Re: [math-fun] circular-arc splines, again
I'm not constructing Reuleaux triangles; they are only C0, not C1, curves. At 09:31 AM 3/14/2012, Fred lunnon wrote:
Why "divisible by 2 or 4", when a Reuleaux triangle has 6 arcs? WFL
On 3/14/12, Henry Baker <hbaker1@pipeline.com> wrote:
I'll see what I can do. On a long bicycle ride yesterday, I began to have doubts about 5; perhaps the number needs to be divisible by 2 or 4. But the construction below doesn't require r1=r2, r3=r4 (the mechanical drawing approach), so this gives me some hope.
At 01:25 AM 3/14/2012, Bill Gosper wrote:
Henry, can you exhibit a smooth loop with five radius changes? A special case of four is the "four point ellipse" from mechanical drawings of yore. These can co-rotate in continuous tangential contact<http://gosper.org/pump1.gif> .
Similarly, "six point Reuleaux triangles <http://gosper.org/reuleaux.gif>". (Rich's observation.) --rwg
hgb> I now think that it is impossible to create a simple closed C1 curve from only 3 circular arc segments. The following construction for 4 segments shows why this is. 1. Draw a circle of radius r1. 2. Draw a circle of radius r2 that intersects circle #1. 3. Draw a circle of radius r3 inside the intersection that is tangent to the first 2 two circles. 4. Draw another circle of radius r4 inside the intersection of #1 & #2 that is tangent to #1 and #2. A circular arc segment is taken from each of the 4 circles to produce a closed C1 curve. Basically, it is the boundary of the intersection region, with both sharp ends cut off by circular arcs from circles #1 & #2. The construction shows that r3<r1, r3<r2, r4<r1, r4<r2. There are probably interesting relationships between the centers of these circles, considered as complex numbers, and the various radii. There is a paper by someone at Bell Labs that showed some similar relationships of tangent circles & complex coordinates.
Wait -- if you have any convex C0 curve made of circle segments, can't you always change it into a C1 of the type Henry wants, changing each corner into a circle arc, by pushing sufficiently small circles into each corner until they are tangent to both arcs that meet there? If I'm understanding this discussion correctly, this makes a 5-segment loop easily: take any pair of different-curvature circle arcs A and B joined without a corner, connect their remote end-points with any circle arc C, then smooth out the AC and BC corners using small circles D and E. --Michael On Wed, Mar 14, 2012 at 12:46 PM, Henry Baker <hbaker1@pipeline.com> wrote:
I'm not constructing Reuleaux triangles; they are only C0, not C1, curves.
At 09:31 AM 3/14/2012, Fred lunnon wrote:
Why "divisible by 2 or 4", when a Reuleaux triangle has 6 arcs? WFL
On 3/14/12, Henry Baker <hbaker1@pipeline.com> wrote:
I'll see what I can do. On a long bicycle ride yesterday, I began to have doubts about 5; perhaps the number needs to be divisible by 2 or 4. But the construction below doesn't require r1=r2, r3=r4 (the mechanical drawing approach), so this gives me some hope.
At 01:25 AM 3/14/2012, Bill Gosper wrote:
Henry, can you exhibit a smooth loop with five radius changes? A special case of four is the "four point ellipse" from mechanical drawings of yore. These can co-rotate in continuous tangential contact<http://gosper.org/pump1.gif> .
Similarly, "six point Reuleaux triangles < http://gosper.org/reuleaux.gif>". (Rich's observation.) --rwg
hgb> I now think that it is impossible to create a simple closed C1 curve from only 3 circular arc segments. The following construction for 4 segments shows why this is. 1. Draw a circle of radius r1. 2. Draw a circle of radius r2 that intersects circle #1. 3. Draw a circle of radius r3 inside the intersection that is tangent to the first 2 two circles. 4. Draw another circle of radius r4 inside the intersection of #1 & #2 that is tangent to #1 and #2. A circular arc segment is taken from each of the 4 circles to produce a closed C1 curve. Basically, it is the boundary of the intersection region, with both sharp ends cut off by circular arcs from circles #1 & #2. The construction shows that r3<r1, r3<r2, r4<r1, r4<r2. There are probably interesting relationships between the centers of these circles, considered as complex numbers, and the various radii. There is a paper by someone at Bell Labs that showed some similar relationships of tangent circles & complex coordinates.
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participants (2)
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Henry Baker -
Michael Kleber