Dot products
The dot products between two unitized objects of the same type are listed in the following table. The vector $$\mathbf v$$ is the difference between their center positions, and the scalars $$r_1$$ and $$r_2$$ are their radii. In the case of dipoles and circles, the vectors $$\mathbf n_1$$ and $$\mathbf n_2$$ correspond to the directions of the carrier lines or the normals of the carrier planes.
| Type | Dot Product |
|---|---|
| Round points $$\mathbf a_1$$ and $$\mathbf a_2$$ | $$\mathbf a_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf a_2 = -\dfrac{1}{2}(\mathbf v^2 + r_1^2 + r_2^2)$$ |
| Dipoles $$\mathbf d_1$$ and $$\mathbf d_2$$ | $$\mathbf d_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf d_2 = -\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 + r_1^2 + r_2^2) + (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$ |
| Circles $$\mathbf c_1$$ and $$\mathbf c_2$$ | $$\mathbf c_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf c_2 = +\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 - r_1^2 - r_2^2) - (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$ |
| Spheres $$\mathbf s_1$$ and $$\mathbf s_2$$ | $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = +\dfrac{1}{2}(\mathbf v^2 - r_1^2 - r_2^2)$$ |
In the case of two spheres $$\mathbf s_1$$ and $$\mathbf s_2$$, the dot product gives the product of the radii $$r_1$$ and $$r_2$$ times the cosine of the angle $$\phi$$ between the tangent planes where they intersect, as shown in the following figure. This can be demonstrating by considering the law of cosines for the angle $$\gamma$$, which states
- $$\mathbf v^2 = r_1^2 + r_2^2 - 2r_1r_2 \cos \gamma$$ .
Plugging this into the formula for the dot product yields $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = r_1r_2 \cos \phi$$.
