I'm trying to make a strophoid in Desmos and I'm having some issues. For reference, a strophoid is defined by the following:
Take some curve $C$ with two points, $O$ (the origin) and$A$ (the pole). Then, construct a line $L$ that passes through $O$ and some point $C$ at $K$. Now construct a circle centered at $K$ that passes through $A$. Call the two points that this circle intersects with $L$ as $P_1$ and $P_2$. The strophoid of a curve is the set of all $P_1$ and $P_2$ for all $K$ on $C$.
Here's what I have so far. In this case, $C$ is just the parabola $y=x^2$, but I have it set up so you can make it anything. So far, I have the circle centered at $K$ that passes through $A$, but I don't know how to graph all the intersections for all $k$ in the graph (or, in fact, if it can be done). I've also tried $\operatorname{distance}(K,A)=\operatorname{distance}(K,(x,m(x-k)+C(k))$, but this just gives two vertical lines at $P_1$ and $P_2$, and again, it doesn't actually give the set of all such points. Any pointers would be appreciated!
Best Answer
Suppose the plane curve $C$ is parametrized by some function $$C : (x(t), y(t)), \quad a \le t \le b.$$ Let $A = (h,k)$. Then the equation of the line through $O$ and point $K = (x(t_k), y(t_k))$ is simply $$y = mx, \quad m = y(t_k)/x(t_k), \tag{1}$$ and the equation of the circle with center at $K$ passing through $A$ is $$(x - x(t_k))^2 + (y - y(t_k))^2 = (h - x(t_k))^2 + (k - y(t_k))^2. \tag{2}$$ Thus $P_1$ and $P_2$ are solutions to the system $(1)$, $(2)$, which yields the corresponding parametrizations of the loci as the point $K$ is allowed to vary as a function of $t_k$.