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#1 2007-02-25 03:55:22

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

2nd deriv of circle versus parabola

Now I was hoping that
the second derivative
of a circle was going
to be a constant, but
I was wrong.  I guess
it is the parabola.  I
can begin to guess
this is because slope
of vertical is infinity
and slope of 45 degrees
is one, and slope of
horizontal is zero, so the
x-y system makes the
parabola's 2nd derivative
simpler than that of
a circle.  Any comments?
Help me learn something?


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#2 2007-02-25 08:34:15

John E. Franklin
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Registered: 2005-08-29
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Re: 2nd deriv of circle versus parabola

What I am getting at
is that the second
derivative is typically
thought of as describing
the way a curve curves.
Like if it curves downward
or upward, then the
2nd derivative is negative
and positive, respectively.
However, it is
important to note
that the curvature of
a circle is very uniform,
however the 2nd
derivative changes on
this curve, which tells
us perhaps that if the
slope is very large,
then the second derivative
might have to be large
too to make a dent in
the curving.  But if you
are close to the horizontal,
and the curve curves upward
a bit, then maybe the
2nd derivative there is not
all that big.  Anyone know??


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#3 2007-02-25 12:51:36

JaneFairfax
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Registered: 2007-02-23
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Re: 2nd deriv of circle versus parabola

The second derivative does not describe the curvature of a curve; rather it describes the rate of at which the tangent to the curve is changing with respect to the x-axis.

The concept of curvature is different: it is defined as

                   

Yes, a circle does have a constant* curvature (

) equal to the reciprocal of its radius. A circle, however, does not have a constant second derivative.

*Constant apart from sign.

Last edited by JaneFairfax (2008-07-09 22:29:19)

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#4 2007-02-25 15:48:58

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Re: 2nd deriv of circle versus parabola

So wickid!! You're right, I tried 67.5 degrees, 78.75 degrees, and 0.1 degrees, and
the first two came out exactly 1/r for kappa curvature, and the last came out 1.000005/r or something (rounding error on calculator or me typing it.

edit addition below later.
tiny error, I actually was getting minus one over r, positive 1/r.
I was so elated how it was cancelling out, I forgot the minus sign.
But while reading on the subject, I see if it curves clockwise, it's minus.
And counter-clockwise is positive.  And I also skimmed a little about torsion as
well in 3-D, but this turned me off because now torsion (Tau), is separate from
curvature (kappa), and the symmetry is all messed up.  It's a shame, we can
see things from such retarded points of view, when we know deep down, it is more magical and symmetrical than that, but we don't know how to describe the 3-D curve with only one variable, we think we need two, which is our first mistake.

Last edited by John E. Franklin (2007-02-26 09:34:39)


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#5 2007-02-25 16:01:41

John E. Franklin
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Re: 2nd deriv of circle versus parabola

Out of interest for the second derivative still, here are some values I computed before I saw the kappa thing you posted.

Last edited by John E. Franklin (2007-02-28 08:09:45)


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#6 2007-02-25 17:46:17

JaneFairfax
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Registered: 2007-02-23
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Re: 2nd deriv of circle versus parabola

Well, actually curvature is defined as

where ψ is the angle the tangent makes with the x-axis (i.e. tan ψ = dy/dx) and s is the arc length of the curve. In other words, curvature is the rate of change of the direction of the tangent with respect to arc length.

On the other hand, the second derivative is rate of change of the tangent itself (i.e. dy/dx) with respect to x. Hence the two are different. Hope this makes sense. tongue

Last edited by JaneFairfax (2007-03-13 05:45:42)

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#7 2007-02-26 02:20:14

Dross
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Registered: 2006-08-24
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Re: 2nd deriv of circle versus parabola

The second derivative of a circle is constant, if only you use polar coordinates:


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#8 2007-02-26 03:12:04

JaneFairfax
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Re: 2nd deriv of circle versus parabola

Hey, I didn’t think of polar co-ordinates! That’s neat. up

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#9 2007-02-26 09:34:13

John E. Franklin
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Re: 2nd deriv of circle versus parabola

See my editting addition in post # 4 if want to.


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#10 2007-02-26 10:24:36

JaneFairfax
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Re: 2nd deriv of circle versus parabola

John E. Franklin wrote:

tiny error, I actually was getting minus one over r, positive 1/r.
I was so elated how it was cancelling out, I forgot the minus sign.
But while reading on the subject, I see if it curves clockwise, it's minus.
And counter-clockwise is positive.

Yes, the sign of

depends of that of the second derivative (with respect to x). For the circle, this is positive below the horizontal axis and negative above it (i.e. the tangent is increasing on the lower semicircle and decreasing on the upper semicircle).

If you like, you can consider the related concept of radius of curvature instead. This is defined as the reciprocal of the absolute value of

and will always be positive.

Last edited by JaneFairfax (2007-02-26 10:25:48)

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#11 2007-02-26 10:48:43

John E. Franklin
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Re: 2nd deriv of circle versus parabola

Yes, I think I would prefer to talk about the denominator of kappa.
But then when the curve is straight, we have the problem of mentioning
infinity!!, ikes!! Get in trouble with some people around here.  And zero is
of course no problem.


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#12 2007-02-26 14:21:15

JaneFairfax
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Re: 2nd deriv of circle versus parabola

That’s right, a straight line has zero curvature. In other words, a straight line does not “curve”! wink

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#13 2007-02-26 15:08:21

John E. Franklin
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Re: 2nd deriv of circle versus parabola

What I meant was, when you talk about kappa to someone, you could say "kappa is one over six".
Kappa is one over 300.
kappa is one over 2.5.
kappa is one over zero, woops, what did I say?
You know, why can't people just stop getting so hung up on these undefined things?
So it's infinity, or something close, or you'd rather not say, so what?


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#14 2007-02-27 03:48:19

Dross
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Re: 2nd deriv of circle versus parabola

If you were being all "proper" and such, you'd probably just state that there was no radius of curvature. Of course, if you said the radius of curvature was infinite, everyone would know what you really meant...


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#15 2007-02-27 04:56:03

John E. Franklin
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Re: 2nd deriv of circle versus parabola

Very diplomatic and logical in the midsts of an emotional comment, thank you.


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#16 2007-02-27 05:13:28

JaneFairfax
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Re: 2nd deriv of circle versus parabola

I would think of a straight line as having infinite (rather than zero) radius of curvature. On the other hand, I would regard a point of inflexion (where the curvature is also 0) as having zero radius of curvature. This is just based on my intuitive geometric visualization, of course.

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#17 2007-02-27 05:24:41

Dross
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Re: 2nd deriv of circle versus parabola

JaneFairfax wrote:

On the other hand, I would regard a point of inflexion (where the curvature is also 0) as having zero radius of curvature.

I wouldn't say so - I instinctively envisage the radius of curvature as a circle with an edge on the curve and corresponding radius. As the circle approaches the point of inflection from one side, it will grow and grow, expanding to a straight line (in my mind, anyway tongue) as it hits the point of inflection, and then appearing as a circle of decreacing radius when it's on the other side of the point of inflection.

I would think of something that has a zero radius of curvature as a corner (and hence non-differentiable) rather than a straight line/point of inflection.

I think this explains what I mean - I'll do some diagrams when I get home to illustrate how I see it. I agree it does depend on how you visualise it, though.


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#18 2007-02-27 05:26:26

John E. Franklin
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Registered: 2005-08-29
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Re: 2nd deriv of circle versus parabola

Yes, like a saw-tooth wave, go up at 45, hit inflexion point, turn 90 degrees clockwise, go down at a 45.
Now I like the first formula you posted for kappa with the 2nd deriv in numerator and the 3/2's power and first deriv in denominator.  I should probably try out that formula somemore soon on some curves and post the resulting graphs.  And if anyone knows of some equations with inflexion points in them, post those equations, so I can try to graph them.  Post your first and second derivatives, if they are too hard for me to calculate, bear in mind, my calculus is too rusty, but I could use pointers and links to lessons on calculus.
Off the subject a bit, parametric equations, I'd like any nice links to read on that that are better than average.


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#19 2007-02-27 05:51:28

JaneFairfax
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Registered: 2007-02-23
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Re: 2nd deriv of circle versus parabola

Er, sorry I’m changing my mind. I would regard both the straight line and point of inflexion as having infinite radius. (I forgot that radius of curvature, unlike curvature, is always positive.) tongue

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#20 2007-02-27 08:52:30

Dross
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Posts: 325

Re: 2nd deriv of circle versus parabola

John E. Franklin wrote:

And if anyone knows of some equations with inflexion points in them, post those equations, so I can try to graph them.  Post your first and second derivatives, if they are too hard for me to calculate, bear in mind, my calculus is too rusty, but I could use pointers and links to lessons on calculus.

Well, a point of inflection is any point where the second derivative changes sign. So start with the second derivative and intigrate. As a simple example of what I mean, let:

Then we know that

must change sign at x = 0 and hence this is a point of inflection. Intigrating, you can add any constants of intigration you want:

You could also start with trigonometric functions and you'd end up with something like

with inflection points at
for any
.


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#21 2007-02-27 17:51:12

John E. Franklin
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Registered: 2005-08-29
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Re: 2nd deriv of circle versus parabola

Here is the graph of y = (0.22/6)x^3 + 0.11x + 4

Now the curvature graphs in red and green
correspond vertically with the tightest
parts of the cubic curve shown in black.

And where x = 0, the inflexion point is shown well
with infinite (green curve) radius of curvature, and zero kappa (red curve).

The green curve is not absolute valued because I like
it better that way, goes with the other data better more intuitively.

Last edited by John E. Franklin (2007-02-28 06:54:00)


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