`123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106` `````` A tutorial on explicit/parametric and everything you did not dare to ask about curves and surfaces in gnuplot Several types of curves and surface are supported in gnuplot. Of those not every operation is supported for every curve or surface type and it can be therefore useful to understand the different types, their advantages and limitations. Curves in gnuplot are almost always planar (with one exception which we will deal with in the end) and are assumed to be in the XY plane. Therefore only X and Y coordinates are needed for plotting curves. The simplest curve is the `explicit function`. This curve is in fact a function and for each given x, there is one and only one y value associated with it. A gnuplot example for such type is `plot sin(x)` or `plot "datafile" using 1". Note the later is using only a single column from the data file which is assumed to be the y values. Alternatively one can define a `parametric curve` form. In this case x and y are both functions of a third free parameter t, while independent of each other. A circle can be expressed parametrically as x = cos(t), y = sin(t) and be plotted using gnuplot as 'set parametric; plot cos(t),sin(t)'. This form is not a function since there can be unlimited number of y values associated with same x. Furthermore the explicit form is a special case of the parametric representation by letting x equal to t. The curve y = sin(x) can be written in parametric form as y = sin(t), x = t. We are used to think of the plane in cartesian coordinate system. In practice, some coordinate systems may be easier to use then others under some circumstances. The polar form uses a different basis to span the XY plane. In this representation the cartesian x coordinate is equal to r cos(t) and the cartesian y coordinate is equal to r sin(t). To draw a unit circle using the polar coordinate system in gnuplot use the following simple command: 'set polar; plot 1'. To better understand this explicit form lets backup a little. When we plot a regular explicit function like `y = sin(x)` we march in equal steps in x, evaluate the provided function and plot a piecewise linear curve between the sampled points approximating the real function. In the polar explicit form we do exactly the same thing, but we march along the angular direction - we turn around the origin, computing the length of the radius at that angle. Since for the unit circle, this radius is a constant 1, `plot 1` in polar form plots a circle (if t domain is from 0 or 2Pi). Note the polar form is explicit in that for each angle there is only a single radius. Surprisingly (or maybe not so surprising) surfaces share the same representations. Since surfaces are two dimensional entities, they require two free parameters (like t for curves). A surface explicit function uses x and y as the free parameters. For each such pair it provides a single z value. An example for this form can be `splot sin(sqrt(x**2+y**2))/sqrt(x**2+y**2)` for a three dimensional sinc function or `splot 'datafile' using 1`. As for curves, the single column used from the data file defines the function value or z in this case. The order of the x and y function values is very strict in this form and simply defines a rectangular grid in the XY plane. Fortunately this strict form allows us to apply a very simplistic hidden line algorithm called "the floating horizon". This hidden line algorithm exploits the rectangular XY domain of the surface and therefore may be used for this type of surfaces only. Since in gnuplot this is the only form of hidden lines removing algorithm provided, only explicit surfaces may have their hidden lines removed. Parametric surfaces are the exact extension for explicit surfaces as in the curves case. the x, y, and z are defined in terms of two new free variables and are totally independent of each other as x(u, v), y(u, v), and z(u, v). Again the explicit surface is a special case of the parametric representation where x = u, and y = v. Examples for plotting parametric surfaces in gnuplot can be `splot cos(u)*cos(v),cos(u)*sin(v),sin(u)` which defines a sphere, or `splot "datafile" using 1:2:3`. Since these are parametric surfaces, gnuplot must be informed to handle them by issuing `set parametric`. The curve polar form takes the obvious extensions in the surface world. The first possible extension is spherical coordinate system, while the second is the cylindrical one. These modes currently work for data files only and both requires two parameters, theta and phi for mapping onto the unit sphere, and theta and z form mapping on a unit radius cylinder as follow: Spherical coord. Cylin. coord. ---------------- ------------- x = cos( theta ) * cos( phi ) x = cos( theta ) y = sin( theta ) * cos( phi ) y = sin( theta ) z = sin( phi ) z = z This subject brings us back to non planar curves. When surfaces are displayed under gnuplot, isocurves are actually getting plotted. An isocurve is a curve on the surface in which one of the two free parameters of the surface is fixed. For example the u isolines of a surface are drawn by setting u to be fixed and varying v along the entire v domain. The v isolines are similarly drawn by fixing v. When data files are specified they are classified internally into two types. A surface is tagged to have grid topology if all its specified isolines are of the same length. A data mesh of five isolines, seven points each is an example. In such a case the surface cross isolines are drawn as well. Seven isolines with five points each will be automatically created and drawn for grid type data. If however, isolines of different length are found in the data, it is tagged as nongrid surface and in fact is nothing more than a collection of three dimensional curves. Only the provided data is plotted in that case (see world.dem for such an example). ``````