Context
- Definition: An ungula is the solid obtained by cutting a cone with a plane and keeping the part between the base of the cone and the plane
I couldn't find the formulas to obtain the upper surface area, the lateral surface area and the volume of an ungula anywhere, so I calculated them myself, but as I don't have software to check that the formulas are correct I would need either one verify the correctness of the following formulas (numerically or analytically) or who can suggest (free) software to use.
Question
Given a cone with base radius $R$ and height $H$, I cut it through a cone which has a slope $\phi$ with respect to the circular base and which forms an angle $\theta$ with respect to the projection of the apical point of the oblique section on the base (see figures)
Depending on the angle $\phi$ the upper surface can be an ellipse (figure 1), a parabola (figure 2), a hyperbola (figure 3) or a triangle (figure 4)
Following various calculations (which I am not going to rewrite because they are extremely long and involve the calculation of 3 different quantities and I would like to avoid asking an excessively long question) I ended up writing the following formulas:
Let, $$\text{atanr}(x):=\int_{0}^{1}\frac{1}{1-xt^{2}}\mathrm{d}t=\begin{cases}\frac{\arctan\left(\sqrt{-x}\right)}{\sqrt{-x}}&x>0\\ 1&x=0\\ \frac{\operatorname{arctanh}\left(\sqrt{x}\right)}{\sqrt{x}}&x<0\end{cases}$$
- $\displaystyle\text{Sup}_{\sup.}=\frac{R^2}{\cos(\phi)}\sin(\theta)\frac{(\lambda+1)\cos(\theta)+\lambda-1}{4\lambda}\left((\lambda-1)^2\text{atanr}(\lambda)-\lambda-1\right)$
- $\displaystyle\text{Sup}_{\text{lat.}}=R\ell\left(\theta-\frac{\sin(2\theta)}{2}-\sin(\theta)\frac{(\lambda+1)\cos(\theta)+\lambda-1}{4\lambda}\left((\lambda-1)^2\text{atanr}(\lambda)-\lambda-1\right)\right)$
- $\displaystyle\text{Vol.}=\frac{R^2 H}{3}\left(\theta-\frac{\sin(2\theta)}{2}+\frac{\sin(\theta)^3}{4\lambda}\left(\lambda^2-1-(\lambda-1)^3\text{atanr}(\lambda)\right)\right)$
Where:
- $\text{Sup}_{\sup.}$ is the area of the upper surface (the conic/triangle)
- $\text{Sup}_{\text{lat.}}$ is the lateral surface area (essentially the total surface area minus the area of the conic/triangle and the area of the bottom base)
- $\text{Vol}.$ is the volume
- $\displaystyle\ell:=\sqrt{R^2+H^2}$ is the length of the apothem of the cone.
- $\displaystyle\lambda:=\frac{R\tan(\phi)-H}{R\tan(\phi)+H}\cdot\frac{1-\cos(\theta)}{1+\cos(\theta)}$ it is a dimensionless quantity that describes the behavior of the upper section
At first I thought it was linked with the eccentricity, but eccentricity is a $$\varepsilon=\frac{\ell}{H}\sin(\phi)$$
In particular
- Figure 2 (parabola) occurs when $\displaystyle\phi=\arctan\left(\frac{H}{R}\right)$, equivalently when $\lambda=0$
- Figure 4 (triangle) occurs when $\displaystyle\phi=\phi_{\max}=\text{arccot}\left(-\frac{R}{H}\cos(\theta)\right)$, equivalently when $\lambda=1$