(MAXIMAL) SLOPE LINE

 Attention: "slope line" is also used for "helix".

 Differential condition: where is the normal vector at M. Differential equation of the projections on xOy for the surface : where . The slope of the slope line at (x, y) is . Special cases: for the translation surface : . for : . Differential equation for the surface parametrized by : .

Given a vertical direction, the (maximal) slope lines are the curves traced on the surfaces perpendicular to the contour lines (that are the sections by horizontal planes). This definition comes from the fact that among all the lines included in a given non horizontal plane, those that have the maximal slope are those that are perpendicular to the horizontal.

A moving body, subject to the only force of a vertical gravitational field, placed with no initial speed at a point M on the surface, follows a trajectory tangent to the slope line at M.

The projections on the plane xOy of the contour and slope lines form a double lattice of orthogonal lines.

Note that the projections on the plane xOy of the slope lines of the surface are the field lines of the vector field .

Examples:
- for a surface of revolution with vertical axis, the slope lines are the meridians.
- when the slope lines are linear, we are faced with a surface of equal slope.
- see numerous other examples at topographic lines.

Do not mistake the slope lines for the flow lines.

See also the thalweg and crest lines.

© Robert FERRÉOL  2018