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How does one measure the smoothness of a curve?

I have a curve defined by (x,y) points on a Cartesian grid.

I have applied a couple different algorithms to smooth the curve.

(code written in Java fwiw)

Now I want to measure the smoothness of the curves.

I have a curve defined by (x,y) points on a Cartesian grid.

I have applied a couple different algorithms to smooth the curve.

(code written in Java fwiw)

Now I want to measure the smoothness of the curves.

Of course is you could take an average of all your y values and declare your smoothed curve to be Y(x) = average. You would have a very smooth curve, which in addition has zero slope.

I doubt very much that that is what you are looking for. Hence the importance of answering my first post.

For each point (xi,yi), calculate the tangent line and get the slope of that line.

If the difference between the greatest and least slopes is > some value, then its not very smooth.

This involves derivates, and it may be saying what mandelia is saying in a different way.

(1,1)

(2,2)

(12,13)

(11,11)

(30,30)

(31,31)

to be just as smooth as or the same curve as

(1,1)

(6,6)

(11,11)

(16,16)

(21,21)

?

Would

(1,1)(2,2)(3,1)(4,2)

have the same smoothness as

(1,1)(2,2)(1,3)(2,4)

If you have a collection of discrete points, a unique tangent line to a single point does NOT exist. You can calculate a tangent line IF you use the points on either side of the point of interest.

Again define what you mean by smoothness.

Discrete Points are not lines. So there is no concept of smoothness there. Lines/curves are represented by exations.

There are infinite number of ways i can draw lines connecting your points. And of varying smoothness

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ie smoothest curve is a straight line : x = my + c

it is more smooth than a curve where dx/dy = c

which is more smooth than d2x/dy2 = c