How to build the line of levels

How to build the line of levels

The line of level of function call a set of points of space in which the values accepted by function are identical. Such lines there can be an infinite set within the area of values determined by a formula. Except mathematics and physics of the line of level are used, for example, in cartography for the indication of levels of identical heights (isohypse) or depths (isobath). In meteorology by such lines designate levels of identical temperatures and pressure (isotherm and an isobar).


1. At creation of lines of level you recognize that they are projections to the plane with zero z-coordinate of lines of crossing of the schedule of the set function with some horizontal plane. Z-coordinate of this plane of section is also a constant to which it is necessary to equate the function equation to receive coordinates of points of the line. It can changes with the step set in statements of the problem if a set of lines is required to construct. And if it is necessary to build only one line of levels, in conditions coordinates of the point lying on it can be given. Schedules from this page can be kept or edited in the interactive mode.

2. Lead the function set in statements of the problem to a type of f (x, y) = const. For example, if formula z = x² + is given to y² - 4*y, it can be written down in an alternative form better to present a function graph form, and to equate to constant c: c+4 = x²+(y-2)². The volume schedule of such function represents an infinite paraboloid, and all its sections the horizontal plane lifted on different levels (i.e. required lines of levels) will be concentric circles with a radius determined by a formula √ (c+4).

3. Substitute instead of a constant about the value set in conditions for the line of level. If it is not given - choose, proceeding from area of values of function. For example, for the example given above number-4 can be the minimum value of a constant. The constant can be equated to 5 and in this case the circle with a radius √ (5+4) = 3 and the center in a point with the abscissa equal 0 and the ordinate equal 2 will be a function graph.

4. If it is necessary to build several lines of levels, repeat the previous step the necessary number of times.

5. It is possible to find services which will help with creation of lines of levels in the Internet. For example, the reference to WolframAlpha service is given below. Into the entry field on its page enter a formula of function and click on the button with an equality badge. Function z used in an example = x² + y² - 4*y should be entered in such look: x^2+y^2-4*y. In several seconds on the page there will be two- and three-dimensional color schedules with lines of levels and also classification of the figure described by a formula, alternative forms of its record and other properties of function which can be used at creation of lines of levels.

Author: «MirrorInfo» Dream Team