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'[PICLIST] [PIC]Find center point of circle?'
2002\01\10@140036 by

Hello,

Can someone tell me how to find the center point of a circle when given the
radius and two points which lie on the circle.

Thanks,

James

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Can't be done, given two points and a radius there are two circles that can
be defined (Unless the distance between the two points is greater than 2R,
in which case there are no circles that can be defined.  In the case where
the distance between the two points is exactly twice the radius, there is
one circle and it's center is midway between the two points.

Kirk

> {Original Message removed}

----- Original Message -----
From: "James Williams" <jlw.creditviewVERIZON.NET>
To: <PICLISTMITVMA.MIT.EDU>
Sent: Thursday, January 10, 2002 1:48 PM
Subject: [PIC]Find center point of circle?

> Hello,
>
> Can someone tell me how to find the center point of a circle when given
the
> radius and two points which lie on the circle.

You can draw a circle of the same radius around each point. These circles
have one or two points of intersection. (1 if the points are on a diameter,
else 2). Then you have to decide which of the two points is the true center,
because your problem is not well constrained -- you need one more point on
the circle to be sure.
/\/\/\/*=Martin

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Let's try this:
1.    Draw a line through the 2 points
2.    Where the line intersects the circle, draw a circle around each point
3.    Draw a line through the intersection of these circles
4.    Where this line intersects the 1st circle, draw a circle around each
point
5.    Draw a line between the intersections of these circles
6.    Where the line in (5) and (3) intersect, is that not the center?

Regards

Chris A

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2002\01\10@152218 by
> -----Original Message-----
>
> Let's try this:
> 1.    Draw a line through the 2 points
> 2.    Where the line intersects the circle, draw a circle
> around each point
> 3.    Draw a line through the intersection of these circles
> 4.    Where this line intersects the 1st circle, draw a
> circle around each
> point
> 5.    Draw a line between the intersections of these circles
> 6.    Where the line in (5) and (3) intersect, is that not the center?
>
> Regards
>
> Chris A

I think you're close, but the intersection of the two lines is just inside the
real circle, not the center.

Here's one way you might solve the problem:
2. If the circles have one intersection, you're lucky and that is your center
and you're done.
3. Draw a line between your two original points.
4. The point of intersection between the two circles you drew that is closest to

-Mike

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1) draw a line between the two points.
2) draw a perpendicular bisector to this line segment
3) You now have a diameter line. Bisect it.
4) the bisect point of the diameter is the center of the circle

Fr. Thomas McGahee

{Original Message removed}
----- Original Message -----
To: <PICLISTMITVMA.MIT.EDU>
Sent: Thursday, January 10, 2002 2:58 PM
Subject: Re: [PIC]Find center point of circle?

> Let's try this:
> 1.    Draw a line through the 2 points
> 2.    Where the line intersects the circle, draw a circle around each
point
> 3.    Draw a line through the intersection of these circles
> 4.    Where this line intersects the 1st circle, draw a circle around each
> point

Which was the first circle? The one we don't know?

> 5.    Draw a line between the intersections of these circles
> 6.    Where the line in (5) and (3) intersect, is that not the center?

For a given radius there are always two possible circles that go through two
arbitrary points unless the points are exactly two radii apart. There is
just not enough information available to resolve the matter no matter how
many circles and lines you draw.
/\/\/\/*=Martin

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> > Let's try this:
> > 1.    Draw a line through the 2 points
> > 2.    Where the line intersects the circle, draw a circle around each
> point
> > 3.    Draw a line through the intersection of these circles
> > 4.    Where this line intersects the 1st circle, draw a circle around
each
> > point
>
> Which was the first circle? The one we don't know?

Well, no one said that we don't know the circle ... We don't know it's
center ....

>
> > 5.    Draw a line between the intersections of these circles
> > 6.    Where the line in (5) and (3) intersect, is that not the center?
>
> For a given radius there are always two possible circles that go through
two
{Quote hidden}

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----- Original Message -----
To: <PICLISTMITVMA.MIT.EDU>
Sent: Thursday, January 10, 2002 4:40 PM
Subject: Re: [PIC]Find center point of circle?

{Quote hidden}

Well the original question was:
"Can someone tell me how to find the center point of a circle when given
the radius and two points which lie on the circle."
I guess I interpreted that too strictly...
/\/\/\/*=Martin

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{Quote hidden}

Actually, if you open the problem up to drawing circles of a given radius around
two arbitrary points, there are four possibilities:

1. The two points are the same, so both circles intersect at all of their
points.
2. The two points are twice the radius apart, so the circles intersect at one
point.
3. The two points are unique and less than twice the radius apart, so the
circles intersect at two points.
4. The two points are greater twice the radius apart, so the circles don't
intersect.

The way the original problem was stated, the two points are on a circle of a
given radius.  That allows for case 1, 2, or 3.

If it's case 1, you can't solve the problem.

If it's case 2, the center of the original circle is the point of intersection
between the two circles.

Case 3 is the interesting one.  I haven't come up with a formal proof, but I
believe that the center of the original circle is the point of intersection
closest to the line through your two initial points.

-Mike

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But ... the two points of intersection are always going to be the same distance from the line joining the two points we were given.

Now if we had 3 points there could be no doubt - except in the case you mentioned where all given points are in the same place

>>> mdippersHARRIS.COM 01/10/02 04:11PM >>>
<snip>

3. The two points are unique and less than twice the radius apart, so the
circles intersect at two points.

<snip>

Case 3 is the interesting one.  I haven't come up with a formal proof, but I
believe that the center of the original circle is the point of intersection
closest to the line through your two initial points.

-Mike

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> From: Dipperstein, Michael
>
> Case 3 is the interesting one.  I haven't come up with a
> formal proof, but I believe that the center of the original
> circle is the point of intersection closest to the line
> through your two initial points.
>
> -Mike

Forget that.  I just came up with a class of counter examples.

-Mike

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On Thu, 10 Jan 2002, Dipperstein, Michael wrote:

> Actually, if you open the problem up to drawing circles of a given radius around
> two arbitrary points, there are four possibilities:
>
> 1. The two points are the same, so both circles intersect at all of their
> points.

> If it's case 1, you can't solve the problem.

Sure you can.  Draw a circle with radius r around the point. It will
intersect the original circle at two points.  Connect these two points
with a line l.  Now draw a line m perpendicular to line l at its midpoint.
That will be a diameter line, simple to find the center from there
(thanks, Fr.Tom!!)

Dale
(so easy to see farther when you can stand on someone's shoulders!)

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Hi all,

As an aside, is there a standard way of doing this in machining? In other
words, I have always wondered how  machinists find the exact center of a
part of circular cross-section, considering that the cross-section may only
be approximately circular. I have always done this by measuring across the
piece to find the longest line segment that I could across it (a diameter
rather than a general chord), and then measure half way across that, or
find two diameters and find their intersection, or draw a circle of the
same radius on paper with a compass, make a hole where the compass point
was, lay the circle on top of the part, and mark through the hole. However,
I was never sure if there was a better way to do it.

Sean

At 04:50 PM 1/10/02 -0500, you wrote:
{Quote hidden}

----------------------------------------------------
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> As an aside, is there a standard way of doing this in machining?

Yes, use a combination square with a center head.

www.suncoasttools.com/shop/CombinationSquare-Protractor-CenterGage-Ru
le.html

The, "V" shaped, center head mounts on the rule so that the edge of the rule
runs dead center in the "V".

Here's a really nice one with the head installed.
http://www.starrett.com/squares.htm

Paul

=========================================
Paul Hutchinson
Chief Engineer
Maximum Inc., 30 Samuel Barnet Blvd.
New Bedford, MA 02745
phutchinsonimtra.com
http://www.maximum-inc.com
=========================================

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I stand corrected.  I PRESUMED the circle has been drawn already!

Chris A

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I'm sorry, I left out that the direction is also know.  In either of the two
possibilities, the direction is also know.

{Original Message removed}
In that case a trig solution would go like this:

1. Draw a line between the two known points.
2. From the midpoint of this line, draw a perpendicular line in the known
direction of the circle's center point.  The endpoint of this line segment
will be the center of the circle.  Let's call it's unknown  length A.
3. A right triangle is now defined by the line segment you just drew (of
unknown length A.), half of the line segment drawn in step 1 of known
length, let's call B (B is half the distance beween the 2 original points)
and finally the hypotenuse is R.
4. For a right triangle, A^2 + B^2 = C^2 or in our case A^2 + B^2 = R^2,
solving for R yields: A = sqrt(R^2 - B^2)
5. Now you have the distance to draw the line segment described in step 2 to
the center of the circle.

You didn't mention what type of coordinate system you were using, but doing
this in either a rectangular or polar coordinate system is fairly
straightforward.

Kirk

> {Original Message removed}
The centre is equidistant from both points on the circumference; just draw
two arcs, one each from point; the intersection willgive u the centre. Note
that there would be two of these, one on either side of the imaginary line
drawn thru' the 2 points on the circle

anand

{Original Message removed}
> Can someone tell me how to find the center point of a circle
> when given the radius and two points which lie on the circle.

Recall that two points and a radius are defined by two
circles -- mirror images -- on either side of the chord
line connecting the two given points.  Further data is
needed to decide which circle is the desired one.

method 1
--------

1) set a compass (not magnetic, point & pencil type) to the
exact given radius of the circle

2) put the compass' pivot on either given point, draw a half
circle arc on the side towards the other given point

3) put the compass' pivot on the other given point, draw an
arc on the side towards step 2's arc -- two intersections
are the centers of the two possible desired circles.

method 2
--------

1) draw a straight line through the two points

2) set a compass (not magnetic, point & pencil type) to a
reasonable distance, usually slightly larger than the
chord generated in step 1

3) put the pivot on either given point, draw a half circle
arc on the side towards the other point

4) put the compass' pivot on the other given point, draw a
second half circle arc on side towards step 3's arc

5) the arcs in steps 3 & 4 intersect in two places; use the
straight edge to draw a line through these two intersection
points -- this line bisects the chord (which may be handy)

6) set the compass for the exact radius of the circle

7) put the compass' pivot on either given point, draw an arc
which intersects the chord bisector line -- intersection
is the center of the desired circle.

Or use both methods together -- skew between the "answers"
gives you a measurement of your error in finding the center.

Lee Jones

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> As an aside, is there a standard way of doing this in machining? In other
> words, I have always wondered how  machinists find the exact center of a
> part of circular cross-section, considering that the cross-section may
only
> be approximately circular. I have always done this by measuring across the

I don't know the name of the tool but it's essentially like a "T" square
except that the "T" is a "V" shape. The ruler comes out of the inside of the
"V". You scribe a line across the end of the piece, move it around roughly
90 degrees and scribe another line. If your stock is round the two lines
cross at the center.

Tim

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>Case 3 is the interesting one.  I haven't come up with a formal proof, but
I
>believe that the center of the original circle is the point of intersection
>closest to the line through your two initial points.

Trouble is there are 2 circles that still correspond to this definition -
consider the symbol for a current generator, the line you talk of is between
the intersection points of the two circles. Now is it the right hand or left
hand (or upper or lower depending on how its drawn) that is the original
circle?

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>Let's try this:
>1.    Draw a line through the 2 points
>2.    Where the line intersects the circle, draw a circle around each
>point
^^^ What circle ?
>3.    Draw a line through the intersection of these circles
>4.    Where this line intersects the 1st circle, draw a circle around
>each
>point
>5.    Draw a line between the intersections of these circles
>6.    Where the line in (5) and (3) intersect, is that not the center?

Graphically you use a compass, set it to the known radius value using a
ruler, then apply it on each point and draw a circle. The two circles
intersect in two points which are the centers of the two possible circles
you are looking for. But I suspect that this is not what he wants...

Peter

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There are two circles that satisfy your conditions. The centers of both
circles are on a line that cuts the middle of the distance between the
tangent points at right angles. To find the centers apply the hypothenuse
theorem to the radius (is hyp) and half the distance between the points (a
cathete), to find the distance to the center(s). The 'side' on which they
are is unedfined (+/-sqrt(0.25*distance^2+r^2)). Then you probably want to
adapt this to the actual coordinate system you are using (cartesian,
bitmap etc).

Peter

PS: Are you in an exam ? ;-)

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> Hi all,
>
> As an aside, is there a standard way of doing this in machining? In
> other
> words, I have always wondered how  machinists find the exact center of a

One way is to use a steel ruler and a scribing needle to find large
diameters as you suggest and scribe a line for each such found diameter.
After 20 or so scribeds line the center will be pretty easy to see with
the bare eye. This is not very accurate.

Another way involves a compass with steel needles. It is set to slightly
more than half the diameter and used to scribe arcs near the presumed
center with a point on the rim (you can use a tight fitting tube around
the piece to keep the outer point from slipping off). You scribe 3 arcs
about 120 degrees apart, then take 3 positions between the previous 3 120
degree arc's centers and radius the middle of the previously traced
triangle.  This results in diminishing triangles being traced near the
center of the piece. The center is in the middle of the smallest one.
Usually you find it to 1% in 9 scribed arcs.  This also works with long
strings and over long distances by other means (radio df probably). It
works because the radius picked in the 'next' try is an improvement vs.
the previous estimate.

Peter

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> I'm sorry, I left out that the direction is also know.  In either of the
> two possibilities, the direction is also know.

Ah. Is the radius pretty large wrt. the distance between the points ? ;-)

Peter

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The chord of the two points can be small or as large as twice the radius(for
an arc with 180 swing).

{Original Message removed}

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