Drawing circles with Bezier curves
Author Message
Drawing circles with Bezier curves

If you are interested in knowing how to draw a circle (or an ellipse)
using Bezier curves only (that is the only way to do it when, for
example, you are creating a Type 1 font), take a look at a new page in
which I describe how to do it.

The URL is http://www.*-*-*.com/

Thu, 05 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves
If you are interested in knowing how to draw a circle (or an ellipse)
using Bezier curves only (that is the only way to do it when, for
example, you are creating a Type 1 font), take a look at a new page in
which I describe how to do it.

The URL is http://www.whizkidtech.net/bezier/circle/

Thu, 05 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

> If you are interested in knowing how to draw a circle (or an ellipse)
> using Bezier curves only (that is the only way to do it when, for
> example, you are creating a Type 1 font), take a look at a new page in
> which I describe how to do it.

> The URL is http://www.whizkidtech.net/bezier/circle/

It might be mildy interesting if you solved the general case,
that is for an arc other than pi/2.  It's not much harder.
Or if you included an error analysis and showed that your value
of kappa' did not produce the absolute minimum error
(Hint: your error is always on one side).

Thu, 05 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:
>> The URL is http://www.whizkidtech.net/bezier/circle/

>It might be mildy interesting if you solved the general case,
>that is for an arc other than pi/2.  It's not much harder.

Yes, I suppose it would be interesting. It, however, would certainly
be more complicating than multiplying the radius by a constant. I am
not quite sure of the usefulness of solving the general case, though:
The need for emulating an arc with a Bezier curve applies mostly to
Type 1 fonts; outside of that, we can use "arc," which is built into
postscript but not available in Type 1 fonts.

Personally, I have never needed the general case in font design, but I
have used the pi/2 arc many times over.

If there is a genuine need (other than mere intellectual curiosity)
for the general case, I would consider working on it, although not in
the immediate future as I am currently working on several different
projects.

Quote:
>Or if you included an error analysis and showed that your value
>of kappa' did not produce the absolute minimum error
>(Hint: your error is always on one side).

I am not sure I follow what you mean by my error being always on one

Unless you are talking about a round-off error I have seen in at least
one commercial package which clearly uses kappa. It would appear that
the program multplies the radius by kappa and keeps the result as a
floating point number, then adds it to and subtracts from start/end
points, and only then rounds it to an integer.

Because of that, the lengths of half of the control line segments
differ from the other half by one. The problem is not caused by using
kappa but by rounding off the result at the wrong time. If the program
rounded off the length to integer before adding/subtracting it, the
problem would not be there.

If you are referring to some other error, kindly enlighten me by
telling me straight what it is you are talking about.

P.S. My apologies to everyone for posting the original message twice.
Apparently my news server accepted the message on the first attempt
but failed to acknowledge it, so my mail software sent it out a second
time.

Fri, 06 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves
This will prove to be an interesting thead to watch...

I have yet to see a any of the clone intrepreters do this
correctly. The results are always close, but never produce
the same results as Adobe across the board.

I ended up paying my brother the "ex-rocket-scientist/professor"
quite a bit to reaserch this and develop a solution.

Joe
--
Joe C. Hecht

(850) 689-0877 121 Louise Dr. Crestview, FL 32536
http://home1.gte.net/joehecht/index.htm

Fri, 06 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

>>
>>>> The URL is http://www.whizkidtech.net/bezier/circle/
>>>
>>>It might be mildy interesting if you solved the general case,
>>>that is for an arc other than pi/2.  It's not much harder.
>>
>>Yes, I suppose it would be interesting. It, however, would certainly
>>be more complicating than multiplying the radius by a constant. I am
>>not quite sure of the usefulness of solving the general case, though:
>>The need for emulating an arc with a Bezier curve applies mostly to
>>Type 1 fonts; outside of that, we can use "arc," which is built into
>>PostScript but not available in Type 1 fonts.
>>
>>Personally, I have never needed the general case in font design, but I
>>have used the pi/2 arc many times over.
>>
>>If there is a genuine need (other than mere intellectual curiosity)
>>for the general case, I would consider working on it, although not in
>>the immediate future as I am currently working on several different
>>projects.
>
>Actually, there is a genuine need for implementing "arc" with Bezier
>curve(s) for drawing PDF (Acrobat).
>
>PDF is a superset of a subset of PostScript, and it doesn't have
>the "arc" operator.  So, if you need to draw a circle or arc using
>PDF, you must come up with a solution.
>
>One generalizable solution is available at
>http://www.tinaja.com/cubic01.html
>
>but I don't know where this appeared originally.
>Does anyone know?

Ps2pdf must also be doing something similar.  So ghostscript source code
must have the code to do this.
--

Fri, 06 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

> >Or if you included an error analysis and showed that your value
> >of `kappa' did not produce the absolute minimum error
> >(Hint: your error is always on one side).
> I am not sure I follow what you mean by my error being always on one

The specific approximation you arrive it makes the Bezier curve
touch the circle at the center of the arc(45 degree).  This makes the
Bezier curve always lie on one side of the circle.  The sign
of the error is constant. If you adjust kappa slightly you can move the
point at 45 degree in or out relative to the circle.  If you
move it so that the Bezier curve crosses the circle, then you
can reduce the maximum error.  I can't remember too many
decimal digits of the optimal value, but  it starts .5541...
compared to your kappa = 0.5522847498...

Of course for most purposes either approximation of the circular
arc is good enough...

--

Fri, 06 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

>>> The URL is http://www.whizkidtech.net/bezier/circle/

>>It might be mildy interesting if you solved the general case,
>>that is for an arc other than pi/2.  It's not much harder.

>Yes, I suppose it would be interesting. It, however, would certainly
>be more complicating than multiplying the radius by a constant. I am
>not quite sure of the usefulness of solving the general case, though:
>The need for emulating an arc with a Bezier curve applies mostly to
>Type 1 fonts; outside of that, we can use "arc," which is built into
>PostScript but not available in Type 1 fonts.

>Personally, I have never needed the general case in font design, but I
>have used the pi/2 arc many times over.

>If there is a genuine need (other than mere intellectual curiosity)
>for the general case, I would consider working on it, although not in
>the immediate future as I am currently working on several different
>projects.

Actually, there is a genuine need for implementing "arc" with Bezier
curve(s) for drawing PDF (Acrobat).

PDF is a superset of a subset of PostScript, and it doesn't have
the "arc" operator.  So, if you need to draw a circle or arc using
PDF, you must come up with a solution.

One generalizable solution is available at
http://www.tinaja.com/cubic01.html

but I don't know where this appeared originally.
Does anyone know?

Sat, 07 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:
> If you are interested in knowing how to draw a circle (or an ellipse)
> using Bezier curves only (that is the only way to do it when, for
> example, you are creating a Type 1 font), take a look at a new page in
> which I describe how to do it.

It might be worth noting that the corresponding value for a 2-knot "circle"
is 4/3, versus the 4-knot case value of 4/3 * (sqrt(2)-1).

Richard J. Kinch
Publisher, TrueTeX brand typesetting software.
See http://truetex.com

Sat, 07 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

> >> The URL is http://www.whizkidtech.net/bezier/circle/

> >It might be mildy interesting if you solved the general case,
> >that is for an arc other than pi/2.  It's not much harder.

> Yes, I suppose it would be interesting. It, however, would certainly
> be more complicating than multiplying the radius by a constant. I am
> not quite sure of the usefulness of solving the general case, though:

The general case solution  (useful or not :-) is
(for H = half the angle of the arc):

kappa = (4/3) (1 - cos H) / sin H

or

kappa = (4/3) \sqrt{ (1 - cos H) / (1 + cos H) }

or

kappa = (4/3) sin H / (1 + cos H)

Take your pick --- the last one is the better one to use when H is small,
(while the first one is the better one to use when H is near pi)

Regards, Berthold.

Tue, 10 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:
>[...]
>>>> The URL is http://www.whizkidtech.net/bezier/circle/
>[...]
>One generalizable solution is available at
>http://www.tinaja.com/cubic01.html

Thank you for the information. I have now posted a link to that site
at the end of my page. I have given you credit for informing me about
it, if you'd rather not be mentioned, please let me know.

---
Whiz Kid Technomagic
http://www.whizkidtech.net/

Sat, 14 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

>>[...]
>>>>> The URL is http://www.whizkidtech.net/bezier/circle/
>>[...]
>>One generalizable solution is available at
>>http://www.tinaja.com/cubic01.html

>Thank you for the information. I have now posted a link to that site
>at the end of my page. I have given you credit for informing me about
>it, if you'd rather not be mentioned, please let me know.

I don't mind, but you don't have to.  Though, I think you should
credit Don Lancaster who put together that collection and the
original author Jim Fitzsimmons who is mentioned there.

Who is Jim Fitzsimmons, BTW?
I did a web search, but nothing turns up.
Where was that TeXT tutorial published originally?

Izumi

Sat, 14 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:
>> I am not sure I follow what you mean by my error being always on one

>The specific approximation you arrive it makes the Bezier curve
>touch the circle at the center of the arc(45 degree).  This makes the
>Bezier curve always lie on one side of the circle.  The sign
>of the error is constant. If you adjust kappa slightly you can move the
>point at 45 degree in or out relative to the circle.  If you
>move it so that the Bezier curve crosses the circle, then you
>can reduce the maximum error.  I can't remember too many
>decimal digits of the optimal value, but  it starts .5541...
>compared to your kappa = 0.5522847498...

Hmmm... Interesting observation. But wouldn't the error vary with
various implementations of PostScript interpreters?

Quote:
>Of course for most purposes either approximation of the circular
>arc is good enough...

Indeed, in most cases 0.55 alone gives a very good approximation. And
I like the idea of my kappa being "as easy as 1-2-3-4." I do not need
to remember (or look up) the value of kappa, I can always calculate it
when I need it. :-)

FWIW, I have used that kappa for years, and it always LOOKED like a
perfect circle to me.

BTW, where did you find that value? I calculated my own simply because
I could not find that information anywhere in literature (this was
pre-www).

---
Whiz Kid Technomagic
http://www.whizkidtech.net/

Sun, 15 Apr 2001 03:00:00 GMT
Drawing circles with Bezier curves

Quote:

> >point at 45 degree in or out relative to the circle.  If you
> >move it so that the Bezier curve crosses the circle, then you
> >can reduce the maximum error.  I can't remember too many
> >decimal digits of the optimal value, but  it starts .5541...
> >compared to your kappa = 0.5522847498...
> Hmmm... Interesting observation. But wouldn't the error vary with
> various implementations of PostScript interpreters?

No.  This is the correct optimum value independent of implementation.
If the computer arithmetic is not exact you will get a departure from
this which however will be totally insignificant.

Quote:
> BTW, where did you find that value? I calculated my own simply because
> I could not find that information anywhere in literature (this was
> pre-www).

You start with the value that puts the Bezier curve right on the
circle at the middle (see at end), then change the value and look
at the maximum departure of the Bezier from the circle.

From earlier posting by Berthold K.P. Horn

The general case solution  (useful or not :-) is
(for H = half the angle of the arc):

kappa = (4/3) (1 - cos H) / sin H

or

kappa = (4/3) \sqrt{ (1 - cos H) / (1 + cos H) }

or

kappa = (4/3) sin H / (1 + cos H)

Take your pick --- the last one is the better one to use when H is small,
(while the first one is the better one to use when H is near pi)

--

Sun, 15 Apr 2001 03:00:00 GMT

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