INTERSECTING CYLINDERS

Pipe clearance calculations are critical to determine the structural stability of manholes and to make sure holes meet clearance requirements of municipalities.  Often ellipses or even circles are used to determine clearances prior to production.  These methods are close but in the early 1900’s Charles Proteus Steinmetz developed the Steinmetz Curve to model the intersection of two cylinders.  Using parametric equations we can calculate the exact path of the cylinder intersection in three dimensions.

Front View
Front View

For all calculations below a= Hole Radius,  b = Manhole Radius and all dimensions are in inches

From the front view we can make the following determinations for \[0<=t<=2\pi\]

\[cos(t)= \frac{x}{a}\]

\[x(t) = a * cos(t)\]

and

\[sin(t) = \frac{z}{a}\]

\[z(t) = a * sin(t)\]

 

 

 

Top View of Manhole Pipe Entry
Top View

From the top view we can solve for the y value.

\[b^2 = y^2 + x^2\]

\[y = \pm \sqrt{b^2-x^2}\]

\[y(t) = \pm \sqrt{b^2-(a * cos(t))^2}\]

 

 

 

 

Side View of Manhole Pipe Entry
Side View

 

We now have the following parametric equations:

\[0<=t<=2\pi\]

\[x(t) = a * \cos(t)\]

\[y(t) = \pm \sqrt{b^2-(a * cos(t))^2}\]

\[z(t) = a * \sin(t)\]

 

 

 

 

Now that we have the basic parametric equations we can begin to think about rotated holes, elevation and holes offset from center line.  We will start by adding offset to the holes.  The z(t) will remain the same since offset only affects the horizontal dimensions.

\[0<=t<=2\pi\]

\[x(t) = a * \cos(t) + Offset\]

\[y(t) = \pm \sqrt{b^2-[(a * cos(t))+Offset]^2}\]

\[z(t) = a * \sin(t)\]

Now we need to add elevation.  This will only affect the z(t) dimension.  Remember the elevation is given in inches.  To use feet simply multiply the elevation X 12 in the formula below.

\[0<=t<=2\pi\]

\[x(t) = a * \cos(t) + Offset\]

\[y(t) = \pm \sqrt{b^2-[(a * cos(t))+Offset]^2}\]

\[z(t) = a * \sin(t)+Elevation+\frac{a}{2}\]

Finally we need to rotate the points to the correct position.  For single hole structures this is not an issue but when dealing with multiple holes this will be necessary.  We can simply rotate about the z axis to get the points in the correct position.  Also, remember in the precast manhole business we always measure angles in the clockwise direction so our rotation angle in the formula will be 360-Angle or 2*pi-Angle.

\[Rz(Angle) \begin{pmatrix} x(t) \\y(t) \\z(t) \end{pmatrix}=\begin{pmatrix} cos(Angle)&sin(Angle)&0 \\-sin(Angle)&cos(Angle)&0 \\0&0&1 \end{pmatrix}\begin{pmatrix} a * \cos(t) + Offset \\\pm \sqrt{b^2-[(a * cos(t))+Offset]^2} \\a * \sin(t)+Elevation+\frac{a}{2} \end{pmatrix}\]

Expanded to the solution below.

\[\begin{pmatrix} x(t) \\y(t) \\z(t) \end{pmatrix}=\begin{pmatrix} ((a * \cos(t) + Offset)*(cos(360-Angle))) -(\sqrt{b^2-[(a * cos(t))+Offset]^2} *sin(360-Angle))\\(a * \cos(t) + Offset )*sin(360-Angle)+(\sqrt{b^2-[(a * cos(t))+Offset]^2}*cos(360-Angle)) \\a * \sin(t)+Elevation+\frac{a}{2} \end{pmatrix}\]

We can use the above equation to plot three dimensional points at the intersection of two cylinders.