When you're leaning against a seat cushion in an accelerating vehicle, the acceleration of the vehicle is roughly proportional to your displacement of the cushion. In such a function, you lose two derivatives. Therefore, the 3rd derivative of position, how fast you're jerked forward or backward, is roughly proportional to how fast your cushion squishes or unsquishes
I always liked to think that "curvature" itself was "how far away a curve was from being a circle" since the circle is the curve with constant curvature. Never heard of the term aberrancy before. Very nice.
Aberrancy could also reasonably have been called Lopsidedness since it’s sort of signifying how far the curve is from being symmetric about the point under consideration. But in all fairness Aberrancy is a cooler sounding word. 🙂
16:56 y = c1 x + ε seems to be sloppy! If ε is the distance between the chord and the tangent, the chord crosses the y-axis at x=0 and y = ε / sqrt(1 + c1²). That means, that the chord has the equation y = c1 x + ε / sqrt(1 + c1²) not y = c1 x + ε.
9:48 if u take b/a which is y/x you get the tangent of the angle ccw from the x axis. Theta is measured clockwise from y-axis so tangent theta should be x/y. 10:49 Anyway later on by substituting m = 0, we see that Sc = -1/A(c)
Aberrancy at a point should be how far a curve is from being symmetrical about it's normal to that point. A quadratic at it's extremum or ellipse at it's pointy end or any even function at origin also had aberrancy 0. The argument used for the circle works here too.
Kind of confusing because the tangent of theta is a/b not b/a. And if m is close to zero don’t you get A = -1/S? Seems like you need a minus sign in there.
A 3 got dropped in the end. I think what you have in the end is d/dx of the curvature. To be invariant, I think it should rather be d/ds of the curvature, where d/ds = (1+y’^2)^(-1/2) d/dx. Very interesting to see the chord interpretation!
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When you're leaning against a seat cushion in an accelerating vehicle, the acceleration of the vehicle is roughly proportional to your displacement of the cushion. In such a function, you lose two derivatives. Therefore, the 3rd derivative of position, how fast you're jerked forward or backward, is roughly proportional to how fast your cushion squishes or unsquishes
Thats why its called the jerk 😂
I always liked to think that "curvature" itself was "how far away a curve was from being a circle" since the circle is the curve with constant curvature. Never heard of the term aberrancy before. Very nice.
Aberrancy could also reasonably have been called Lopsidedness since it’s sort of signifying how far the curve is from being symmetric about the point under consideration. But in all fairness Aberrancy is a cooler sounding word. 🙂
A statistician would call this skew. The fourth derivative measures kortosis, how heavy the tails are.
16:56 y = c1 x + ε seems to be sloppy! If ε is the distance between the chord and the tangent, the chord crosses the y-axis at x=0 and y = ε / sqrt(1 + c1²). That means, that the chord has the equation y = c1 x + ε / sqrt(1 + c1²) not y = c1 x + ε.
9:48 if u take b/a which is y/x you get the tangent of the angle ccw from the x axis. Theta is measured clockwise from y-axis so tangent theta should be x/y.
10:49 Anyway later on by substituting m = 0, we see that Sc = -1/A(c)
Aberrancy at a point should be how far a curve is from being symmetrical about it's normal to that point.
A quadratic at it's extremum or ellipse at it's pointy end or any even function at origin also had aberrancy 0. The argument used for the circle works here too.
You could also talk about the third derivative test for inflection points,
The aberrancy is zero for all even functuons, so more specifically you could say it measures oddness around a point.
Just splendid sir.... thanks for your valuable learning video
The Aberrancy of Plane Curves
Russell A. Gordon
The Mathematical GaZette
Vol. 89, No. 516 (Nov., 2005), pp. 424-436 (13 pages)
2:50 I assume, you've also set f'(0) = 0 as the x-axis is the tangent to f(x) at x=0.
I know in physics we call the third derivative of displacement the jerk. :)
is it the rate of change of the curvature ?
Kind of confusing because the tangent of theta is a/b not b/a. And if m is close to zero don’t you get A = -1/S? Seems like you need a minus sign in there.
A 3 got dropped in the end. I think what you have in the end is d/dx of the curvature. To be invariant, I think it should rather be d/ds of the curvature, where d/ds = (1+y’^2)^(-1/2) d/dx.
Very interesting to see the chord interpretation!
Skewness ?
17:37 what does a negative intersection mean?
Oh ok nvm i got it. It's just means the x coordinate of the intersection is negative.
Is there an aberranncy equivalent to the circle of curvature?
I think it's some kind of spiral
14:16, skip.
I showed this to my evangelical neighbor and he told me that “Trump is going to end all this”!!!!
As far as I can tell, he thinks that MAGA is going to put an end to mathematics!!!
Cool geometric interpretation from Azerbaijan
I thought second derivative is instantaneous rate of change
That's just the derivative lol
Second derivative would be instantaneous rate of change in velocity, which is acceleration.
First derivative is rate of change. Second derivative is the rate of change of the rate of change.
Thank you for acknowledging that other countries exist :)
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29:28 I somehow calculated a different formula, instead of (1+c1^2)c3 I calculated (c1^2-1)c3
me too