I am trying to express the segment of a curve in terms of $t$. For example, for a straight line between $(1,1)$ and $(2,2)$, I can express it like:
$$ (x,y) = (1,1) + t (1,1), \space 0\le t \le 1 $$
My question is how do I do the same for a curve? (of say a certain curvature, start point and end point).
Thanks.
I'm a little uncertain about what you mean by "radius" (is your curve necessarily a circle?), but the general method is to define a curve $\gamma:[a,b]\to\mathbb{R}^2$ by $$ \gamma(t) = (x(t), y(t)).$$ For example, $(x,y) = (\cos t, \sin t)$ with $t\in[0,2\pi]$ is a circle; $(x,y) = (t^2, t)$ for $t\in\mathbb{R}$ is an (inverse) parabola, and so on. If you want a circle of a given radius $r$ about a center point $(h,k)$, we multiply to increase the size of the "radius vector" and translate the curve (by adding vectors): $$(x,y) = (h,k)+r\,(\cos t, \sin t)\quad 0\leq t\leq 2\pi.$$