Can Manifold Angle Determine Concavity and Convexity of a Curve?

[sniper]

I have a small query to make regarding concavity and convexity...I have three consecutive points on a curve.and given the angle between the lines joining 1st and 2nd point,2nd and 3rd point respectively.I am supposed to define a function which accounts for concavity and convexity of the curve.It should be a linear function of the angle that is calculated

i tried using equation of the line passing through 1st and 3rd point and had one arbitrary point inside the curve.My logic was that,based on the sign of the result we get by substituting the point number 2 and the arbitrary point,i can determine the concavity and convexity.But this seems to fail in my case as there are points which are close to each other,so,this logic is not able to differentiate properly between a concave and convex side.

instead,i thought of using the manifold angle which was calculated as mentioned in the first few sentences,as the basis to determine the nature of the curve.just wanted to know if this is feasible or not...the function should assume a high value in convex region and small value in concave regions.and one more function to be defined such that it assumes a positive value in concave regions and negative in convex...both should be functions of manifold angle...please help me in sorting this...at least give me an idea of how i can go about it...

 

[fresh_42]

If we use the cross product of the direction vectors of the two lines, we get an oriented angle, i.e. its sine value. This allows to distinguishes between concave and convex: greater than 90° or less than 90°. But this isn't linear in the angle. Linearity is in my opinion not necessary and makes geometrically no sense. What should an addition of angles stand for with respect to the curvature? Of course we can add the projection on the sign of the sine value we get, so the outcome is directly concave or convex. But additivity makes still no sense.