I have a very basic question,
I normally do linear interpolation between 2 endpoints $a$ and $b$, where both $a$ and $b$ $\in R^n , n>10$ as below:
$ (1-t)*a + t * b $ , as $t$ moves from $0$ to $1$.
How do I modify above, to also linearly extrapolate beyond $a$ and $b$. I want to control the amount of extrapolation within some bounds..eg I may only want to extrapolate upto $2b$ or up to $0.5a$
Is it possible to do this, by just changing the range of $t$? If not, what's the right way to do this?
Thanks!
Your idea is correct. If you set
$$(1-t)\cdot a+t\cdot b, \quad t\in [0,1]$$
then when $t$ moves from $0$ to $1$, the point $D$ is moving from $B$ to $C$. It means that if you take $t>1$ then $D$ pass through $C$ and goes away from it. In the same way, if you take $t<0$, it means that $D$ goes through $B$ and goes away from it.
Maybe a better way to understand that is writing
$$(1-t)a+tb=a+t(b-a)=a+ t\cdot\vec{BC}$$
where $t\cdot\vec{BC}=\vec{BD}$.
Can you see that?