity reparametrizations, that is the curvature is a property of the curve and not the way in which you traverse it. | ||
Revision as of 12:22, 17 April 2008
Curvature is a a number associated with every point on a smooth curves that describes "how curvy" the curve is at that point. In particular, the "least curvy" curve is a line, and fittingly lines have zero curvature. For a circle of radius 
, the curvature at every point is 
. Intuitively, this grows smaller as grows larger because one must turn much more sharply to follow the path of a circle of small radius than to follow the path of a circle with large radius.
Given a twice-differentiable function 
, the curvature of the graph 
of the function at the point 
is given by the formula
![\[\kappa(x) = \dfrac{f''(x)}{(f'(x)^2+1)^{3/2}}.\]](http://latex.artofproblemsolving.com/d/0/b/d0b863969c855002143f753e178e5cab7e8a54ba.png)
For a curve given in parametric form by the pair 
, the curvature at a point is
\[\kappa(t) = \?.\] (Error compiling LaTeX. Unknown error_msg)
This expression is invariant under positive-velocity reparametrizations, that is the curvature is a property of the curve and not the way in which you traverse it.
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