pcap
Oct31-09, 01:57 AM
Hello,
I am trying to understand how to reparameterize a Hermite curve described by the parametric vector function \vec{P}(t) to a curve described by \vec{Q}(T) where T = at + b. In particular, I am having trouble finding the derivative of the reparameterized curve.
We know T_i = at_{i} + b and T_j = at_j + b. We also know, \frac{dT}{dt} = a.
The source (http://books.google.com/books?id=m0Je92uycVAC&pg=PA128&lpg=PA128&dq=hermite+curve+parameter&source=bl&ots=-RGIDOdZEw&sig=oNi3c0qmm6cWF63ntAC7wf8ws5Q&hl=en&ei=9q_rSszGOonYtgOxoLX1Aw&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBUQ6AEwBQ#v=onepage&q=hermite%20curve%20parameter&f=false) I am looking at arrives at the following equation:
\frac{d\textbf{Q}(T)}{dT} = \frac{d\textbf{P}(t)}{dt} \frac{dt}{dT}
I do not understand how they arrived at this derivative, so I would appreciate any insight into this.
My thinking is a bit foggy now, so hopefully some rest will help. At any rate, I can provide more clarification as needed. Thanks!
I am trying to understand how to reparameterize a Hermite curve described by the parametric vector function \vec{P}(t) to a curve described by \vec{Q}(T) where T = at + b. In particular, I am having trouble finding the derivative of the reparameterized curve.
We know T_i = at_{i} + b and T_j = at_j + b. We also know, \frac{dT}{dt} = a.
The source (http://books.google.com/books?id=m0Je92uycVAC&pg=PA128&lpg=PA128&dq=hermite+curve+parameter&source=bl&ots=-RGIDOdZEw&sig=oNi3c0qmm6cWF63ntAC7wf8ws5Q&hl=en&ei=9q_rSszGOonYtgOxoLX1Aw&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBUQ6AEwBQ#v=onepage&q=hermite%20curve%20parameter&f=false) I am looking at arrives at the following equation:
\frac{d\textbf{Q}(T)}{dT} = \frac{d\textbf{P}(t)}{dt} \frac{dt}{dT}
I do not understand how they arrived at this derivative, so I would appreciate any insight into this.
My thinking is a bit foggy now, so hopefully some rest will help. At any rate, I can provide more clarification as needed. Thanks!