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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
level curve
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(Definition)
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The level curves (in German Niveaukurve, in French ligne de niveau) of a surface
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(1) |
in $\mathbb{R}^3$ are the intersection curves of the surface and the planes $z = \,\mathrm{constant}$ . Thus the projections of the level curves on the $xy$ -plane have equations of the form
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(2) |
where $c$ is a constant.
For example, the level curves of the hyperbolic paraboloid $z = xy$ are the rectangular hyperbolas $xy = c$ (cf. this entry).
The gradient $f'_x(x,\,y)\,\vec{i}\!+\!f'_y(x,\,y)\,\vec{j}$ of the function $f$ in any point of the surface (1) is perpendicular to the level curve (2), since the slope of the gradient is $\displaystyle\frac{f'_y}{f'_x}$ and the slope of the level curve is $\displaystyle-\frac{f'_x}{f'_y}$ , whence the slopes are opposite inverses.
Analogically one can define the level surfaces (or contour surfaces)
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(3) |
for a function $F$ of three variables $x$ , $y$ , $z$ . The gradient of $F$ in a point $(x,\,y,\,z)$ is parallel to the surface normal of the level surface passing through this point.
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"level curve" is owned by pahio.
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(view preamble | get metadata)
See Also: level set, convex angle
| Other names: |
contour curve, isopleth |
| Also defines: |
level surface, contour surface |
This object's parent.
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Cross-references: passing through, surface normal, parallel, variables, opposite inverses, slope, perpendicular, point, function, gradient, rectangular hyperbolas, equations, projections, planes, curves, intersection, surface
There are 3 references to this entry.
This is version 9 of level curve, born on 2007-10-17, modified 2008-06-11.
Object id is 10003, canonical name is ContourLine.
Accessed 3924 times total.
Classification:
| AMS MSC: | 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries) | | | 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space) | | | 53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space) |
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Pending Errata and Addenda
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