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Science Forum Index » Mathematics Forum » finding circle radius from other measurements
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| JB |
 Posted: Thu Dec 18, 2003 6:17 pm |
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My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't. |
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| Rob Johnson |
| Posted: Thu Dec 18, 2003 6:17 pm |
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In article <eee5452.0312181517.33cc14b2@posting.google.com>,
jbbrown9@hotmail.com (JB) wrote:
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
Since you don't have the altitude of the arc from the big circle, you
can try drawing the tangents from the very outside edges of the two arcs
of the big circle to where they intersect. Call the length of one of
these tangent segments "t" and the length of half the chord between the
outside edges of the two arcs "s". If a is the angle subtended by the
whole arc, then t = r tan(a/2) and s = r sin(a/2). The radius of the
circle is then t*s/sqrt(t^2-s^2).
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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| David W. Cantrell |
| Posted: Thu Dec 18, 2003 6:48 pm |
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jbbrown9@hotmail.com (JB) wrote:
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
See <http://mathforum.org/dr.math/faq/faq.circle.segment.html>. What you
called b, it calls c for chord length. Things would have been easy if you
had also measured height h, the distance from the midpoint of a chord to
the midpoint of an arc. See Case 8 at the link above. Then you could have
said
radius r = (c^2 + 4 h^2)/(8h).
David |
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| Jeremy Brown |
| Posted: Thu Dec 18, 2003 7:04 pm |
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I got another good "if only you had measured ..." answer that assumed I
had measured the arc length instead of chord length. That's an
interesting site, though.
Quote:
See <http://mathforum.org/dr.math/faq/faq.circle.segment.html>. What you
called b, it calls c for chord length. Things would have been easy if you
had also measured height h, the distance from the midpoint of a chord to
the midpoint of an arc. See Case 8 at the link above. Then you could have
said
radius r = (c^2 + 4 h^2)/(8h).
David |
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| phil |
| Posted: Thu Dec 18, 2003 7:28 pm |
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On 18 Dec 2003, JB wrote:
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
a href="http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg">http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg</a
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
a href="http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg">http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg</a
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
To find the center of a circle (or circular arc)
draw any two chords--the longer the better. Then
erect a perpendicular bisector on each chord. The
two bisectors meet a the center of the circle.
phil
O
/\
-\-\--o o |
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| Russell Blackadar |
| Posted: Thu Dec 18, 2003 9:22 pm |
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JB wrote:
[requesting solution for r, given diagram in:]
Quote: http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
[snip]
Of course there is a solution -- indeed you have slightly more
information than you need, to fix r. But no *easy* solution
that I can see.
The brute force method is to note that (x-x_0)^2 + (y-y_0)^2 = r^2
is the general equation for a circle, and so, by plugging in x and
y of any three known points you will get 3 quadratic equations in
the 3 unknowns x_0, y_0, and r, which you then (er... um... this is
the hard part!) solve. The coordinates of your known points depend
on how you draw your axes; one possible choice gives three of the
coordinate pairs as (a,0), (a+b,0), ((a+b)cos c, (a+b)sin c). |
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| Virgil |
| Posted: Thu Dec 18, 2003 11:03 pm |
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In article <eee5452.0312181517.33cc14b2@posting.google.com>,
jbbrown9@hotmail.com (JB) wrote:
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle.
If your diagram has left right symmetry, as would appear from your
diagram, then it is possible to calculate the diameter from direct
measurements.
With obvious meanings, I shall refer to the "head" as the arc of the
smaller circle and the "shoulders" as arcs of the larger circle.
Connect by a line segment the nearest point on one "shoulder" to the
nearest point on the other "shoulder", the points where the
"shoulders" meet the "head", or the left most point on the right
shoulder to the right most point on the left shoulder. Let A be the
length of this line segment and the distance betweeen these points.
Connect in a similar way by another line segment the leftmost point
of the left shoulder to the rightmost point on the right shoulder,
and call the length of this segemnt B. If done right, both segments
should be horizontal, perpendicular to the axis of symmetry of your
figure, and parallel to each other.
Let H represent the (vertical) distance between these line segments
(the vertical height of the "shoulders").
Then the diameter of the larger circle is given by
D = sqrt( (A^2+B^2+H^2)^2 - (A^2+B^2)^2 + (A^2-B^2)^2 ) / H
where "sqrt" means "square root of" and "^2" means "square of".
Hope this is of some use. |
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| Rob Johnson |
| Posted: Fri Dec 19, 2003 1:23 am |
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In article <20031218.181214@whim.org>,
rob@trash.whim.org (Rob Johnson) wrote:
Quote: In article <eee5452.0312181517.33cc14b2@posting.google.com>,
jbbrown9@hotmail.com (JB) wrote:
My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
Since you don't have the altitude of the arc from the big circle, you
can try drawing the tangents from the very outside edges of the two arcs
of the big circle to where they intersect. Call the length of one of
these tangent segments "t" and the length of half the chord between the
outside edges of the two arcs "s". If a is the angle subtended by the
whole arc, then t = r tan(a/2) and s = r sin(a/2). The radius of the
circle is then t*s/sqrt(t^2-s^2).
In the diagram at
http://www.whim.org/nebula/math/images/tangent-semichord.gif
it is simple to see why r = t*s/sqrt(t^2-s^2). First notice that
h = sqrt(t^2-s^2). Then using similar triangles, we can derive that
r/s = t/h. Therefore, r = s*t/h.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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| Ron Larham |
| Posted: Fri Dec 19, 2003 3:59 am |
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"JB" <jbbrown9@hotmail.com> wrote in message
news:eee5452.0312181517.33cc14b2@posting.google.com...
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
Don't try to calculate it. Solve it by
geometric construction.
Produce a 1-1 scale diagram of what you
have measured.Bisect the two cords, erect
perpendiculars to the cords through their
mid points. Where they intersect is the
centre of the big circle,
The distance from the centre to
one end of one of the cords is
the radius.
RonL |
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| Jyrki Lahtonen |
| Posted: Fri Dec 19, 2003 7:48 am |
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Pick any three points from the circle, and connect them
to form a triangle. Measure one side 'a' and the angle
alpha that is opposite to 'a'. Then by the law of sines
r= a/(2*sin(alpha))
Cheers,
Jyrki Lahtonen, Turku, Finland |
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| Rob Johnson |
| Posted: Fri Dec 19, 2003 8:46 am |
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Guest
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In article <eee5452.0312181517.33cc14b2@posting.google.com>,
jbbrown9@hotmail.com (JB) wrote:
Quote: My wife wanted to frame a project she is working on and she had a
picture of a sample mat. I figured I could take measurements from the
picture of the original and create a plan that would match it very
closely. Everything was easy until I tried to figure out the radius of
a circular cut. The way it was cut, it is the top portion of a circle
(less than half) with about 2/3 of another circle covering part of the
top. So, I can only measure two small arcs of the original circle. I
made a picture showing the relevant parts here:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir1.jpg
I am after the diameter of the larger circle. The smaller circle is
irrelevant for the problem I'm trying to solve and its diameter was
easy to measure anyway. I took some measurements and made the
following diagram of the larger circle:
http://wwwcsif.cs.ucdavis.edu/~brownjb/cir2.jpg
b is the measurement from one end of the arc to the other end. I
connected the ends of the arcs and continued the lines to where they
met. The distance from the edge of the circle to the intersection
point is a. I also measured the angle between the lines, c. Now, I had
a, b, and c. I figured this was more than enough information to find
r, the radius of the circle, in terms of a, b, and c. But, I could
never figure it out. I spent more time than really should have been
necessary and ran it by someone else who also should have been able to
solve it and we never came up with a solution.
I don't need an answer to this for the mat any more, but the problem
has really been bugging me. It seems like there should be a
straightforward, quick solution, but I'm starting to think there
isn't.
I am starting a new subthread since I was finally able to download the
images referenced above. In particular, cir2.jpg.
The construction is pretty simple: form the right triangle with vertices
at the center of the circle, the midpoint of one of the chords of length
b, and the intersection of the lines containing the chords. Let d be
the distance from the center of the circle to the midpoint of the chord.
d/(a+b/2) = tan(c/2). Thus, d = (a+b/2) tan(c/2). The radius is then
simply given by
r^2 = (b/2)^2 + (a+b/2)^2 tan^2(c/2)
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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