It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and others.

Let’s however look at the spiral defined by the equation: $r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$

I am looking for a method ( a tool) which could help to plot the spiral on paper ( I named it as Archimedean-Galileo spiral. For large $\theta$, the curve represents Archimedean spiral: $r=\theta$. When $\theta$ is small it transforms in Galileo spiral $r=\theta^2$) .

The spiral has a property that the junction point of the curve and the ray uniformly rotated in the origin coordinates when the junction point moves with uniform acceleration.

Do you think that there is a way to draw it without computer, but with other special curves (tools)?

I thought about the spiral of Theodorus, but I am not sure how the spiral of Theodurus is connected with the equation.