Difference between revisions of "Curvature"
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| − | '''Curvature''' is a a number associated with every point on each [[smooth | + | '''Curvature''' is a a number associated with every point on each [[smooth curve]] 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 | ||
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For a curve given in [[parametric form]] by the pair <math>(x(t), y(t))</math>, the curvature at a point is | 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>\kappa(t) = \dfrac{x'(t)y''(t) - y'(t)x''(t)}{(x'(t)^2+y'(t)^2)^{3/2}}.</cmath> |
| + | Given any vector-valued function <math>\mathbf{r}(t)</math>, the curvature at a given time is | ||
| + | <cmath>\kappa(t) = \frac{d\mathbf{t}}{ds} = \frac{\mathbf{v} \times \mathbf{a}}{|\mathbf{v}^3|}</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. | ||
Latest revision as of 15:31, 13 November 2024
Curvature is a a number associated with every point on each smooth curve 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
![\[\kappa(t) = \dfrac{x'(t)y''(t) - y'(t)x''(t)}{(x'(t)^2+y'(t)^2)^{3/2}}.\]](http://latex.artofproblemsolving.com/e/f/a/efaba9529b4c4c102357c95774132c634f4a1aff.png)
Given any vector-valued function 
, the curvature at a given time is
![\[\kappa(t) = \frac{d\mathbf{t}}{ds} = \frac{\mathbf{v} \times \mathbf{a}}{|\mathbf{v}^3|}\]](http://latex.artofproblemsolving.com/9/a/5/9a58cb49d0acfe6eb8362bba3ffdd25adc21f5bf.png)
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.
Curvature of surfaces
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