Say I have a general tetrahedron, or in fact only the tip of a tetrahedron, being three triangles in 3-space sharing a single vertex and sharing edges pairwise.
Given that I only know the three angles inside those triangles at the single common vertex, is there a simple way to compute the three dihedral angles between each of the neighbouring triangles?
Best Answer
This is to address one of the comments in how to derive the formula for the dihedral angles.
Given a tip of a tetrahedron, label the 3 edges attached to it as $1, 2, 3$ in such a way when you view the tip within the tetrahedron, the edges $1, 2, 3$ are arranged in counterclockwise manner.
Let $\vec{v}_1$, $\vec{v}_2$, $\vec{v}_3$ be the 3 unit vectors pointing away from the tip along the direction of edge $1, 2, 3$ respectively. Let us look at one of edge, say edge $2$. It is in contact with two faces. One bounded by vectors $\vec{v}_1$ and $\vec{v}_2$. Another bounded by vectors $\vec{v}_2$ and $\vec{v}_3$. The inward normal vectors of these two faces are given by:
$$ \vec{n}_{12} = \frac{\vec{v}_1 \times \vec{v}_2}{|\vec{v}_1 \times \vec{v}_2|} \,\,\,\text{ and }\,\,\, \vec{n}_{23} = \frac{\vec{v}_2 \times \vec{v}_3}{|\vec{v}_2 \times \vec{v}_3|} $$ In terms of the normal vectors, the dihedral angle at edge $2$, $\phi_2$, satisfies:
$$\begin{align} \cos(\phi_2) &= -\vec{n}_{12}\cdot\vec{n}_{23}\\ &= -\frac{( \vec{v}_1 \times \vec{v}_2 )\cdot(\vec{v}_2 \times \vec{v}_3)}{ |\vec{v}_1 \times \vec{v}_2||\vec{v}_2 \times \vec{v}_3|}\\ &= -\frac{(\vec{v}_1\cdot\vec{v}_2)(\vec{v}_2\cdot\vec{v}_3) - (\vec{v}_1\cdot\vec{v}_3)|\vec{v}_2|^2}{|\vec{v}_1 \times \vec{v}_2||\vec{v}_2 \times \vec{v}_3|}\\ &= \frac{\vec{v}_1\cdot\vec{v}_3 - (\vec{v}_1\cdot\vec{v}_2)(\vec{v}_2\cdot\vec{v}_3)}{|\vec{v}_1 \times \vec{v}_2||\vec{v}_2 \times \vec{v}_3|}\tag{*} \end{align}$$ Let $\theta_{ij}$ be the vertex angle between edge $i$ and $j$. We can simplify R.H.S of (*) to get: $$\cos(\phi_2) = \frac{\cos(\theta_{13}) - \cos(\theta_{12})\cos(\theta_{23})}{\sin(\theta_{12})\sin(\theta_{23})}$$
The formula of other dihedral angles can be derived in same manner.