first of all many thanks for your time and efforts!
I would like to ask about how to implement the known end slopes boundary conditions into the curve fitting using the least squares.
Suppose I have a collection of points and I would like to best fit a Bezier curve of a given degree to these points. Suppose, also, I would like to treat the first and the last point as actual first and last control points. Starting from the error definition in matrix notation
E(C) = (P - TMC)^T (P - TMC)
d(E(C))/dC = -2(TM)^T (P - TMC) <--- your equation lacks M^T
after rearrangement and abbreviating TM as b, we get
b^T bC = b^T P.
Let's define b^T b => B and b^T P => A, such that we now have
BC = A.
If C(1) and C(end) are known, we get
B(2:end-1, 2:end-1)C(2:end-1) = A(2:end-1) - C(1)B(2:end-1,1) - C(end)B(2:end-1,end)
and finally
C(2:end-1) = B(2:end-1, 2:end-1)C(2:end-1)^-1 [A(2:end-1) - C(1)B(:,1) - C(end)B(:,end)]
This works like a charm. However, I don't understand how to derive/implement the slope conditions (let say in the form of unit tangents) at the first and last points...
Would really appreciate your thoughts on this.
Victor
victorsalit
commented
1 year ago
Hi Pomax,
first of all many thanks for your time and efforts! I would like to ask about how to implement the known end slopes boundary conditions into the curve fitting using the least squares. Suppose I have a collection of points and I would like to best fit a Bezier curve of a given degree to these points. Suppose, also, I would like to treat the first and the last point as actual first and last control points. Starting from the error definition in matrix notation E(C) = (P - TMC)^T (P - TMC) d(E(C))/dC = -2(TM)^T (P - TMC) <--- your equation lacks M^T after rearrangement and abbreviating TM as b, we get b^T bC = b^T P. Let's define b^T b => B and b^T P => A, such that we now have BC = A. If C(1) and C(end) are known, we get B(2:end-1, 2:end-1)C(2:end-1) = A(2:end-1) - C(1)B(2:end-1,1) - C(end)B(2:end-1,end) and finally C(2:end-1) = B(2:end-1, 2:end-1)C(2:end-1)^-1 [A(2:end-1) - C(1)B(:,1) - C(end)B(:,end)] This works like a charm. However, I don't understand how to derive/implement the slope conditions (let say in the form of unit tangents) at the first and last points...
Would really appreciate your thoughts on this. Victor