Topographic line

TOPOGRAPHIC LINE TRACED ON A SURFACE

Contrary to the geometric lines, such as the curvature lines, the asymptotic lines and the geodesics, that are intrinsic to the surface, the topographic lines of a surface are the lines the definition of which is associated to a direction, called vertical; in particular, we define the contour lines, the maximal slope lines, the lines of constant slope (helices), the lines forming a constant angle with respect to the contour lines (rhumb lines), the lines of extreme slant, the thalweg lines and the crest lines, the flow lines.

Examples (in red, the crests, in blue the thalwegs):

Hyperbolic paraboloid: xy = z.
Horizontal projections of the contour lines: hyperbolas xy = constant.
Horizontal projections of the slope lines: hyperbolas x² - y² = k (turned by 45° with respect to the previous ones).
(write )
Thalweg (blue): y = -x.
Crest (red): y = x.

Elliptic cone: xy = z².
Projections of the contour lines: portions of the hyperbolas xy = k.
Projections of the slope lines: portions of the hyperbolas x² - y² = k .
Remark: the projections of the contour and slope lines are, in the domain xy >0, the same as those of the previous surface, which proves that the projections of the contour and slope lines cannot characterize the surface.

Elliptic paraboloid: x² /2 + y² = - z.
Projections of the contour lines:
ellipses x² /2 + y² = k.
Projections of the slope lines:
parabolas y = k x² and x = 0.
(write )
The singular slope lines (in red) are x = 0 and y = 0 but only the latter is a crest line.

Elliptic cone: x² = zy.
Projections of the contour lines: parabolas constant.y = x².
Projections of the slope lines: ellipses x² /2 + y² = k.
Contour and slope lines are swapped in comparison with the previous surface!

Surface of equation: - z = PA + PB, where P is the projection of M on xOy.
Projections of the contour lines: ellipses with foci A and B.
Projections of the slope lines: hyperbolas with foci A and B.

Surface of equation: z = PA -PB, where P is the projection of M on xOy.
Projections of the contour lines: hyperbola with foci A and B.
Projections on the contour lines: hyperbolas of foci A and B.
Again, there is a swap contour/slope lines in comparison with the previous surface.

Egg box: z = sin x sin y.
Projections of the contour lines:
sin x sin y = k.

Projections of the slope lines:
cos x = k cos y.
(write )

Surface z = y sin x.
Projections of the contour lines: the secantoids: y = k/sin(x).
Projections of the slope lines:
exp(y²) cos² x = k.
()
Lines with singular slope: cos x = 0.
The slope lines passing by the saddles: exp(y²) cos² x = 1 are not singular slope lines...

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Hyperbolic paraboloid: xy = z. Horizontal projections of the contour lines: hyperbolas xy = constant. Horizontal projections of the slope lines: hyperbolas x² - y² = k (turned by 45° with respect to the previous ones). (write ) Thalweg (blue): y = -x. Crest (red): y = x.
Elliptic cone: xy = z². Projections of the contour lines: portions of the hyperbolas xy = k. Projections of the slope lines: portions of the hyperbolas x² - y² = k . Remark: the projections of the contour and slope lines are, in the domain xy >0, the same as those of the previous surface, which proves that the projections of the contour and slope lines cannot characterize the surface.
Elliptic paraboloid: x² /2 + y² = - z. Projections of the contour lines: ellipses x² /2 + y² = k. Projections of the slope lines: parabolas y = k x² and x = 0. (write ) The singular slope lines (in red) are x = 0 and y = 0 but only the latter is a crest line.
Elliptic cone: x² = zy. Projections of the contour lines: parabolas constant.y = x². Projections of the slope lines: ellipses x² /2 + y² = k. Contour and slope lines are swapped in comparison with the previous surface!
Surface of equation: - z = PA + PB, where P is the projection of M on xOy. Projections of the contour lines: ellipses with foci A and B. Projections of the slope lines: hyperbolas with foci A and B.
Surface of equation: z = PA -PB, where P is the projection of M on xOy. Projections of the contour lines: hyperbola with foci A and B. Projections on the contour lines: hyperbolas of foci A and B. Again, there is a swap contour/slope lines in comparison with the previous surface.
Egg box: z = sin x sin y. Projections of the contour lines: sin x sin y = k. Projections of the slope lines: cos x = k cos y. (write )
Surface z = y sin x. Projections of the contour lines: the secantoids: y = k/sin(x). Projections of the slope lines: exp(y²) cos² x = k. () Lines with singular slope: cos x = 0. The slope lines passing by the saddles: exp(y²) cos² x = 1 are not singular slope lines...