PlanetMath: level curve

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level curve

(Definition)

The level curves (in German Niveaukurve, in French ligne de niveau) of a surface

$\displaystyle z = f(x,\,y)$

(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

-plane have equations of the form

$\displaystyle f(x,\,y) = c$

(2)

where

is a constant.

For example, the level curves of the hyperbolic paraboloid are the rectangular hyperbolas (cf. this entry).

The gradient $f'_x(x,\,y)\,\vec{i}\!+\!f'_y(x,\,y)\,\vec{j}$ of the function 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)

$\displaystyle F(x,\,y,\,z) = c$

(3)

for a function

of three variables

. The gradient of

in a point $(x,\,y,\,z)$ is parallel to the surface normal of the level surface passing through this point.

"level curve" is owned by pahio.

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See Also: level set, convex angle

Other names:

contour curve, isopleth

Also defines:

level surface, contour surface

Keywords:

level set

This object's parent.

Attachments:: tilt curve (Definition) by pahio

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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 1193 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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