Difference between revisions of "Curvature"
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− | '''Curvature''' is a a number associated with every point on | + | '''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 | ||
<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>\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. | ||
+ | |||
+ | == Curvature of surfaces == | ||
+ | |||
{{stub}} | {{stub}} | ||
[[Category:Calculus]] | [[Category:Calculus]] |
Latest revision as of 21:53, 20 November 2015
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
For a curve given in parametric form by the pair , the curvature at a point is
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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