[Math] parametrize the boundary of a region

Question

I need to parametrize the boundary of this region :

$D=\{y^2+z^2\le x^2+18,x^2+y^2\le 16\}$

So It's a one-sheet hyperboloid (radius=$\sqrt{18}$)+ cylinder with radius 4

I know how to parametrize them separately:

hyperboloid : $r(\theta,\phi)=(\sinh\phi,\sqrt{18}sin\theta\cosh{\phi},\sqrt{18}cos\theta\cosh{\phi})$

cylinder: $r(\theta,z)=(4\cos\theta,4\sin\theta,z)$

But how do I connect them? Ho do I parametrize the surface boundary?

Or maybe "parametrize the boundary" implies giving two parametrizations?If it is so, are my parametrizations right?

Answer 1

We consider at first equalities. There are two surfaces of revolution $(a=\sqrt{18},b=4 )$

$$ y^2+z^2-x^2=a^2 $$

$$ y^2+z^2=b^2 $$

which are hyperboloid of one sheet and a circular cylinder respectively.

Intersection of surfaces is the boundary of region for further purposes of integration etc.

Eliminate $y$ we get for $z$

$$ z = \sqrt{ 2x^2+a^2-b^2 } \tag1 $$

Circle of radius $b$ in projection is parameterized as $ (x,y)= b(\cos \theta, \sin \theta) $ and for $z$ we obtain by plugging in from above (1)

$$ z=\pm \sqrt{ 2 b^2 \cos^2 \theta +(a^2-b^2)}$$

The connected regions and the surfaces that created them along with (white) lines of intersection are plotted in Mathematica below .

Note that when two conicoids intersect we can generally have

1. Null real intersection ( disjunct surfaces)

2. One real closed loop intersection

3. Two real intersections

depending on distance of centers of central conicoids. In each case we can refer to the patches so created arbitrarily connecting them as internal or external to intersection contour made or defined by the equality. Each sign defines its own boundary of demarcated intersection, which enables determining regions where inequalities apply.

To find demarcation and connection a sketch is often helpful in guiding the region continuity of the patch in reference.

So the parameterizations are same but for $\pm$ sign where each sign represents a region.

The situation is no different if you consider intersection of a big cylinder and a smaller cylinder regions so created!