Pick-up your elliptical paint brush and make like a Van Gogh
If you consider that the BBC Micro's angular graphics are strictly
for squares, why not think again ...
By ERIC J. SIMPSON
MOST pictures and diagrams produced by BBC graphics have an
angular quality about them, based as they are on triangles, rectangles
and squares.
To mimic paintings it is necessary to use an underlying technique
to imitate an artist's brush strokes, and the best shape for this
is the ellipse, which can be round as a circle or long and thin
like a line.
This program produces a screen filling four colour owl by over-painting
ellipses just as a painter in oils superimposes his brush strokes.
Most readers by now will have come across programs to draw circles
using the parametric form for a circle of radius (RADIUS) and
centre (XC, YC):
X=XC+RADIUS*COS(A)
Y=YC+R ADIUS*SIN(A)
with A varying from 0 to 360 degrees to draw a full circle,
with GCOL commands to vary its colour and PLOT 85 to infill triangles.
A varies in steps so that the shape produced is not a true circle
but a polygon with many sides. If the step size is 15 then the
shape is a 24 sided regular polygon indistingushable from a circle
on a VDU.
An ellipse has two measurements to describe it, its major semi-axis
MAJ and it minor semi-axis MIN. In a circle MAJ equals MIN, but
if their ratio is a long way from one the ellipse is very long
and thin, so the ellipse is:
X=XC+MAJ*COS(A)
Y=YC+MIN*SIN(A)
The ellipse so far has its axes horizontal and vertical, which
is of little use for mimicking painting, and now we need to rotate
it anticlockwise about its centre by an angle INC. To rotate a
point we use a rotation matrix:
This matrix only applies to rotation about the origin, so we
omit XC,YC for the time being, giving:
X=MAJ*COS(A)*COS(INC)-MIN*SIN(A)*SIN(INC)
Y=MAJ*COS(A)*SIN(INC)+MIN*SIN(A)*COS(INC)
Finally we add XC back to X giving XT and YC back to Y giving
YT to produce an ellipse where we can alter its shape, size, centre
and inclination to the horizontal as we wish.
When the program was first written it had a run time of between
six and seven minutes. I looked for ways to shorten this and it
occurred to me that a look-up table was the obvious answer.
Each point involves the calculation of eight sines and cosines
for INC and A. With 24 points to an ellipse, and more than 100
ellipses in the finished picture, we would need more than 20,000
such steps.
By putting a table of sines and cosines at the start of the
program at five degree intervals, less than 150 calculations are
needed.
So long as A and INC are always in multiples of five, their
sines and cosines as elements of the array T at the start of the
program, then:
COS(INC) is replaced by T(INC/5,1)
SIN(INC) is replaced by T(INC/5,2)
COS(A) is replaced by T(A/5,1)
SIN(A) is replaced by T(A/5,2)
To complete the speeding-up, MAJ, MIN, INC, XC and YC are entered
as integers. The final expressions for plotting the curves become:
X=MAJ%*T(A%/5,1)
Y=MIN%*T(A%/5,2)
XT=X*T(INC%/5,1)-Y*T( INC%/5,2)+XC%
YT=X*T(INC%/5,2)+Y*T( INC%/5,1)+YC%
With these improvements the run time drops to about two minutes.
The look-up table occupies lines 5 to 30 in the listing. The
green and white background is between lines 35 and 115. Line 55
changes red into green from the normal colours in Mode 1.
Lines 120 to 205 contain the routine for drawing an ellipse,
drawing data from lines 245-845.
The data is split by REM statements to show which part of the
owl is drawn by that section to facilitate modification in the
final picture.
Lines 210 to 240 hold a print routine to put spots on the owl's
breast.
