Is the arc length always irrational between two rational points?

Solution 1:

Obviously, a straight line between two rational points can have rational length $-$ just take $(0,0)$ and $(1,0)$ as your rational points.

But a curved line can also have rational length. Consider parabolas of the form $y=\lambda x(1-x)$, which all pass through the rational points $(0,0)$ and $(1,0)$. If $\lambda=0$, then we get a straight line, with arc length $1$. And if $\lambda=4$, then the curve passes through $(\frac12,1)$, so the arc length is greater than $2$.

Now let $\lambda$ vary smoothly from $0$ to $4$. The arc length also varies smoothly, from $1$ to some value greater than $2$; so for some value of $\lambda$, the arc length must be $2$, which is a rational number.

Solution 2:

An example of a curve with rational arc lengths between at least some pairs of rational points is a cardioid.

Down to scaling and rotation, a cardioid may be rendered in polar coordinates by the equation

$$r=1-\cos\theta$$

with arc length differential

$$ds=\left(\sqrt{r^2+(dr/d\theta)^2}\right)d\theta=\sqrt{2-2\cos\theta}~d\theta=2\sin(\theta/2)d\theta$$

Integrating this from $\theta=0$ to an arbitrary value of $\theta$ gives the arc length function

$$s=4(1-\cos(\theta/2))$$

Thus the arc length from the origin to $(-2,0)$ ($\theta=\pi$) is $4$. Moreover, suppose we select $\theta=2\cos^{-1}(a/c)$ where $a^2+b^2=c^2$ is a Pythagorean triple. Then we have

$$\cos\theta=2(a^2/c^2)-1$$

$$\sin\theta=2(b/c)(a/c)=2ab/c^2$$

Clearly giving rational values for the Cartesian coordinates $x=(1-\cos\theta)\cos\theta$ and $y=(1-\cos\theta)\sin\theta$. The arc length from the origin is then the rational quantity

$$s=4(1-\cos(\theta/2))=4(1-a/c)$$

Solution 3:

So, my question is that do all curved path have irrational lengths?

Of course not. A circle with radius $\frac{1}{2\pi}$ is a curved path and has length $1$ which is a rational number. If you put the center of the circle to $(-\frac1{2\pi}, 0)$, then $(0,0)$, a "rational" point, is on the circle, and the circle can be seen as a path from $(0,0)$ to $(0,0)$.

Solution 4:

Consider the two points $(-\frac12,0)$ and $(\frac12,0)$. For any real value of $y_0$, we can draw a circular arc between these two points which is centered at $(0,y_0)$ and which lies entirely in the upper half-plane. As $y \to - \infty$, the length of this arc approaches 1 (since the arc approaches a straight line); as $y \to +\infty$, the arc length approaches $\infty$. Since the arc length varies continuously with $y_0$, it must be the case that the arc length can be any real number greater than 1, including all rational lengths greater than 1.