Learning PostScript  ~ Part 5 ~  By Peter Goed  <> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <>  Arcs and Curves Drawing Curves  In this tutorial we will look at the simplest and most difficult graphic shapes that can be constructed using PostScript; namely arcs and curves. These are the most versatile PostScript operators. The 'arc' operator is one of the simplest of all the PostScript operators for drawing graphic shapes. It is much easier to draw circles using the 'arc' operator than it is to construct rectangles. The 'arc' operator draws arcs of any length or radius at any position. You can use it in conjunction with 'lineto' to draw semicircles or pie wedges of any radius or orientation, 'fill' and 'stroke' can be used to paint the resulting shape with any colour. When used in conjunction with 'scale', the 'arc' operator can also be used to draw an ellipse. A variation of the 'arc' operator, 'arcto' may be used to connect intersecting lines with curved line segments, and although more difficult to use than the 'arc' operand, is ideal when constructing rectangles with rounded corners. The 'curveto' operator is the most powerful PostScript operator and also the hardest to use. At the end of this tutorial we will briefly discuss 'curveto' and in tutorial 7 we will use this operator to design some elaborate symmetrical designs.  'arc'  In Tutorial 1 we constructed a simple rectangle using five PostScript operations; one 'moveto', three 'lineto' or 'rlineto' and a 'closepath' instruction. Compared to this circles are a piece of cake. A circle can be of any size at any location and can be easily drawn using a single 'arc' operator, together with its five operands. These five operands describe in order, the x and y coordinates for the centre of the circle, the radius of the circle and the beginning and ending angles of the arc. For circles the beginning angle is 0 degrees and the ending angle is 360 degrees. The following example will produce a circle of 2 inch radius centred on the page. 306 396 translate % sets the point of origin at the page centre 0 0 144 0 360 arc % the 0,0 coordinates set the x and y coordinates % at the "translated" centre of 306, 396 stroke showpage  Donut  Using the same circle as previously, we now use the 'fill' operator to fill the 2 inch circle with the default colour (black) and the 1 inch radius circle with white. Because PostScript painting operators apply opaque paint, the inside circle obscures the portion of the black circle underneath, thus creating a "donut" effect. 306 396 translate 0 0 144 0 360 arc fill 1 setgray 0 0 72 0 360 arc fill showpage  Yin and Yan  The following code produces in a fairly simple way, a very complex graphic that would be hard to draw by hand. The Yin and Yan symbol is created using only six 'arc' operators, five of which are circles and one a semicircle. One of the arcs is stroked and five are filled - three with black and two with white. The page description for this example must be constructed in the proper sequence for it to come out correctly. 306 396 translate % set page centre as 0,0 0 0 144 90 270 arc % step one; black semicircle fill % with black (default) 0 -72 72 0 360 arc % step two; lower black circle fill gsave % step three, draw two white circles 1 setgray % set colour to white 0 72 72 0 360 arc % upper white circle fill % paint white 0 -72 12 0 360 arc % white dot fill grestore 0 72 12 0 360 arc % step four, black dot fill 0 0 144 0 360 arc % black outer circle stroke /Times-Roman findfont 20 scalefont setfont -60 -200 moveto 1 2 scale (Yin and Yan) show showpage There is no need to use 'closepath' to finish the semicircle in step one. The use of the 'fill' command tells the interpreter to to close the path implicitly.  Semicircle  The following code will construct a semicircle from 90 degrees to 270 degrees, fill the semicircle with the default colour, draw a circle around the semicircle and show the cardinal compass points. 306 396 translate /Helvetica-Bold findfont 24 scalefont setfont 0 0 72 90 270 arc fill .1 setlinewidth 0 0 144 0 360 arc stroke 110 -5 moveto (0/360\312) show % the "\nnn" is used to access characters from the % expanded character set. Here we used "octal" % \312, which gives us the degree symbol. For a % further explanation, see the PostScript Language % Reference Manual, pages 254 and 255 give a full % listing of the character set and its "octal" % numbers. -15 138 moveto (90\312) show -165 -5 moveto (180\312) show -20 -150 moveto (270\312) show showpage Pie Wedge  The following instruction will create a pie wedge comprising 20% of a circle. The arc begins at 0 degrees and ends at 72 degrees. 306 396 translate /Times-Bold findfont 36 scalefont setfont 2 setlinewidth 0 0 144 0 72 arc % 360 degrees x 20% = 72 degrees 0 0 lineto closepath gsave fill grestore stroke gsave 1 setgray 35 30 moveto (20%) show grestore gsave /Helvetica-Bold findfont 24 scalefont setfont 150 -5 moveto % you will need to remove these two instructions (0\312) show % when completing the whole Pie Chart in part three 35 144 moveto (72\312) show grestore showpage % used here only to print out part one of this code. % Remove to allow parts one and two to function together.  Second Pie Wedge  We can issue the following instructions to create a second pie wedge to our page instructions. % continued from previous code before the 'showpage' instruction. 0 0 144 72 arc % 360 degrees x 35% = 126 degrees + 72 degrees = 198 0 0 lineto closepath gsave .9 setgray fill grestore stroke -90 40 moveto (35%) show gsave /Helvetica-Bold findfont 24 scalefont setfont -200 -50 moveto (198\312) show grestore showpage % don't forget to remove this instruction to enable % part three to function together with parts one and two. Complete Pie Chart % continued from previous code before the 'showpage' instruction. 0 0 144 198 360 arc % 360 x .45 = 162 + 198 = 360 0 0 lineto closepath stroke -20 -80 moveto (45%) show gsave Helvetica-Bold findfont 24 scalefont setfont 150 -5 moveto (0/360\312) show % don't forget to remove the "(0\312) show from part one of the Pie Wedge exercise. grestore showpage  Pie Slice  The versatility of PostScript makes it very easy to "slice" a piece of pie to emphasise any of the pie wedges. This is easily accomplished by adding a 'gsave/grestore' around the wedge and using translate to reposition the pie slice. I have added a few new instructions to the previous code and removed the degree markings around the outside of the pie. 306 396 translate /Times-Bold findfont 36 scalefont setfont 2 setlinewidth gsave % added 25 20 translate % added 0 0 144 0 72 arc % 360 degrees x 20% = 72 degrees 0 0 lineto closepath gsave fill grestore stroke gsave 1 setgray 35 30 moveto (20%) show grestore grestore % added gsave % added 0 0 144 72 arc % 360 degrees x 35%=126 degrees + 72 degrees=198 0 0 lineto closepath gsave .9 setgray fill grestore stroke -90 40 moveto (35%) show grestore % added gsave newpath % added because there was no explicit 'moveto' % instruction preceding the 'arc' instruction. % To see the difference, remove the 'newpath' % instruction and see the difference that it % creates. 0 0 144 198 360 arc % 360 x .45 = 162 + 198 = 360 0 0 lineto closepath stroke -20 -80 moveto (45%) show grestore showpage  Adding Labels to Your Pie Chart  If you remove the 'showpage' instruction from the previous example and add the following instructions, you will produce an attractive pie chart complete with descriptive labels. This is a very handy piece of code and can be reused for many business applications by minor manipulation of this code to suit a specific set of circumstances. gsave /Helvetica findfont 24 scalefont setfont 150 110 moveto (FEES) show -215 150 moveto (SALES TAXES) show -90 -180 moveto (INCOME TAXES) show grestore -175 250 moveto 2 2 scale (REVENUE) show showpage  Transforming Circles into.....  You can easily transform circular shapes into elliptical shapes by scaling them nonuniformly. Firstly let us create four shaded circular shapes. 306 396 translate 0 252 translate 0 0 72 0 360 arc stroke 0 -180 translate 0 0 72 0 360 arc fill 0 -180 translate gsave 0 0 72 0 360 arc .5 setgray fill grestore 0 -180 translate 0 0 72 0 180 arc closepath gsave .9 setgray fill grestore stroke showpage  Ellipses  We now use the 'scale' operator to stretch our four circular shapes into various elliptical shapes. Some experimentation is encouraged to get the feel of this section. 306 396 translate 0 252 translate gsave .5 1 scale 0 0 72 0 360 arc stroke grestore 0 -180 translate gsave 2 1 scale 0 0 72 0 360 arc fill grestore 0 -180 translate gsave .25 .50 scale 0 0 72 0 360 arc .5 setgray fill grestore 0 -180 translate gsave 4 1 scale 0 0 72 0 180 arc closepath gsave .9 setgray fill grestore stroke grestore showpage  Atomic Patterns  We can combine 'arc' and 'scale' with 'rotate' to change the orientation of each ellipse to create an interesting construction for the atom. 306 396 translate gsave 0 rotate .33 1 scale 0 0 144 0 360 arc stroke grestore gsave 60 rotate .33 1 scale 0 0 144 0 360 arc stroke grestore gsave -60 rotate .33 1 scale 0 0 144 0 360 arc stroke grestore showpage Ellipses with Text You can draw an ellipse at any location specified in a 'translate' instruction. Always use 0, 0 for the x/y coordinates for the 'arc' operator when drawing ellipses. Otherwise the 'scale' operator used for the ellipse will relocate the ellipse elsewhere on the page. /Times-BoldItalic findfont 72 scalefont setfont gsave 204 666 translate 2 .9 scale 0 0 72 0 360 arc fill grestore gsave 204 666 translate 2.1 1 scale 0 0 72 0 360 arc stroke grestore 1 setgray 72 648 moveto (Typeface) show showpage  'arcto'  The 'arcto' operator connects two line segments with a circular arc of any radius and angle. It accomplishes this by adding an arc to the end of one line segment, from which an other line can be added. This makes 'arcto' ideal for connecting two line segments with rounded corners which can then be extended to form either a rectangle with rounded corners or two parallel lines connected with rounded ends. The five operands required by 'arcto' are the x and y coordinates tangent to the arc on both the first and the second line, and the radius of the arc (x1 y1 x2 y2 r). We use the 'clear' operator here to discard some residual information generated by the PostScript interpreter after executing 'arcto'. Whether or not the arc is preceded by a straight-line segment is determined by the distance between the points and the radius of the arc. Arcs both with and without straight line segments are used in the following example, in which two parallel lines are connected by curved ends. 108 108 translate newpath % sets a new current point to prevent an % error code being generated 0 36 moveto % moves the current point up 36 points on the % page to give us starting coordinates for % x0 (108) and y0 (144) 0 72 216 72 36 arcto clear % 0=x1 72=y2 (x1, y1 axis point set % at 108, 180 [108+72=180]) % 216=x2 72=y2 (x2, y2 axis point set at % 324, 180 [108+216=324, 108+72=180]) % from the x1, y1 and x2, y2 coordinates % above, the starting point for our % radius will be at the only point % located at a distance of 36 points (r) % in a direction perpendicular to both % tangent lines and is located within % the inner angle between the tangent % lines, this point can only fall at % 144, 144, and an arc of radius 36 is % drawn from 108, 144 to 144, 180. 216 72 216 0 36 arcto clear % x1,y1=324, 180 x2,y2=324, 108 and % hence our radius centre point will be % at 288, 144. This instruction will % now draw a straight line from 144, 180 % to 288, 180 and then draw an arc from % this point to 324, 144. 216 0 0 0 36 arcto clear % x1,y1=324, 108 x2,y2=108, 108 and the % radius point will be at 288, 144 [take % into account where our last radius % ended, as this is the current point % this arc will start from. The % endpoint for the arc is at 324, 144] 0 0 0 72 36 arcto clear % x1,y1=108 ,108 x2,y2=108, 180 and % the radius point will be 144, 144. % A line will now be drawn from the end % of the previous arc at 288, 108 to the % start of this arc at 144, 108, and the % arc will finish at 108, 144 - which % was our starting point of the 1st arc. stroke showpage I suggest that you look at the explanation given in the PostScript Language Reference Manual (the red book) on page 119 for a more detailed description of 'arcto', as this instruction is fairly hard to come to terms with. It is a good idea to draw this example out on a sheet of paper marked with a 72 point line grid to get a better understanding, I found that this helped no end.  Round Box with Text  You can combine rounded boxes created with 'arcto' with text to create a pleasing title page, as shown by the following page description. /Times-BoldItalic findfont 72 scalefont setfont 76 424 moveto 76 472 528 472 48 arcto clear 528 472 528 376 48 arcto clear 528 376 76 376 48 arcto clear 76 376 76 472 48 arcto clear gsave fill % black box grestore gsave 12 setlinewidth stroke % additional 12 point black box around perimeter grestore 1 setgray gsave 4 setlinewidth stroke % white ring around original perimeter grestore 100 400 moveto (ROUNDBOX) show showpage In tutorial seven we will discuss how to create a similar rounded box around any string of text of any size and typeface.  'curveto'  The 'curveto' operator is the foundation of the PostScript language, it gives you access to the Bezier curves discussed in tutorial 2. This powerful operator adds a Bezier curve section to the current path between the current point and the end point of the curve. The 'curveto' operator requires an initial current point and six operands. The initial 'moveto' (new current point) establishes the beginning of the curve, and the last set of coordinates establishes the end of the curve. The other two sets of x/y coordinates are the control points required to shape the curve. This section is not an easy one to come to terms with and will require a bit of work with pencil and paper on a page layout graph to fully comprehend. The PostScript Language reference Manual (page 140) has a fairly good explanation that may help you better understand this operator. The page description below will give you two Bezier curves, each has very similar code. Note the effect that changing the 150 to -150 has on the curve. Some experimentation with changing the sets of coordinates will help to make the picture clearer. The 'curveto' operator expects input as follows "x1 y1 x2 y2 x3 y3 curveto" 306 396 translate gsave 0 144 translate % set centre point at 0, 144 -200 0 moveto % set new current point -200, 144 -100 100 100 150 200 0 curveto % x1=-100 y1=244 x2=100 y2=294 % x3=200 y3=144 stroke grestore gsave 0 -144 translate % set centre point at 0, -144 -200 0 moveto % set new current point -200, -144 -100 100 100 -150 200 0 curveto % x1=-100 y1=-44 x2=100 y2=-294 % x3=200 y3=-144 stroke grestore showpage  Brush Stroke Underline  The Bezier model is ideally suited for use in interactive graphics, because the curve can be easily manipulated by moving the control points with a mouse, but the 'curveto' operator is very difficult to use directly in a program, which is why most PostScript language tutorials say very little about this operator. However, you can use the 'curveto' operator to perform some functions that could not be done any other way, such as simulating a brush stroke underlining of text, as the following page description will show you. /Times-BoldItalic findfont 72 scalefont setfont 72 648 moveto (Typeface) show 4 setlinewidth 77 605 moveto 144 620 216 630 325 620 curveto 78 606 moveto 145 620 216 632 327 624 curveto 78 607 moveto 145 621 217 634 329 626 curveto stroke showpage As you can see it is not easy, but with a little bit of thought you can create some stunning underlines which cannot be done in any other way except a structured drawing program (maybe).  Sine Wave  There is however one area in which 'curveto' is easier to use directly than it is interactively: for creating symmetrical designs where a single 'curveto' that creates the simple sine wave is repeated many times. 306 396 translate 0 0 moveto 72 72 72 -72 144 0 curveto stroke showpage In the next tutorial we will delve into repeat loops to create some wonderful symmetrical graphics patterns, we will also introduce a new painting operator in this tutorial, as well as a host of new and exciting operators. Again this month there will be no homework, but make sure that you understand the concept of the operators discussed in this month's tutorial and try out some examples of your own; the easiest way to do this is to use the "POST" interpreter from the Fish Disk Library to display your page descriptions on screen and/or send them to any dot matrix printer. From here on in we are starting to get to some fairly complex coding and it would be a good idea to save the majority of the code for later use in a more serious page description that you will no doubt want to write. I save mine with a name that describes the function of the code and give it a .par cod to identify it as a piece of "stand alone" code that I can send out the parallel port to my PostScript printer and get a result. See you next tutorial.  <> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <%> 34 <>