2 A tutorial on explicit/parametric
4 everything you did not dare to ask
10 Several types of curves and surface are supported in gnuplot. Of those
11 not every operation is supported for every curve or surface type and it
12 can be therefore useful to understand the different types, their advantages
15 Curves in gnuplot are almost always planar (with one exception which we
16 will deal with in the end) and are assumed to be in the XY plane.
17 Therefore only X and Y coordinates are needed for plotting curves.
18 The simplest curve is the `explicit function`. This curve is in fact a
19 function and for each given x, there is one and only one y value associated
20 with it. A gnuplot example for such type is `plot sin(x)` or
21 `plot "datafile" using 1". Note the later is using only a single column from
22 the data file which is assumed to be the y values.
24 Alternatively one can define a `parametric curve` form. In this case
25 x and y are both functions of a third free parameter t, while independent
26 of each other. A circle can be expressed parametrically as x = cos(t),
27 y = sin(t) and be plotted using gnuplot as
28 'set parametric; plot cos(t),sin(t)'.
29 This form is not a function since there can be unlimited number of y values
30 associated with same x. Furthermore the explicit form is a special case of
31 the parametric representation by letting x equal to t. The curve y = sin(x)
32 can be written in parametric form as y = sin(t), x = t.
34 We are used to think of the plane in cartesian coordinate system.
35 In practice, some coordinate systems may be easier to use then others
36 under some circumstances. The polar form uses a different basis
37 to span the XY plane. In this representation the cartesian x coordinate
38 is equal to r cos(t) and the cartesian y coordinate is equal to r sin(t).
39 To draw a unit circle using the polar coordinate system in gnuplot use the
40 following simple command: 'set polar; plot 1'. To better understand this
41 explicit form lets backup a little.
42 When we plot a regular explicit function like `y = sin(x)` we march in equal
43 steps in x, evaluate the provided function and plot a piecewise linear curve
44 between the sampled points approximating the real function. In the polar
45 explicit form we do exactly the same thing, but we march along the angular
46 direction - we turn around the origin, computing the length of the radius
47 at that angle. Since for the unit circle, this radius is a constant 1,
48 `plot 1` in polar form plots a circle (if t domain is from 0 or 2Pi).
49 Note the polar form is explicit in that for each angle there is only a
52 Surprisingly (or maybe not so surprising) surfaces share the same
53 representations. Since surfaces are two dimensional entities, they
54 require two free parameters (like t for curves).
56 A surface explicit function uses x and y as the free parameters. For
57 each such pair it provides a single z value. An example for this form
58 can be `splot sin(sqrt(x**2+y**2))/sqrt(x**2+y**2)` for a three dimensional
59 sinc function or `splot 'datafile' using 1`. As for curves, the single column
60 used from the data file defines the function value or z in this case.
61 The order of the x and y function values is very strict in this form and
62 simply defines a rectangular grid in the XY plane. Fortunately this
63 strict form allows us to apply a very simplistic hidden line algorithm
64 called "the floating horizon". This hidden line algorithm exploits the
65 rectangular XY domain of the surface and therefore may be used for this
66 type of surfaces only. Since in gnuplot this is the only form of hidden
67 lines removing algorithm provided, only explicit surfaces may have their
70 Parametric surfaces are the exact extension for explicit surfaces as in
71 the curves case. the x, y, and z are defined in terms of two new free
72 variables and are totally independent of each other as x(u, v), y(u, v),
73 and z(u, v). Again the explicit surface is a special case of the parametric
74 representation where x = u, and y = v. Examples for plotting parametric
75 surfaces in gnuplot can be `splot cos(u)*cos(v),cos(u)*sin(v),sin(u)` which
76 defines a sphere, or `splot "datafile" using 1:2:3`. Since these are
77 parametric surfaces, gnuplot must be informed to handle them by issuing
80 The curve polar form takes the obvious extensions in the surface world.
81 The first possible extension is spherical coordinate system, while the
82 second is the cylindrical one. These modes currently work for data files
83 only and both requires two parameters, theta and phi for mapping onto the
84 unit sphere, and theta and z form mapping on a unit radius cylinder as follow:
86 Spherical coord. Cylin. coord.
87 ---------------- -------------
88 x = cos( theta ) * cos( phi ) x = cos( theta )
89 y = sin( theta ) * cos( phi ) y = sin( theta )
92 This subject brings us back to non planar curves. When surfaces are displayed
93 under gnuplot, isocurves are actually getting plotted. An isocurve is a
94 curve on the surface in which one of the two free parameters of the
95 surface is fixed. For example the u isolines of a surface are drawn by
96 setting u to be fixed and varying v along the entire v domain. The v isolines
97 are similarly drawn by fixing v. When data files are specified they are
98 classified internally into two types. A surface is tagged to have grid
99 topology if all its specified isolines are of the same length. A data mesh
100 of five isolines, seven points each is an example. In such a case the
101 surface cross isolines are drawn as well. Seven isolines with five points
102 each will be automatically created and drawn for grid type data. If
103 however, isolines of different length are found in the data, it is
104 tagged as nongrid surface and in fact is nothing more than a collection
105 of three dimensional curves. Only the provided data is plotted in that
106 case (see world.dem for such an example).
projects / gnuplot / blob