Re: [math-fun] circular-arc splines, again
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.
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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On Wed, Mar 14, 2012 at 10:53 AM, 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.
I don't have a proof yet, but I think that it might be true that if an arc of radius r1 lies between arcs of radius r2 and r3, then either r1 is less than both of r2 and r3, or greater than both. This would imply that the total number of arcs must be even. Andy
participants (3)
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Andy Latto -
Fred lunnon -
Henry Baker