Curve fitting by curvature

M

MarkRaz2

New member
Joined
Jun 15, 2022
Messages
7
Hi, I found this example below on how to fit a curve by points and derivatives. Is it possible to something similar if If I have some points and curvature k? or if I know the osculating circle? y/n

thanks in advance

1655348125818.png
1655348074221.png

1655348155931.png
 
The system the book is describing is known as Ordinary Least Square (OLS), more commonly known as Multiple Linear Regression (MLR).
[math]\hat{y}=X\beta+\epsilon[/math]
As the name suggests, the OLS seeks the least square. In other words, minimize the residual error [imath]\epsilon[/imath] of the curve, so it doesn't take the curvature into account.
However, you can compare the fitted and the known curvature simply by looking at their ratios. If it's close to 1, it indicates a good fit.
 
MarkRaz2 said:
Hi, I found this example below on how to fit a curve by points and derivatives. Is it possible to something similar if If I have some points and curvature k? or if I know the osculating circle? y/n

thanks in advance

View attachment 33107
View attachment 33106

View attachment 33108
Click to expand...
Do you mean you have curvatures instead of second derivatives or in addition to them?
Since curvature can be expressed through the first two derivatives you can, conversely, express the second derivatives through the first one and the curvature and thus reduce the problem to the one you've posted.
 
blamocur said:
Do you mean you have curvatures instead of second derivatives or in addition to them?
Since curvature can be expressed through the first two derivatives you can, conversely, express the second derivatives through the first one and the curvature and thus reduce the problem to the one you've posted.
Click to expand...
Hi thanks, I am trying to find "g2" continuity:

1655419908951.png

according to this http://www.viewmold.com/ug_html_files/modeling/apx_continuity.html

it states:

"G1 implies that the tangent vectors are equal in direction, but not magnitude. G2 implies the curvature is the same, but the second derivatives are not."

So I guess I'm asking if you can fit 2 polynomials using least squares based on their curvature ?? in addition to 1st and second. Since as you know you need 1st and second to find curvature.

1655420123926.png


thanks appreciate it
 
BigBeachBanana said:
The system the book is describing is known as Ordinary Least Square (OLS), more commonly known as Multiple Linear Regression (MLR).
[math]\hat{y}=X\beta+\epsilon[/math]
As the name suggests, the OLS seeks the least square. In other words, minimize the residual error [imath]\epsilon[/imath] of the curve, so it doesn't take the curvature into account.
However, you can compare the fitted and the known curvature simply by looking at their ratios. If it's close to 1, it indicates a good fit.
Click to expand...

thanks for the reply, what do you mean by "ratios"

thanks
 
blamocur said:
Do you mean you have curvatures instead of second derivatives or in addition to them?
Since curvature can be expressed through the first two derivatives you can, conversely, express the second derivatives through the first one and the curvature and thus reduce the problem to the one you've posted.
Click to expand...
Just realized that my previous post is not really correct because in this thread the derivatives are given in different points, i.e. [imath]x_{2,i}[/imath] for the 1st and [imath]x_{3,j}[/imath] for the second.

To make sure I understand the problem you are trying to solve: you are given 4 sets of [imath]x[/imath]'s, one for which the values are defined, another with defined first derivatives, then the set with defined second derivatives and the fourth set for which curvatures are defined. And you need to find a polynomial (3rd degree? 4th degree?) which minimizes sum of squared errors. Is this correct?
 
You must log in or register to reply here.
Top