The [url=https://en.wikipedia.org/wiki/Contour_line]contour lines[/url] (or curves) of a surface are the sections of this surface by the horizontal planes (i.e. perpendicular to the vertical direction). The [url=https://en.wikipedia.org/wiki/Contour_line]contour lines[/url] of a plane are called horizontal (principal) lines.[br]Contours are one of several common methods used to denote [url=https://en.wikipedia.org/wiki/Elevation]elevation[/url] or [url=https://en.wikipedia.org/wiki/Altitude]altitude[/url] and depth on maps. [br][url=https://en.wikipedia.org/wiki/Contour_line]Contour lines[/url] of topographic surface are closed curves surrounding summits or immits.[br][url=https://en.wikipedia.org/wiki/Contour_line]Contour lines[/url] at [url=https://mathcurve.com/surfaces.gb/talus/talus.shtml]surface of equal slope[/url] (plane, cone,...) are [url=https://www.mathpages.com/home/kmath724/kmath724.htm]equidistant[/url].[br][br]The slope lines ( = lines of the steepest descent) cut at right angles the contour lines; and this property applies also to the [url=https://en.wikipedia.org/wiki/Multiview_projection]top view projection[/url].

Plane [math]\alpha[/math] is given by points [color=#3c78d8][i]GHI[/i][/color] on the cube. Horizontal projection plane [math]\pi[/math] is [color=#3c78d8]blue face[/color] of a cube, frontal projection plane [math]\nu[/math] is [color=#e06666]red face[/color]. The base (ground) line[i] x12[/i] is intersection line of projection planes, i.e.[math]x12=\pi\cap\nu[/math]. Trace lines of the planes [i][color=#ff0000]n, p[/color][/i] are lines where the plane [math]\alpha[/math] meets the planes of projections, i.e. [math]p=\alpha\cap\pi[/math] and [math]n=\alpha\cap\nu[/math].[br][list][*]Horizontal line [i][color=#cc0000]h[/color][/i] of a plane is parallel to the blue face. [/*][*]The line of steepest descent [color=#1155Cc][i]s[/i][/color] (=slope line) is perpendicular to the horizontal line [color=#ff0000][i]h[/i][/color]. This line determines the slope angle of the plane. The right angle between lines [color=#1e84cc][i]s[/i][/color] and [color=#ff0000][i]h[/i][/color] has true size in top view, i.e. [math]s1\perp h1[/math] - tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_orthogonal.png[/icon].[br][/*][/list]

In given point A of plane α construct the first steepest line s (i.e. both projections s1, s2). A horizontal line [color=#ff0000][i]h[/i][/color] and frontal line [i]f[/i] of the plane are given. ([url=https://geotest.geometry.cz/?lang=EN]GeoTest[/url])

Solution:[br] First slope line [color=#1e84cc][i]s[/i][/color] is perpendicular to horizontal line [i][color=#ff0000]h[/color][/i] in top view. Choose arbitrary point [math]H\in h[/math] and determine auxiliary first slope line [i]as[/i] passing through a point [i]H[/i]. First slope line is perpendicular to horizontal line in top view. Lines of steepest descent are mutually parallel, [math]as||s[/math].[br]Step 2: Arbitrary [math]H1\in h1[/math], top view of auxiliary first slope [i]as[/i] [math]as1\perp h1;H1\in as1[/math][br]Steps 3, 4: Front view of intersection points [i]H, F[/i] [math]H2\in h2;F2\in f2[/math].[br]Step 5: Front view of auxiliary slope line [i]as = H2F2[/i].[br]Step 6: Slope line [color=#1e84cc][i]s[/i][/color] is parallel with auxiliary slope line [i]as[/i] in both views [math]as1||s1;as2||s2[/math].

Construct a line of the steepest descent (both projections [i]s1, s2[/i]) in the point [i]A[/i] of given plane α orthogonal to the vertical plane of projection. The trace n of the plane α is given ([url=https://geotest.geometry.cz/?lang=EN]GeoTest[/url]).