# Parameterize and find the arc length

Parameterize the following curves in $$\mathbb{R}^3$$ and find their arc length:

• the intersection of the cylinder $$x^2+y^2=1$$ and the plane $$x+y+z=1$$
• the intersection of the sphere $$x^2+y^2+z^2=1$$ and the plane $$x+y+z=1$$

For the first part, I parameterized by $$x = \sqrt{1-y^2}$$ and $$z = 1-x-y$$. Then I substituted $$y = \sin(t)$$ for $$0 \le t \le 2\pi$$. So I got $$r(t): \left<\cos(t),\sin(t),1-\cos(t)-\sin(t)\right>$$. Is this a right way to approach to get the r(t)?

Also, when I am trying to use the arc length formula, how should I determine the "range" of an integral? We need some sort of range like from a to b in order to get arc length.

For second part, I just put $$x^2 + y^2 + z^2 = x+y+z$$ then made the equation $$x(x-1) + y(y-1) + z(z-1)=0$$, but then after here I am stuck on how to parameterize like the first part. Also, if I have to get the arc length for this curve, how should I set the "range"?

The intersection of the cylinder $$x^2+y^2=1$$ with the plane $$x+y+z=1$$ is an ellipse $$E$$ going around the cylinder. As $$E$$ projects bijectively onto the unit circle in the $$(x,y)$$-plane we use the representation $$t\mapsto(\cos t,\sin t)\qquad(0\leq t\leq2\pi)$$ for this circle and then compute $$z=1-x-y=1-\cos t-\sin t\ .$$ This leads to your parametrization $$E:\quad t\mapsto{\bf r}(t)=(\cos t,\sin t, 1-\cos t-\sin t)\qquad(0\leq t\leq2\pi)\ .$$ For the length of $$E$$ we need $${\bf r}'(t)=(-\sin t,\cos t,\sin t-\cos t)\ ,$$ so that $$L(E)=\int_0^{2\pi}|{\bf r}'(t)|\>dt=\int_0^{2\pi}\sqrt{2-2\cos t\sin t}\>dt\ .$$ This is an elliptic integral. It cannot be evaluated in terms of elementary functions.

• the intersection of the sphere $$x^2+y^2+z^2=1$$ and the plane $$x+y+z=1$$

Eliminate $$z$$

$$x^2+y^2+(1-x-y)^2= 1\rightarrow x^2+y^2 +xy -x-y =0 \tag1$$

Note that a surface is 2-parametered, while intersection is 1- parametered. So we find one parameter as a function of the other.

Solve for $$y(x)$$

$$2 y(x)= (1-x) \pm \sqrt {(1-x)(1-3x)} \tag2$$

suggested parametrization

$$[x,2y,2z] =\big[t, (1-t)-\sqrt{(1-t)(1+3t)}, (1-t)+\sqrt{(1-t)(1+3t)}\big] \tag3$$

The parametrization can exist only if $$(1>t>-\dfrac13)$$

Arc length can be found by integration between these limits.

$$\int_0^{2 \pi}(\sqrt{x^{'2}+y^{'2}+z^{'2} })\,dt$$

The small circle circumference can be analytically evaluated.

Sketched the plane and sphere to visually verify the parametrization.

Same ( easier) procedure for cylinder/plane intersection.

Using direction cosines the slant plane works out as $$\cos^{-1}\frac{1}{\sqrt3}$$. On flat development the intersection elliptic arc becomes a sine-curve $$z=\sqrt{3} \sin \, (s)$$ whose analytical solution is $$a E(\theta, e)$$ in terms of second order elliptic integrals.