$\mathbb N = \{1,2,3,4\dots\}$
Let us suppose we are starting at a point with coordinates $(0,0)$. Now draw a line from $(0,0)$ to $(1,0)$ and from $(1,0)$ to $(1,2)$. Now by the Pythagorean theorem, I will draw the hypotenuse i.e. a line from $(0,0)$ to $(1,2)$ of length $\sqrt{5}$. Now I will draw a line of length $3$ from the last point $(1,2)$ and perpendicular to the last hypotenuse we got. It will give another point, suppose $(x_1,y_1)$. Now again I will draw the hypotenuse and again draw a perpendicular line of length $4$ from this hypotenuse and so on. The values of length of the lines are coming from the set $\mathbb N$ of natural numbers.
Now the question is: How can I find out the equation of the curve satisfying the points $(0,0), (1,0), (1,2) , (x_1,y_1) , (x_2,y_2), \dots$ ?
It seems that the curve will be a spiral, but again I don't know how to find out the equation.
A similar figure can be -