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.



How to draw Archimedean-Galileo spiral?
This is not a complete solution, but a suggestion on what you can try. First, note that the ["planimeter"](https://en.wikipedia.org/wiki/Planimeter) will "compute" for you integrals
$$
F(x)=\int ydx,
$$
where $y=f(x)$ is the given curve. This "computation" is in the form of a numerical output measured by a rotating wheel, but, using Paucellier-Lipkin inversor you can convert this to a linear motion which will lead to a geometric drawing of the graph of $F(x)$.


On the other hand, "Kempe's universality theorem" allows you to trace arbitrary planar algebraic curves using mechanical linkages. Kempe's original proof suffered from a number of deficiencies, its description could be found in e.g.


* Anupam Saxena, *[Kempe's Linkages and the Universality Theorem](https://www.ias.ac.in/article/fulltext/reso/016/03/0220-0237)*, Resonance – Journal of Science Education **16** Issue 3 (2011) pp 220-237.


The deficiencies were fixed by Kapovich and Millson in *[Universality theorems for configuration spaces of planar linkages](https://arxiv.org/abs/math/9803150)* ([published in Topology](https://doi.org/10.1016/S0040-9383(01)00034-9)).


(Note that David Khudaverdyan has a [computer-aided implementation](https://david.wf/linkage/) of a curve-drawing algorithm for some planar curves.) Combination of planimeter and Kempe-like linkages, allows you to "compute integrals" of algebraic functions using a mechanical device. By playing with these constructions, maybe you can construct a device (impractical, of course), which will trace the curve you are interested in.


An interesting question coming out of this is if there is a generalization of "Kempe's universality" for *1-st order algebraic ODEs* (or even more general ODEs/PDEs), e.g.:


Suppose that $P$ is, say, a polynomial, function of several variables and you have an ODE of the form:
$$
P(u', u, t)=0,
$$  

where $u=u(t)$ is an unknown function ${\mathbb R}\to {\mathbb R}^2$. Is there a "mechanical device" (where one would allow both mechanical linkages and "planimeter-like" gadgets, yet to be specified) which will trace solution curves of this ODE?


Edit: Indeed, Claude Shannon in


* *Mathematical theory of the differential analyzer*, J. Math. Phys. Mass. Inst. Tech. 20 (1941), 337-354, doi:[10.1002/sapm1941201337](https://doi.org/10.1002/sapm1941201337)


proved that one can "draw" solutions of arbitrary algebraic differential equations using mechanical devices ("analog computers"). A gap in
Shannon's proof was noted and fixed by


* Marian Boykan Pour-el, *Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)*, Trans. Amer. Math. Soc. **199** (1974), 1-28 doi:[10.1090/S0002-9947-1974-0347575-8](https://doi.org/10.1090/S0002-9947-1974-0347575-8).


On the practical side, there are real [differential analyzers](https://en.wikipedia.org/wiki/Differential_analyzer) (like the one by V. Bush), built up to 1950s, which were used to solve algebraic differential equations.


