I am facing a problem of the transformation of a line to a circle curve as shown in the attached figure.
The line segment $A1Z1$ which has many points$(A1(x_A,y_A), B1(x_B,y_B),...P1(x_P,y_P),...)$ is needed to transform to a circle curve $A2Z2 (\theta,R$).
So far I do not know how to get the coordinate of the corresponding points of the circle curve.
Hope for your help and thank you very much. Dave.
@ndhanson3, @Moti, Thank you very much for your comments. Parametrization is really a key word. Let me answer myself.
The parametrization of a line segment is written as: $$\begin{cases} x = (x_Z-x_A)t + x_A, \quad where \; t\in[0,1]\\ y = (y_Z-y_A)t + y_A \end{cases}$$ so $$ t = \frac{x-x_A}{x_Z-x_A} , \quad where \; t\in[0,1]$$ while the parametrization of an arc is written as $$\begin{cases} \hat{x} = Rsin(\hat{t}), \quad where \; \hat{t} \in [0,\theta]\\ \hat{y} = Rcos(\hat{t}) \end{cases}$$ It is a transformation from $t\in[0,1]$ to $\hat{t}\in[0,\theta]$, so $\hat{t}$ is computed $$\begin{aligned} \hat{t} &= (\theta-0)\bigg(\frac{t-0}{1-0}\bigg) + 0 \\ &= \theta t \\ &= \theta \bigg(\frac{x-x_A}{x_Z-x_A}\bigg) \end{aligned}$$ Subtitution into the arc equation, the transformation of the arc has the parametric equation $$\begin{cases} \hat{x} = Rsin\Big(\theta \big(\frac{x-x_A}{x_Z-x_A}\big) \Big), \\ \hat{y} = Rcos\Big(\theta \big(\frac{x-x_A}{x_Z-x_A}\big) \Big) \end{cases}$$
Best,-Dave.