# geometry

## Primitives

### Base Classes

 Primitive Base class for geometric primitives.

### Classes

 Circle A circle is defined by a plane and a radius. Ellipse A ellipse is defined by a plane and a major. Frame A frame is defined by a base point and two orthonormal base vectors. Line A line is defined by two points. Plane A plane is defined by a base point and a normal vector. Point A point is defined by XYZ coordinates. Polygon An object representing an ordered collection of points in space connected by straight line segments forming a closed boundary around the interior space. Polyline A polyline is a sequence of points connected by line segments. Quaternion Creates a Quaternion object. Vector A vector is defined by XYZ components and a homogenisation factor.

The following representations of primitives can be used interchangeably as input in methods and functions. The representations using native Python objects also correspond to the required input parameters of the default constructor functions of the corresponding COMPAS objects.

 Object COMPAS Python point Point(float, float, float) [float, float, float] vector Vector(float, float, float) [float, float, float] line Line(point, point) [[float, float, float], [float, float, float]] plane Plane(point, vector) [[float, float, float], [float, float, float]] circle Circle(plane, float) [[[float, float, float], [float, float, float]], float] polygon Polygon(points) [[float, float, float], … [float, float, float]] polyline Polyline(points) [[float, float, float], … [float, float, float]] ellipse Ellipse(plane, float, float) [[[float, float, float], [float, float, float]], float, float] frame Frame(point, vector, vector) [[float, float, float], [float, float, float], [float, float, float]]

COMPAS primitives also support indexing, assignment, and iteration according to the above equivalency.

>>> a = Point(0, 0, 0)
>>> x, y, z = a
>>> x = a
>>> a = 0

>>> plane = Plane(Point(0, 0, 0), Vector(0, 0, 1))
>>> a, n = plane
>>> x, y, z = n
>>> plane = Point(1, 0, 0)


## Shapes

### Base Classes

 Shape Base class for geometric shapes.

### Classes

 Box A box is defined by a frame and its dimensions along the frame’s x-, y- and z-axes. Capsule A capsule is defined by a line segment and a radius. Cone A cone is defined by a circle and a height. Cylinder A cylinder is defined by a circle and a height. Polyhedron A polyhedron is defined by its vertices and faces. Sphere A sphere is defined by a point and a radius. Torus A torus is defined by a plane and two radii.

## Predicates 2D

 is_ccw_xy Determine if c is on the left of ab when looking from a to b, and assuming that all points lie in the XY plane. is_colinear_xy Determine if three points are colinear on the XY-plane. is_polygon_convex_xy Determine if the polygon is convex on the XY-plane. is_point_on_line_xy Determine if a point lies on a line on the XY-plane. is_point_on_segment_xy Determine if a point lies on a given line segment on the XY-plane. is_point_on_polyline_xy Determine if a point is on a polyline on the XY-plane. is_point_in_triangle_xy Determine if a point is in the interior of a triangle lying on the XY-plane. is_point_in_polygon_xy Determine if a point is in the interior of a polygon lying on the XY-plane. is_point_in_convex_polygon_xy Determine if a point is in the interior of a convex polygon lying on the XY-plane. is_point_in_circle_xy Determine if a point lies in a circle lying on the XY-plane. is_polygon_in_polygon_xy Determine if a polygon is in the interior of another polygon on the XY-plane.

## Predicates 3D

 is_colinear Determine if three points are colinear. is_coplanar Determine if the points are coplanar. is_point_in_halfspace is_point_in_polyhedron Determine if the point lies inside the given polyhedron. is_point_in_sphere is_point_on_line Determine if a point lies on a line. is_point_on_plane Determine if a point lies on a plane. is_point_on_polyline Determine if a point is on a polyline. is_point_on_segment Determine if a point lies on a given line segment. is_polygon_convex Determine if a polygon is convex.

## Transformations

Classes

 Projection Create a projection transformation. Reflection Creates a Reflection that mirrors points at a plane. Rotation Create a rotation transformation. Scale Creates a scale transformation. Shear Create a shear transformation. Transformation The Transformation represents a 4x4 transformation matrix. Translation Create a translation transformation.

Functions

 mirror_points_line Mirror a point about a line. mirror_points_line_xy Mirror a point about a line. mirror_points_plane Mirror a point about a plane. mirror_points_point Mirror multiple points about a point. mirror_points_point_xy Mirror multiple points about a point. project_points_line Project points onto a line. project_points_line_xy Project points onto a line in the XY plane. project_points_plane Project multiple points onto a plane. reflect_line_plane Bounce a line of a reflection plane. reflect_line_triangle Bounce a line of a reflection triangle. rotate_points Rotates points around an arbitrary axis in 3D. rotate_points_xy Rotates points in the XY plane around the Z axis at a specific origin. scale_points Scale points. translate_points Translate points. translate_points_xy Translate points and in the XY plane.

Conversions

 axis_and_angle_from_matrix Returns the axis and the angle of the rotation matrix M. axis_angle_vector_from_matrix Returns the axis-angle vector of the rotation matrix M. axis_angle_from_quaternion Returns an axis and an angle of rotation from the given quaternion. basis_vectors_from_matrix Returns the basis vectors from the rotation matrix R. euler_angles_from_matrix Returns Euler angles from the rotation matrix M according to specified axis sequence and type of rotation. euler_angles_from_quaternion Returns Euler angles from a quaternion. matrix_from_frame Computes a change of basis transformation from world XY to the frame. matrix_from_euler_angles Calculates a rotation matrix from Euler angles. matrix_from_axis_and_angle Calculates a rotation matrix from an rotation axis, an angle and an optional point of rotation. matrix_from_axis_angle_vector Calculates a rotation matrix from an axis-angle vector. matrix_from_basis_vectors Creates a rotation matrix from basis vectors (= orthonormal vectors). matrix_from_translation Returns a 4x4 translation matrix in row-major order. matrix_from_orthogonal_projection Returns an orthogonal projection matrix to project onto a plane. matrix_from_parallel_projection Returns an parallel projection matrix to project onto a plane. matrix_from_perspective_projection Returns a perspective projection matrix to project onto a plane along lines that emanate from a single point, called the center of projection. matrix_from_perspective_entries Returns a matrix from perspective entries. matrix_from_shear_entries Returns a shear matrix from the 3 factors for x-y, x-z, and y-z axes. matrix_from_shear Constructs a shear matrix by an angle along the direction vector on the shear plane (defined by point and normal). matrix_from_scale_factors Returns a 4x4 scaling transformation. matrix_from_quaternion Calculates a rotation matrix from quaternion coefficients. quaternion_from_matrix Returns the 4 quaternion coefficients from a rotation matrix. quaternion_from_euler_angles Returns a quaternion from Euler angles. quaternion_from_axis_angle Returns a quaternion describing a rotation around the given axis by the given angle.

Quaternion math

 quaternion_norm Calculates the length (euclidean norm) of a quaternion. quaternion_unitize Makes a quaternion unit-length. quaternion_is_unit Checks if a quaternion is unit-length. quaternion_multiply Multiplies two quaternions. quaternion_canonize Converts a quaternion into a canonic form if needed. quaternion_conjugate Conjugate of a quaternion.

## Linear algebra

 add_vectors Add two vectors. add_vectors_xy Add two vectors, assuming they lie in the XY-plane. cross_vectors Compute the cross product of two vectors. cross_vectors_xy Compute the cross product of two vectors, assuming they lie in the XY-plane. divide_vectors Element-wise division of two vectors. divide_vectors_xy Element-wise division of two vectors assumed to lie in the XY plane. dot_vectors Compute the dot product of two vectors. dot_vectors_xy Compute the dot product of two vectors, assuming they lie in the XY-plane. length_vector Calculate the length of the vector. length_vector_xy Compute the length of a vector, assuming it lies in the XY plane. length_vector_sqrd Compute the squared length of a vector. length_vector_sqrd_xy Compute the squared length of a vector, assuming it lies in the XY plane. multiply_matrices Mutliply a matrix with a matrix. multiply_matrix_vector Multiply a matrix with a vector. multiply_vectors Element-wise multiplication of two vectors. multiply_vectors_xy Element-wise multiplication of two vectors assumed to lie in the XY plane. norm_vector Calculate the length of a vector. norm_vectors Calculate the norm of each vector in a list of vectors. normalize_vector Normalise a given vector. normalize_vector_xy Normalize a vector, assuming it lies in the XY-plane. normalize_vectors Normalise multiple vectors. normalize_vectors_xy Normalise multiple vectors, assuming they lie in the XY plane. orthonormalize_vectors Orthonormalize a set of vectors. power_vector Raise a vector to the given power. power_vectors Raise a list of vectors to the given power. scale_vector Scale a vector by a given factor. scale_vector_xy Scale a vector by a given factor, assuming it lies in the XY plane. scale_vectors Scale multiple vectors by a given factor. scale_vectors_xy Scale multiple vectors by a given factor, assuming they lie in the XY plane. square_vector Raise a vector to the power 2. square_vectors Raise a multiple vectors to the power 2. subtract_vectors Subtract one vector from another. subtract_vectors_xy Subtract one vector from another, assuming they lie in the XY plane. sum_vectors Calculate the sum of a series of vectors along the specified axis. transpose_matrix Transpose a matrix. vector_component Compute the component of u in the direction of v. vector_component_xy Compute the component of u in the direction of v, assuming they lie in the XY-plane. vector_average Average of a vector. vector_variance Variance of a vector. vector_standard_deviation Standard deviation of a vector.

## Points, Vectors, Lines, Planes

 angle_points Compute the smallest angle between the vectors defined by three points. angle_points_xy Compute the smallest angle between the vectors defined by the XY components of three points. angle_vectors Compute the smallest angle between two vectors. angle_vectors_xy Compute the smallest angle between the XY components of two vectors. angle_vectors_signed Computes the signed angle between two vectors. angles_points Compute the two angles between two vectors defined by three points. angles_points_xy Compute the two angles between the two vectors defined by the XY components of three points. angles_vectors Compute the the 2 angles formed by a pair of vectors. angles_vectors_xy Compute the angles between the XY components of two vectors. angle_planes Compute the smallest angle between the two normal vectors of two planes. centroid_points Compute the centroid of a set of points. centroid_points_xy Compute the centroid of a set of points lying in the XY-plane. centroid_points_weighted Compute the weighted centroid of a set of points. distance_line_line Compute the shortest distance between two lines. distance_point_line Compute the distance between a point and a line. distance_point_line_xy Compute the distance between a point and a line, assuming they lie in the XY-plane. distance_point_line_sqrd Compute the squared distance between a point and a line. distance_point_line_sqrd_xy Compute the squared distance between a point and a line lying in the XY-plane. distance_point_plane Compute the distance from a point to a plane defined by origin point and normal. distance_point_point Compute the distance bewteen a and b. distance_point_point_xy Compute the distance between points a and b, assuming they lie in the XY plane. distance_point_point_sqrd Compute the squared distance bewteen points a and b. distance_point_point_sqrd_xy Compute the squared distance between points a and b lying in the XY plane. midpoint_point_point Compute the midpoint of two points lying in the XY-plane. midpoint_point_point_xy Compute the midpoint of two points lying in the XY-plane. midpoint_line Compute the midpoint of a line defined by two points. midpoint_line_xy Compute the midpoint of a line defined by two points.

## Polygons & Polyhedrons

 area_polygon Compute the area of a polygon. area_polygon_xy Compute the area of a polygon lying in the XY-plane. area_triangle Compute the area of a triangle defined by three points. area_triangle_xy Compute the area of a triangle defined by three points lying in the XY-plane. centroid_polygon Compute the centroid of the surface of a polygon. centroid_polygon_xy Compute the centroid of the surface of a polygon projected to the XY plane. centroid_polygon_vertices Compute the centroid of the vertices of a polygon. centroid_polygon_vertices_xy centroid_polygon_edges Compute the centroid of the edges of a polygon. centroid_polygon_edges_xy centroid_polyhedron Compute the center of mass of a polyhedron. normal_polygon Compute the normal of a polygon defined by a sequence of points. normal_triangle Compute the normal vector of a triangle. normal_triangle_xy Compute the normal vector of a triangle assumed to lie in the XY plane. volume_polyhedron Compute the volume of a polyhedron represented by a closed mesh.

## Point Sets

 bounding_box Computes the axis-aligned minimum bounding box of a list of points. bounding_box_xy Compute the axis-aligned minimum bounding box of a list of points in the XY-plane. convex_hull Construct convex hull for a set of points. convex_hull_numpy Compute the convex hull of a set of points. convex_hull_xy Computes the convex hull of a set of 2D points. convex_hull_xy_numpy Compute the convex hull of a set of points in the XY plane. oriented_bounding_box_numpy Compute the oriented minimum bounding box of a set of points in 3D space. oriented_bounding_box_xy_numpy Compute the oriented minimum bounding box of set of points in the XY plane.

## Distance

 closest_point_in_cloud Calculates the closest point in a pointcloud. closest_point_in_cloud_xy Calculates the closest point in a list of points in the XY-plane. closest_point_on_line Computes closest point on line to a given point. closest_point_on_line_xy Compute closest point on line (continuous) to a given point lying in the XY-plane. closest_point_on_plane Compute closest point on a plane to a given point. closest_point_on_polyline Find the closest point on a polyline to a given point. closest_point_on_polyline_xy Compute closest point on a polyline to a given point, assuming they both lie in the XY-plane. closest_point_on_segment Computes closest point on line segment (p1, p2) to test point. closest_point_on_segment_xy Compute closest point on a line segment to a given point lying in the XY-plane.

## Intersections

 intersection_circle_circle_xy Calculates the intersection points of two circles in 2d lying in the XY plane. intersection_line_line Computes the intersection of two lines. intersection_line_line_xy Compute the intersection of two lines, assuming they lie on the XY plane. intersection_line_plane Computes the intersection point of a line and a plane intersection_line_triangle Computes the intersection point of a line (ray) and a triangle based on the Moeller Trumbore intersection algorithm intersection_mesh_mesh Compute the intersection of tow meshes. intersection_plane_plane Computes the intersection of two planes intersection_plane_plane_plane Computes the intersection of three planes intersection_ray_mesh Compute the intersection(s) between a ray and a mesh. intersection_segment_segment Compute the intersection of two lines segments. intersection_segment_segment_xy Compute the intersection of two lines segments, assuming they lie in the XY plane. intersection_segment_plane Computes the intersection point of a line segment and a plane is_intersection_line_line Verifies if two lines intersect. is_intersection_line_line_xy Verifies if two lines intersect on the XY-plane. is_intersection_line_plane Determine if a line (ray) intersects with a plane. is_intersection_line_triangle Verifies if a line (ray) intersects with a triangle. is_intersection_plane_plane Verifies if two planes intersect. is_intersection_segment_plane Determine if a line segment intersects with a plane. is_intersection_segment_segment Verifies if two segments intersect. is_intersection_segment_segment_xy Determines if two segments, ab and cd, intersect.

## Offsets

 offset_line Offset a line by a distance. offset_polyline Offset a polyline by a distance. offset_polygon Offset a polygon (closed) by a distance.

## Interpolation

 barycentric_coordinates Compute the barycentric coordinates of a point wrt to a triangle. discrete_coons_patch Creates a coons patch from a set of four or three boundary polylines (ab, bc, dc, ad). tween_points Compute the interpolated points between two sets of points. tween_points_distance Compute an interpolated set of points between two sets of points, at a given distance.

## Fitting

 bestfit_circle_numpy Fit a circle through a set of points. bestfit_plane Fit a plane to a list of (more than three) points. bestfit_plane_numpy Fit a plane through more than three (non-coplanar) points.

## Boolean operations

 boolean_union_mesh_mesh Compute the boolean union of two triangle meshes. boolean_difference_mesh_mesh Compute the boolean difference of two triangle meshes. boolean_intersection_mesh_mesh Compute the boolean intersection of two triangle meshes.

## Triangulation

 conforming_delaunay_triangulation Construct a Conforming Delaunay triangulation of set of vertices, constrained to the specified segments. constrained_delaunay_triangulation Construct a Delaunay triangulation of set of vertices, constrained to the specified segments. delaunay_from_points Computes the delaunay triangulation for a list of points. delaunay_from_points_numpy Computes the delaunay triangulation for a list of points using Numpy. delaunay_triangulation Construct a Delaunay triangulation of set of vertices. voronoi_from_points_numpy Generate a voronoi diagram from a set of points.

## Triangle meshes

 trimesh_gaussian_curvature Compute the discrete gaussian curvature of a triangle mesh. trimesh_geodistance Compute the geodesic distance from every vertex of the mesh to a source vertex. trimesh_harmonic Compute the harmonic parametrisation of a triangle mesh within a fixed circular boundary. trimesh_isolines Compute isolines on a triangle mesh using a scalarfield of data points assigned to its vertices. trimesh_lscm Compute the least squares conformal map of a triangle mesh. trimesh_massmatrix Compute massmatrix on a triangle mesh using a scalarfield of data points assigned to its vertices. trimesh_principal_curvature Compute the principal curvature directions of a triangle mesh. trimesh_remesh Remeshing of a triangle mesh. trimesh_remesh_constrained Constrained remeshing of a triangle mesh. trimesh_slice Slice a mesh by a list of planes.

 quadmesh_planarize Planarize the faces of a quad mesh.
 icp_numpy Align two point clouds using the Iterative Closest Point (ICP) method.
 Primitive Base class for geometric primitives. Shape Base class for geometric shapes.