Any mathematicians out there?

I know what a radius of a circle or a sphere is; quite simple. But what exactly does "radius of curvature" refer?

I'm trying to reconcile two technical translations of a machine part that seem to refer to the same thing, but the original foreign language texts use different expressions, resulting in the English language expressions "radius of curvature" and "radius of an arc of a circle." I need to make sure one of the texts is not erroneus so that I do not approve of the English text that is wrong. That is, I want to make sure one is not describing something differently, such as more broadly, more narrowly, e.g., and therefore erroneously.

In short, the Wikipedia entries (particularly for "radius of curvature") seem to be written by professional mathematicians and after reading them, I still don't know.

FWIW: Read the first complete sentence on this page:

http://books.google.com/books?id=GGBv_Gcy5g8C&pg=PA446&lpg=PA446&dq=%22radius+of+curvature%22+vs.+%22radius+of+a+circular+arc%22&source=bl&ots=Sxt1-QpHnk&sig=clT6OTE2uNXl688PqALlR_Lazt8&hl=en&ei=c2T8Ssu0BYrflAeVoumdBQ&sa=X&oi=book_result&ct=result&resnum=3&ved=0CA0Q6AEwAg#v=onepage&q=&f=false


The slightly different meaning of the two phrases can be inferred from the way they're used/defined in that sentence.

This may help you:
Attachments:

Mister Magoo Wrote:
-------------------------------------------------------
> FWIW: Read the first complete sentence on this
> page:
>
> http://books.google.com/books?id=GGBv_Gcy5g8C&pg=P
> A446&lpg=PA446&dq=%22radius+of+curvature%22+vs.+%2
> 2radius+of+a+circular+arc%22&source=bl&ots=Sxt1-Qp
> Hnk&sig=clT6OTE2uNXl688PqALlR_Lazt8&hl=en&ei=c2T8S
> su0BYrflAeVoumdBQ&sa=X&oi=book_result&ct=result&re
> snum=3&ved=0CA0Q6AEwAg#v=onepage&q=&f=false
>
>
> The slightly different meaning of the two phrases
> can be inferred from the way they're used/defined
> in that sentence.


Thanks, Magoo. I think I understand now. The diagrams are helpful.

What seems confusing to me is that the expression "radius of curvature" is said to refer to "a" radius of a "point" on the curve. Unless the curvature is constant (i.e., a circle), then it would appear "the" radius of the curve changes as one moves along the curve, yet texts refer to "the" radius rather than to the radii (radiuses) of the curve.

NoEinstein Wrote:
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> ...yet texts refer to "the" radius rather than to the radii (radiuses)
> of the curve.


"The radius of curvature of a curve at ANY point is defined as the radius of a
circular arc that best fits the curve at THAT point."

Based on this definition, it does appear that the radius of curvature (i.e., the
radius of a circular arc that best fits the curve at THAT point) could indeed
change as you move along the curve.

Accordingly, I don't think the limiting "the" should be understood to imply that
there are no other possible radiuses of curvature at other points along the curve.

Rather, I think such language is simply referring to a specific radius of curvature,
i.e., the one that's relevant for purposes of the issue being addressed.

Again, the FWIW caveat applies here.

Mister Magoo Wrote:
-------------------------------------------------------
> NoEinstein Wrote:
> --------------------------------------------------
> -----
> > ...yet texts refer to "the" radius rather than
> to the radii (radiuses)
> > of the curve.
>
>
> "The radius of curvature of a curve at ANY point
> is defined as the radius of a
> circular arc that best fits the curve at THAT
> point."
>
> Based on this definition, it does appear that the
> radius of curvature (i.e., the
> radius of a circular arc that best fits the curve
> at THAT point) could indeed
> change as you move along the curve.
>
> Accordingly, I don't think the limiting "the"
> should be understood to imply that
> there are no other possible radiuses of curvature
> at other points along the curve.
>
> Rather, I think such language is simply referring
> to a specific radius of curvature,
> i.e., the one that's relevant for purposes of the
> issue being addressed.
>
> Again, the FWIW caveat applies here.


Thanks again.

You're welcome.

And to think, some people around here call me a troll.

Heaven forfend.

In this forum, it seems to me almost everybody is a troll, i.e., either a registered troll or a non-registered troll.

The poor naive saps who just happen upon this forum to try to get a serious answer or to raise a serious issue (are there any any more? - seem like everyone is a regular to me) might be characterized as "not a troll."

mister magoo, you bastard!

(rips off Magoo mask, whips out shotgun and cocks it)

Who you calling a bastard, son?

Einstein's Penis Wrote:
-------------------------------------------------------
> This may help you:


This is the closest any mathematician has ever been to a woman with breasts that large.

---------------------------------------------------------------------------------
http://bible.cc/1_corinthians/13-11.htm

The Joker Wrote:
-------------------------------------------------------
> (rips off Magoo mask, whips out shotgun and cocks
> it)
>
> Who you calling a bastard, son?


tee hee.

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bloody blisters Wrote:
-------------------------------------------------------
> your addvertisement style sucks.
>
> my family died thanks to you.


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The Joker Wrote:
-------------------------------------------------------
> The ad worked. You bought the product.
>
> Admittedly our exclusive 100% natural organic
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> cartilage, coral calcium, cesium, ellagic acid,
> Cat's Claw, Essiac, and Agaricus Blazeii, among
> other proprietary ingredients) is ONLY effective
> on 98% of cancers.
>
> But no one ever claimed it was 100% effective, or
> certainly not in writing!
>
> In any event, I'm sorry for your loss (all
> purchases are final).

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Harry locked his mother in the closet.

Harold. Please. Not again the TV.

Okay, okay. Harry opened the door, then stop playin games with my head.

He started walking across the room toward the television set.

And dont bug me.

He yanked the plug out of the socket and disconnected the rabbit ears.

Sara went back into the closet and closed the door.

Harry stared at the closet for a moment.

So okay, stay.

He started to push the set, on its stand, when it stopped with a jerk, the set
almost falling.

What the hells goin on here?

He looked down and saw a bicycle chain going from a steel eye on the side of the
set to the radiator. He stared at the closet.

Whatta ya tryin to do, eh? Whats with this chain? You trying to get me to break
my own mothers set? to break the radiator? — she sat mutely on the closet floor —
an maybe blow up the whole house? You tryin to make me a killer? Your own son?
your own flesh and blood?

WHATTA YA DOIN TA ME????

Harry was standing in front of the closet.

YOUR OWN SON!!!!

A thin key slowly peeked out from under the closet door.

Harry worked it out with his fingernail then yanked it up.

Why do you always gotta play games with my head for krists sake, always laying
some heavy guilt shit on me? Dont you have any consideration for my feelings?
Why do you haveta make my life so difficult?

Why do—Harold, I wouldnt. The chain isnt for you. The robbers.

Then why didnt you tell me? The set almost fell. I coulda had a heart attack.

Sara was shaking her head in the darkness. You should be well Harold.

Then why wont you come out?

Harry tugging on the door and rattling the knob, but it was locked on the
inside. Harry threw his hands up in despair and disgust.

See what I mean? See how you always gotta upset me?

He walked back to the set and unlocked the chain, then turned back to the closet.

Why do you haveta make such a big deal outta this? eh? Just ta lay that guilt
shit on me, right? Right???? — Sara continued rocking back and forth — you know
youll have the set back in a couple a hours but ya gotta make me feel guilty.

He continued to look at the closet — Sara silent and rocking — then threw up his
hands, Eh, screw it, and pushed the set, carefully, out of the apartment.

Mr. Powell,
What is that? Requiem for a Dream?

Yes.

I should think a 5-Tuttle book in anyone's estimation (though only a 3.08-Tuttle movie, imho).