Re: How does 'arc' work?

Newsgroups	comp.lang.postscript
Date	2016-04-20 02:06 -0700
References	(3 earlier) <2240fb7d-ff33-4281-b837-a2b119198179@googlegroups.com> <0fc50e67-e520-40cd-b841-f458f58c9290@googlegroups.com> <c5b7ac2f-3af9-41f7-a0a4-00396e69944a@googlegroups.com> <c16e0fb9-1e4e-4eef-93be-344ee3f29c9d@googlegroups.com> <1c1d99b7-0d1b-402b-a160-35747f2d94ff@googlegroups.com>
Message-ID	<eaa95dcf-797d-4a2c-bd2f-751776bf2cf8@googlegroups.com> (permalink)
Subject	Re: How does 'arc' work?
From	jdaw1 <jdawiseman@gmail.com>

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I think you want the half-way point to have exactly the correct radius. Please confirm that is so.

If so, that which I call ‘z’, the distance of the inner control points from the outer, of course along the tangent to the circle, satisfies z == Tan[a/4] * 4/3.

With a=90°, Tan[a/4] = Tan[22½°] = √2-1, so z = (√2-1)*4/3 ≈ 0.55228474983. Adobe’s value of 0.552 won’t have exactly the correct radius at 45°. Indeed, at 45° Adobe’s radius is Sqrt[499849/500000] ≈ 0.999848988597778.

There are several sensible grounds for choosing z, all of which give values close to 0.552. Correct radius at half-way? Correct fill area (z = 2-Sqrt[(22-5π)/3])? Max and min radii symmetrical around ½ (z = root of a order-12 polynomial: 2097152, -23592960, 117129216, -342814720, 668905728, -921504768, 883891456, -515232000, 96817248, 63288000, -28977264, 466560, 18225, so z ≈ 0.551915)? Minimal absolute dRadius/dAngle (z ≈ 0.552031)? They’re all close to 0.552.

But what is Adobe’s formula for non-90° angles?

Thread

How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-11-20 23:23 -0800
  Re: How does 'arc' work? bugbear <bugbear@trim_papermule.co.uk_trim> - 2013-11-21 14:05 +0000
    Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-01 04:03 -0800
  Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2013-11-23 15:06 -0800
    Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-01 03:48 -0800
      Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-01 03:58 -0800
        Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-10 23:02 -0800
          Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-11 00:17 -0800
            Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-11 00:20 -0800
              Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2013-12-11 01:37 -0800
              Re: How does 'arc' work? tlvp <mPiOsUcB.EtLlLvEp@att.net> - 2014-02-28 03:47 -0500
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2014-03-01 00:30 -0800
          Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2014-04-29 13:10 -0700
    Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2019-02-02 16:19 -0800
  Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2014-05-02 06:29 -0700
    Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2014-05-08 02:21 -0700
      Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-01 15:28 -0700
        Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-02 21:35 -0700
          Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-03 13:54 -0700
            Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-18 21:40 -0700
              Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-19 15:04 -0700
                Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-20 02:06 -0700
                Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-20 05:18 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-22 00:40 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-22 11:32 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-23 20:49 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-23 22:46 -0700
                Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-24 04:01 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-30 21:47 -0700
                Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-05-01 02:59 -0700
        Re: How does 'arc' work? Scott Hemphill <hemphill@hemphills.net> - 2016-04-03 10:40 -0400
          Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-05 21:48 -0700
            Re: How does 'arc' work? Scott Hemphill <hemphill@hemphills.net> - 2016-04-06 10:37 -0400
            Re: How does 'arc' work? luser- -droog <mijoryx@yahoo.com> - 2016-04-07 23:59 -0700
            Re: How does 'arc' work? jdaw1 <jdawiseman@gmail.com> - 2016-04-19 02:54 -0700

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