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Other name: isohypse (from the Greek hypsos "height").

Differential equation: .
Equation for the surface : f(x,y) = constant.

Given a vertical direction, the contour lines (or curves) of a surface (S) are the sections of this surface by the horizontal planes (i.e. perpendicular to the vertical direction).

Remark: the projections on the plane xOy of the contour lines of the surface , are the curves solution of the differential equation (where ); therefore, they are the equipotential lines of the vector field ; while the slope lines are its field lines.

See examples at topographic lines.

See the case of the equidistant contour lines at surface of equal slope.

The envelope of the projections of the contour lines on xOy is the visible outline of the surface.

Natural contour lines in a puddle (picture by Alain Juhel), rice field.

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© Robert FERRÉOL  2018