I have a 1-dimensional curve (e.g. Gaussian). What is a measure of its sharpness, i.e. whether the area under it is more concentrated around the center or more 'widespread'? I would like to have a universal measure which is independent of the exact shape.
2007-07-01 23:42:57 · 5 answers · asked by jmatejic 2 in Science & Mathematics ➔ Mathematics
I have a 1-dimensional curve (e.g. Gaussian). What is a measure of its sharpness, i.e. whether the area under it is more concentrated around the center or more 'widespread'? I would like to have a universal _integral_ measure which is independent of the exact shape.
2007-07-02 08:58:02 · update #1
This is differential geometry stuff, and you need to learn a little about curvature, torsion, etc.
The second derivative's norm (euclidean, say) of the curve in parametric form could help you, but you need to get into this stuff to fully understand what is going on.
Regards
Tonio
2007-07-01 23:56:41 · answer #1 · answered by Bertrando 4 · 0⤊ 1⤋
Draw two tangents from the two straight ends of the curve. The measure at the meeting points of these tangents is a measure of the curve.
2007-07-07 14:43:37 · answer #2 · answered by Joymash 6 · 0⤊ 0⤋
The absolute value of the second derivative can usually be considered to be the measure of "sharpness".
2007-07-01 23:50:43 · answer #3 · answered by Helmut 7 · 0⤊ 1⤋
sharpness of a curvature is measured by the radius of curvature
defined by
K = y ' ' / ( 1 + y ' ^2 )^ 3/2 . . . where y ' = dy/dx, . . y ' ' = d^2/dx^2
2007-07-01 23:57:15 · answer #4 · answered by CPUcate 6 · 0⤊ 1⤋
Mean +/- standard deviation, would it not help ?
2007-07-08 19:24:09 · answer #5 · answered by Snoopy 3 · 0⤊ 0⤋