Reference documentation for deal.II version 9.5.0
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Physics::VectorRelations Namespace Reference

## 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)

## Detailed Description

Functions to compute relations between spatial vectors.

## ◆ angle()

template<int spacedim, typename Number >
 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.

$\vec{a} \cdot \vec{b} = \|\vec{a}\| \|\vec{b}\| \cos(\theta)$

## ◆ signed_angle()

template<int spacedim, typename Number >
 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.

$\tan{\theta} = \frac{\sin(\theta)}{\cos(theta)} = \frac{(\vec{a} \times \vec{b}) \cdot \vec{n}}{\vec{a} \cdot \vec{b}}$

Note
Only applicable for three-dimensional vectors spacedim == 3.