We have the three dimensional curve layout complete and need to display this data in two dimensions for easier interpretation and use on standard cut sheets.  This will be a little simpler than our three dimensional curve because we will always start with the hole at 0 degrees and move it to the correct position in two dimensions using s=r*theta to calculate the distance.  We will start with the formula below and you can see the derivation on the cylinder intersection page.  Only the x and z components are necessary for this two dimensional representation.

a= hole radius

b= manhole radius


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

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

Calculate S from r and theta
Top View

We can determine from the top view:





\[s=b*\arcsin(\frac{a * \cos(t) + Offset}{b})\]


To move the x component to the proper two dimensional position we apply the following calculation.  The Angle measurement should be in radians.

\[x(t) = Angle*b+s\]

\[x(t) = Angle*b+b*\arcsin(\frac{a * \cos(t) + Offset}{b})\]

The z component will be the same as derived on the cylinder intersection page so we have the following equations to plot hole layouts in two dimensions.


\[x(t) = Angle*b+b*\arcsin(\frac{a * \cos(t) + Offset}{b})\]

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

Try out the calculations with the following graph hosted on Desmos  https://www.desmos.com/calculator/vcgh6cvthj.