## Basic trigonometry For example, the circumcenter of a triangle, which is the position of the dual Voronoi vertex. ### Coding style notes Throughout, we will (attempt to) provide a type signature for all functions. To do so for array-based functions, we use [jaxtyping](https://docs.kidger.site/jaxtyping). ### JAX The aim is to create a triangulation datastructure compatible with the JAX library for automatic differentiation and numerical computing. In practice, this means that we use `jnp` (=`jax.numpy`) instead of `numpy`, and make sure our code follows JAX’s functional programming paradigm (see [JAX- the sharp bits](https://docs.jax.dev/en/latest/notebooks/Common_Gotchas_in_JAX.html)). ------------------------------------------------------------------------ source ### get_polygon_area ``` python def get_polygon_area( pts:Float[Array, 'n_vertices 2'] )->Float[Array, '']: ``` *Area of 2D polygon assuming no self-intersection.* ------------------------------------------------------------------------ source ### get_triangle_area ``` python def get_triangle_area( a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'] )->Float[Array, '']: ``` *Unoriented area of triangle with vertices a, b, c. Works in dim 2 or 3.* ------------------------------------------------------------------------ source ### get_oriented_triangle_area ``` python def get_oriented_triangle_area( a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'] )->Float[Array, '*']: ``` *Signed area of triangle with vertices a, b, c. If d=2, returns a scalar, if d=3, a vector.* ------------------------------------------------------------------------ source ### get_circumcenter ``` python def get_circumcenter( a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'], zero_clip:float=1e-10 )->Float[Array, 'dim']: ``` *Return circumcenter coordinates of triangle with vertices a, b, c* ------------------------------------------------------------------------ source ### get_cot_between_vectors ``` python def get_cot_between_vectors( a:Float[Array, 'dim'], b:Float[Array, 'dim'] )->Float[Array, '']: ``` *Cotangent of angle between two vectors* ------------------------------------------------------------------------ source ### get_angle_between_vectors ``` python def get_angle_between_vectors( a:Float[Array, 'dim'], b:Float[Array, 'dim'] )->Float[Array, '']: ``` *Angle between two vectors* ------------------------------------------------------------------------ source ### get_signed_angle_between_vectors ``` python def get_signed_angle_between_vectors( a:Float[Array, '2'], b:Float[Array, '2'] )->Float[Array, '']: ``` *Signed angle between two 2d vectors* ------------------------------------------------------------------------ source ### get_oriented_voronoi_corner_area ``` python def get_oriented_voronoi_corner_area( a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'], zero_clip:float=1e-10 )->Float[Array, '*']: ``` *Compute oriented Voronoi area at corner a of triangle abc. Returns a vector in 3d.* Returns zero for a degenerate triangle. ------------------------------------------------------------------------ source ### get_voronoi_corner_area ``` python def get_voronoi_corner_area( a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'], zero_clip:float=1e-10 )->Float[Array, '*']: ``` *Compute Voronoi area at corner a of triangle abc.* Returns zero for a degenerate triangle. ``` python get_voronoi_corner_area(jnp.array([0.,0.]), jnp.array([0.,1.]), jnp.array([1.,0.])) ``` Array(-0.25, dtype=float64) ``` python get_circumcenter(jnp.array([0.,0.]), jnp.array([0.,1.]), jnp.array([1.,0.])) ``` Array([0.5, 0.5], dtype=float64) ``` python get_signed_angle_between_vectors(jnp.array([1.,0.]), jnp.array([0.5,0.5])) * (180/jnp.pi) ``` Array(45., dtype=float64) ``` python get_polygon_area(jnp.array([[0.,0.], [0.,1.], [1.,0.]]) ), get_triangle_area(*jnp.array([[0.,0.], [0.,1.], [1.,0.]]) ) ``` (Array(0.5, dtype=float64), Array(0.5, dtype=float64)) ``` python get_circumcenter(jnp.array([0.,0.]), jnp.array([0.,1.]), jnp.array([1.,0.])) ``` Array([0.5, 0.5], dtype=float64) ``` python get_circumcenter(jnp.array([1.,0.]), jnp.array([1.,0.]), jnp.array([0.,1.])) ``` Array([0., 0.], dtype=float64) ``` python get_circumcenter(jnp.array([1.,0.]), jnp.array([2.,0.]), jnp.array([1.,0.])) ``` Array([0., 0.], dtype=float64) ``` python get_voronoi_corner_area(jnp.array([0.,0.]), jnp.array([0.,1.]), jnp.array([1.,0.])) ``` Array(-0.25, dtype=float64) ### Normals and rotation matrices in 3D ------------------------------------------------------------------------ source ### quaternion_to_rot_max ``` python def quaternion_to_rot_max( q:Float[Array, '4'] )->Float[Array, '3 3']: ``` *Convert unit quaternion into a 3d rotation matrix.* See https://fr.wikipedia.org/wiki/Quaternions_et_rotation_dans_l%27espace ------------------------------------------------------------------------ source ### get_triangle_normal ``` python def get_triangle_normal( a:Float[Array, '3'], b:Float[Array, '3'], c:Float[Array, '3'] )->Float[Array, '3']: ``` *Compute unit normal vector of triangle abc.* ------------------------------------------------------------------------ source ### get_perp_2d ``` python def get_perp_2d( x:Float[Array, '... 2'] )->Float[Array, '... 2']: ``` *Get perpendicular vector.* ------------------------------------------------------------------------ source ### get_rot_mat ``` python def get_rot_mat( theta:float )->Float[Array, '2 2']: ``` *Get 2D rotation matrix from angle in radians.* ### Barycentric coordinates ------------------------------------------------------------------------ source ### get_barycentric_coordinates ``` python def get_barycentric_coordinates( point:Float[Array, 'dim'], a:Float[Array, 'dim'], b:Float[Array, 'dim'], c:Float[Array, 'dim'], zero_clip:float=1e-10, normalize:bool=True )->Float[Array, '3']: ``` *Compute barycentric coordinates of point with respect to triangle abc.* ``` python vertices = jnp.array([[0., 0 ], [0., 1.], [1., 0.]]) point1 = jnp.array([0.5, 0.2]) get_barycentric_coordinates(point1, *vertices, normalize=False) ``` Array([0. , 0.2, 0.5], dtype=float64) ``` python vertices2 = jnp.array([[0., 0, 0], [0., 1., 0], [1., 0., 0]]) point2 = jnp.array([0.5, 0.2, 0.]) get_barycentric_coordinates(point2, *vertices2, normalize=False) ``` Array([0. , 0.2, 0.5], dtype=float64) ``` python point3 = jnp.array([0.5, 0.2, 1.]) get_barycentric_coordinates(point3, *vertices2, normalize=False) ``` Array([0. , 0.2, 0.5], dtype=float64)