Parametric Arc Length

Spiral Arc Length.png — Figure 1: In the left panel, arc Length of the spiral as plotted from $0$ to $20\pi$ is calculated by Sagemath using the parametric form of the arc length equation. In the right panel, arc length is estimated by summing the straight line segments between the points plotted on the spiral. The arc length estimate on the right is 1973.76 and there are 679 points.

With Parametric equations, the arc length formula is written with equation \eqref{1}. $$ArcLength=\int_{a}^{b}\sqrt{1+\left(\frac{dy/dt}{dx/dt}\right)^{2}}\,dx \tag{1} \label{1}$$ and for convenience might be expressed as $$f(t)=\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}dt$$ Example: Let $x(t)=t\cdot cos(t)$ and $y(t)=t\cdot sin(t).$ This curve will be a really nice spiral between $0$ and $20\pi.$

from sage.symbolic.integration.integral import definite_integral
var('t')
x(t)=t*cos(t)
y(t)=t*sin(t)
L1=parametric_plot((x(t),y(t)),(t,0,20*pi),rgbcolor=hue(0.6))
show(L1,figsize=4,axes=false,frame=true)
f(t)=sqrt((diff(x(t),t))^2+(diff(y(t),t))^2)
definite_integral(f(t),t,0,20*pi).n()

Example 1: 1.20.6. Calculate the length of a cycloid curve for a wheel of radius 8 inches. The cycloid parametric equations are $x(t)=r(t-sin(t))$ and $y(t)=r(1-cos(t)).$