deal.II version GIT relicensing-2206-gaa53ff9447 2024-12-02 09:10:00+00:00
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Functions | |
template<int spacedim, typename Number > | |
Number | angle (const Tensor< 1, spacedim, Number > &a, const Tensor< 1, spacedim, Number > &b) |
template<int spacedim, typename Number > | |
Number | signed_angle (const Tensor< 1, spacedim, Number > &a, const Tensor< 1, spacedim, Number > &b, const Tensor< 1, spacedim, Number > &axis) |
Functions to compute relations between spatial vectors.
Number Physics::VectorRelations::angle | ( | const Tensor< 1, spacedim, Number > & | a, |
const Tensor< 1, spacedim, Number > & | b | ||
) |
Calculate the angle \(\theta\) between two vectors a
and b
. The returned angle will be in the range \([0, \pi]\).
This function uses the geometric definition of the scalar product.
\[ \vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta) \]
Number Physics::VectorRelations::signed_angle | ( | const Tensor< 1, spacedim, Number > & | a, |
const Tensor< 1, spacedim, Number > & | b, | ||
const Tensor< 1, spacedim, Number > & | axis | ||
) |
Calculate the angle \(\theta\) between two vectors a
and b
, where both vectors are located in a plane described by a normal vector axis
.
The angle computed by this function corresponds to the rotation angle that would transform the vector a
into the vector b
around the vector axis
. Thus, contrary to the function above, we get a signed angle which will be in the range \([-\pi, \pi]\).
The vector axis
needs to be a unit vector and be perpendicular to both vectors a
and b
.
This function uses the geometric definitions of both the scalar and cross product.
\begin{align*} \vec{a} \cdot \vec{b} &= \|\vec{a}\| \|\vec{b}\| \cos(\theta) \\ \vec{a} \times \vec{b} &= \|\vec{a}\| \|\vec{b}\| \sin(\theta) \vec{n} \end{align*}
We can create the tangent of the angle using both products.
\[ \tan{\theta} = \frac{\sin(\theta)}{\cos(theta)} = \frac{(\vec{a} \times \vec{b}) \cdot \vec{n}}{\vec{a} \cdot \vec{b}} \]
spacedim == 3
.