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2.9.4 Coordinate Systems for Two-Dimensional Graphics

When you set up a graphics object in Mathematica, you give coordinates for the various graphical elements that appear. When Mathematica renders the graphics object, it has to translate the original coordinates you gave into "display coordinates" which specify where each element should be placed in the final display area.

Sometimes, you may find it convenient to specify the display coordinates for a graphical element directly. You can do this by using "scaled coordinates" Scaled[sx, sy] rather than x, y. The scaled coordinates are defined to run from 0 to 1 in and , with the origin taken to be at the lower-left corner of the display area.

Coordinate systems for two-dimensional graphics.

The rectangle is drawn at a fixed position relative to the display area, independent of the original coordinates used for the plot.

In[1]:= Plot[Tan[x], {x, 0, 2Pi},

Prolog ->

Rectangle[Scaled[{0.7, 0.7}], Scaled[{1, 1}]]]

Out[1]=

When you use x, y or Scaled[sx, sy], you are specifying position either completely in original coordinates, or completely in scaled coordinates. Sometimes, however, you may need to use a combination of these coordinate systems. For example, if you want to draw a line at a particular point whose length is a definite fraction of the width of the plot, you will have to use original coordinates to specify the basic position of the line, and scaled coordinates to specify its length.

You can use Scaled[dsx, dsy, x, y] to specify a position using a mixture of original and scaled coordinates. In this case, x, y gives a position in original coordinates, and dsx, dsy gives the offset from the position in scaled coordinates.

Note that you can use Scaled with either one or two arguments to specify radii in Disk and Circle graphics elements.

Positions specified as offsets.

Each line drawn here has an absolute length of 6 printer's points.

In[2]:= Show[Graphics[Table[

Line[{{x, x^2}, Offset[{0, 6}, {x, x^2}]}],

{x, 10}], Frame->True]]

Out[2]=

You can also use Offset inside Circle with just one argument to create a circle with a certain absolute radius.

In[3]:= Show[Graphics[Table[

Circle[{x, x^2}, Offset[{2, 2}]],

{x, 10}], Frame->True]]

Out[3]=

In most kinds of graphics, you typically want the relative positions of different objects to adjust automatically when you change the coordinates or the overall size of your plot. But sometimes you may instead want the offset from one object to another to be constrained to remain fixed. This can be the case, for example, when you are making a collection of plots in which you want certain features to remain consistent, even though the different plots have different forms.

Offset[adx, ady, position] allows you to specify the position of an object by giving an absolute offset from a position that is specified in original or scaled coordinates. The units for the offset are printer's points, equal to of an inch.

When you give text in a plot, the size of the font that is used is also specified in printer's points. A 10-point font, for example, therefore has letters whose basic height is 10 printer's points. You can use Offset to move text around in a plot, and to create plotting symbols or icons which match the size of text.

Options which determine translation from original to display coordinates.

When Mathematica renders a graphics object, one of the first things it has to do is to work out what range of original and coordinates it should actually display. Any graphical elements that are outside this range will be "clipped", and not shown.

The option PlotRange specifies the range of original coordinates to include. As discussed in Section 1.9.3, the default setting is PlotRange -> Automatic, which makes Mathematica try to choose a range which includes all "interesting" parts of a plot, while dropping "outliers". By setting PlotRange -> All, you can tell Mathematica to include everything. You can also give explicit ranges of coordinates to include.

This sets up a polygonal object whose corners have coordinates between roughly .

In[4]:= obj = Polygon[

Table[{Sin[n Pi/10], Cos[n Pi/10]} + 0.05 (-1)^n,

{n, 20}]] ;

In this case, the polygonal object fills almost the whole display area.

In[5]:= Show[Graphics[obj]]

Out[5]=

With the default PlotRange -> Automatic, the outlying point is not included, but does affect the range of coordinates chosen.

In[6]:= Show[ Graphics[{obj, Point[{20, 20}]}] ]

Out[6]=

With PlotRange -> All, the outlying point is included, and the coordinate system is correspondingly modified.

In[7]:= Show[%, PlotRange -> All]

Out[7]=

The option PlotRange allows you to specify a rectangular region in the original coordinate system, and to drop any graphical elements that lie outside this region. In order to render the remaining elements, however, Mathematica then has to determine how to position this rectangular region with respect to the final display area.

The option PlotRegion allows you to specify where the corners of the rectangular region lie within the final display area. The positions of the corners are specified in scaled coordinates, which are defined to run from 0 to 1 across the display area. The default is PlotRegion -> {{0, 1}, {0, 1}}, which specifies that the rectangular region should fill the whole display area.

By specifying PlotRegion, you can effectively add "margins" around your plot.

In[8]:= Plot[ArcTan[x], {x, 0, 10},

PlotRegion -> {{0.2, 0.8}, {0.3, 0.7}}]

Out[8]=

Specifying the shape of the display area.

What we have discussed so far is how Mathematica translates the original coordinates you specify into positions in the final display area. What remains to discuss, however, is what the final display area is like.

On most computer systems, there is a certain fixed region of screen or paper into which the Mathematica display area must fit. How it fits into this region is determined by its "shape" or aspect ratio. In general, the option AspectRatio specifies the ratio of height to width for the final display area.

It is important to note that the setting of AspectRatio does not affect the meaning of the scaled or display coordinates. These coordinates always run from 0 to 1 across the display area. What AspectRatio does is to change the shape of this display area.

This generates a graphic object corresponding to a hexagon.

In[9]:= hex = Graphics[Polygon[

Table[{Sin[n Pi/3], Cos[n Pi/3]}, {n, 6}] ]] ;

This renders the hexagon in a display area whose height is three times its width.

In[10]:= Show[hex, AspectRatio -> 3]

Out[10]=

For two-dimensional graphics, AspectRatio is set by default to the fixed value of 1/GoldenRatio. Sometimes, however, you may want to determine the aspect ratio for a plot from the original coordinate system used in the plot. Typically what you want is for one unit in the direction in the original coordinate system to correspond to the same distance in the final display as one unit in the direction. In this way, objects that you define in the original coordinate system are displayed with their "natural shape". You can make this happen by setting the option AspectRatio -> Automatic.

With AspectRatio -> Automatic, the aspect ratio of the final display area is determined from the original coordinate system, and the hexagon is shown with its "natural shape".

In[11]:= Show[hex, AspectRatio -> Automatic]

Out[11]=

Using scaled coordinates, you can specify the sizes of graphical elements as fractions of the size of the display area. You cannot, however, tell Mathematica the actual physical size at which a particular graphical element should be rendered. Of course, this size ultimately depends on the details of your graphics output device, and cannot be determined for certain within Mathematica. Nevertheless, graphics directives such as AbsoluteThickness discussed in Section 2.9.3 do allow you to indicate "absolute sizes" to use for particular graphical elements. The sizes you request in this way will be respected by most, but not all, output devices. (For example, if you optically project an image, it is neither possible nor desirable to maintain the same absolute size for a graphical element within it.)