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Any two dimensional object, when plotted on your graph,
will change "shape" and "observed" properties, when the
dimensions of the pane change.
Two dimensional objects are circles, squares, rectangles, triangles,
parabolas, ellipses etc.
An angle, although is not a "closed" schema, is also two dimensional.
Only straight lines are one dimensional, and will be straight under any
transformation.
In att. gif the two charts have EXACTLY the same function.
The angle at A is nearly 45 deg.
The angle at B is nearly 90 deg.
The property is the same, the shape is quite different.
We try to preserve properties, not observed shapes.
A property for an angle u is for example its sin(u).
For u=30 deg we have sin(u)=0.5.
When we write sin(w)=0.5, we are sure that w=30 deg, no matter what
"shape" we see on our graph.
This is a safe way to avoid the great confusion between a paper graph and
a computer graph. Paper is inelastic material.
In AFL Function Reference you have sin(), cos() and atan() as built-in
function and you may use them to detect angles in a safe way.
HINT: In the case you deal with negative angles, use sin(w).
For any angle w with -90<w<90 the value of sin(w) is unique and in
1 to 1 correspondance to the value of w.
And if you deal with the future, any angle will be in -90<w<90.
An angle w=135 deg refers to the past of a point A.
Avoid cos(w), because cos(-pi/3)==cos(pi/3)==0.5
Dimitris Tsokakis
P. S.1.
I copy here CIRCLE Statement from MS GW-Basic Manual
"CIRCLE(xcenter,ycenter),radius[,color][,start],[end][,aspect]
where
aspect describes the ratio of the x radius to the y radius (x:y). The
default aspect ratio depends on the screen mode, but gives visual
circle in either graphics mode, assuming a standard monitor screen aspect
ratio of 4:3"
I thing it is more than clear, what is CIRCLE and what is VISUAL CIRCLE.
P. S. 2. In AFL Function Reference, there is a typo in atan() function.
It should be
The formula "atan( 1.00 )" returns PI/4
instead of
a.. ATAN
EXAMPLE The formula "atan( 1.00 )" returns PI/2
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<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>Any two dimensional object, when plotted on your
graph,</FONT></DIV>
<DIV><FONT face=Arial size=2> will change "shape" and
"observed"</FONT><FONT face=Arial size=2> properties, when the </FONT></DIV>
<DIV><FONT face=Arial size=2>dimensions of the pane change.</FONT></DIV>
<DIV><FONT face=Arial size=2>Two dimensional objects are circles, squares,
rectangles, triangles, </FONT></DIV>
<DIV><FONT face=Arial size=2>parabolas, ellipses etc.</FONT></DIV>
<DIV><FONT face=Arial size=2>An angle, although is not a "closed" schema, is
also two dimensional.</FONT></DIV>
<DIV><FONT face=Arial size=2>Only straight lines are one dimensional, and will
be straight under any</FONT></DIV>
<DIV><FONT face=Arial size=2>transformation.</FONT></DIV>
<DIV><FONT face=Arial size=2>In att. gif the two charts have EXACTLY the same
function.</FONT></DIV>
<DIV><FONT face=Arial size=2>The angle at A is nearly 45 deg.</FONT></DIV>
<DIV><FONT face=Arial size=2>The angle at B is nearly 90 deg.</FONT></DIV>
<DIV><FONT face=Arial size=2>The property is the same, the shape is quite
different.</FONT></DIV>
<DIV><FONT face=Arial size=2>We try to preserve properties, not observed
shapes.</FONT></DIV>
<DIV><FONT face=Arial size=2>A property for an angle u is for example its
sin(u).</FONT></DIV>
<DIV><FONT face=Arial size=2>For u=30 deg we have sin(u)=0.5.</FONT></DIV>
<DIV><FONT face=Arial size=2>When we write sin(w)=0.5, we are sure that w=30
deg, no matter what</FONT></DIV>
<DIV><FONT face=Arial size=2>"shape" we see on our graph.</FONT></DIV>
<DIV><FONT face=Arial size=2>This is a safe way to avoid the great confusion
between a paper graph and</FONT></DIV>
<DIV><FONT face=Arial size=2>a computer graph. Paper is inelastic
material.</FONT></DIV>
<DIV><FONT face=Arial size=2>In AFL Function Reference you have sin(), cos() and
atan() as built-in </FONT></DIV>
<DIV><FONT face=Arial size=2>function and you may use them to detect angles in a
safe way.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>HINT: In the case you deal with negative angles,
use sin(w).</FONT></DIV>
<DIV><FONT face=Arial size=2>For any angle w with -90<w<90 the value of
sin(w) is unique and in</FONT></DIV>
<DIV><FONT face=Arial size=2>1 to 1 correspondance to the value of
w.</FONT></DIV>
<DIV><FONT face=Arial size=2>And if you deal with the future, any anglewill be
in -90<w<90.</FONT></DIV>
<DIV><FONT face=Arial size=2>An angle w=135 deg refers to the past ofa point
A.</FONT></DIV>
<DIV><FONT face=Arial size=2>Avoid cos(w), because
cos(-pi/3)==cos(pi/3)==0.5</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>Dimitris Tsokakis</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>P. S.1. </FONT></DIV>
<DIV><FONT face=Arial size=2>I copy here CIRCLE Statement from MS GW-Basic
Manual</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial
size=2>"CIRCLE(xcenter,ycenter),radius[,color][,start],[end][,aspect]</FONT></DIV>
<DIV><FONT face=Arial size=2>where</FONT></DIV>
<DIV><FONT face=Arial size=2>aspect describes the ratio of the x radiusto the y
radius (x:y). The</FONT></DIV>
<DIV><FONT face=Arial size=2>default aspect ratio depends on the screenmode,
but gives visual</FONT></DIV>
<DIV><FONT face=Arial size=2>circle in either graphics mode, assuming astandard
monitor screen aspect </FONT></DIV>
<DIV><FONT face=Arial size=2>ratio of 4:3"</FONT></DIV>
<DIV><FONT face=Arial size=2>I thing it is more than clear, what is CIRCLE
and what is VISUAL CIRCLE.</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>P. S. 2. In AFL Function Reference, there is a typo
in atan() function.</FONT></DIV>
<DIV><FONT face=Arial size=2>It should be </FONT></DIV>
<DIV><FONT face=Arial size=2>The formula "atan( 1.00 )" returns
PI/4</FONT></DIV>
<DIV><FONT face=Arial size=2>instead of </FONT></DIV>
<DIV><FONT face=Arial size=2>
<LI><A name=ATAN></A><B>ATAN</B>
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<TD></TD>
<TD></TD></TR>
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<TD vAlign=top></TD>
<TD width="80%"></TD></TR>
<TR>
<TD>EXAMPLE</TD>
<TD>The formula "atan( 1.00 )" returns PI/2</TD></TR>
<TR>
<TD></TD>
<TD></TD></TR></TBODY></TABLE></LI></FONT></DIV>
<DIV> </DIV></BODY></HTML>
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