Spatial and Geometric Behavioral Representations

Spatial and Geometric Behavioral Representations are structured encodings that explicitly include spatial object identity, dimensionality, coordinate systems, reference frames, landmarks, orientations, poses, trajectories, regions, surfaces, correspondences, geometric constraints, and projection states as integral parts of their meaning. It is essential to understand that terms such as coordinates, geometry, pose, posture, trajectory, kinematics, projection, reconstruction, alignment, and normalization denote distinct concepts and are not interchangeable synonyms. Storing geometric values within vectors or tensors does not erase their intrinsic geometric semantics; likewise, a geometric representation by itself does not imply behavioral meaning, physical forces, dynamical laws, social relations, or causal mechanisms. The representation captures spatial structure and relationships but does not assert causal or behavioral interpretation intrinsically.

Meaning and Geometric Object Identity

A Spatial or Geometric Behavioral Representation explicitly organizes identified behavioral evidence into geometric objects defined within a declared coordinate system, dimensionality, units, entity correspondence, mapping, calibration or reference state where applicable, validity rules, and uncertainty semantics. This representation may encode absolute or relative position, orientation, pose, landmark configuration, trajectory, point set, region, surface, occupancy, or other geometric structures, without requiring all such forms to be present simultaneously.

Distinct geometric objects include:

• Scalar Coordinate: A single numeric value representing a scalar dimension or feature.
• Point: A location in space identified by coordinates within a declared frame and dimensionality.
• Vector: A directed quantity with magnitude and direction, often representing displacement or direction relative to a frame.
• Landmark: A spatial point with semantic identity and correspondence, often annotated by name or role.
• Landmark Set: An ordered or labeled collection of landmarks with preserved correspondence.
• Body Segment: A spatially extended rigid or articulated portion of a body, often defined by landmarks or geometry.
• Orientation: The rotational relation of an object or segment relative to a reference frame.
• Rigid Pose: The combination of position and orientation of a rigid body relative to a declared frame.
• Joint Configuration: A set of joint parameters describing articulated relations among segments.
• Skeleton: A model composed of landmarks and segments connected by anatomical or kinematic relationships.
• Trajectory: An ordered sequence of points or poses with declared temporal or sequential correspondence.
• Point Cloud: A collection of points representing spatial sampling, possibly unordered and without guaranteed correspondence.
• Region: A spatially contiguous or defined subset of space, such as a polygon or volumetric volume.
• Surface/Mesh: A geometric structure combining vertex coordinates with connectivity/topology defining faces or boundaries.
• Occupancy Structure: A spatial encoding of presence or absence, such as occupancy grids or volumetric cells.

These objects differ fundamentally in identity, constraints, correspondence requirements, admissible operations, and topology, even when serialized as numeric arrays.

Spatial and geometric representations differ from Spatial and Kinematic Descriptors. Representations declare the objects (coordinates, poses, trajectories, regions) and their spatial semantics. Descriptors characterize numerical properties derived from these objects, such as distance, displacement, angle, extent, curvature, speed, or acceleration. Multiple descriptors can be computed from one representation without altering the representation's core definition or semantics.

Coordinate Frames, Dimensions, and Units

Coordinate-frame identity is defined by origin location, axis directions, axis order, handedness (right- or left-handed system), dimensionality, units, and relation to the represented entity or environment. Representative coordinate frames include:

• World-Centered: Fixed global frame, often georeferenced or lab-based.
• Room-Centered: Local environment frame anchored to a room or bounded space.
• Sensor/Camera-Centered: Origin and axes aligned with a sensor or camera device.
• Body-Centered: Frame fixed to a living organism's body.
• Segment-Centered: Frame anchored to a particular body segment.
• Object-Centered: Frame relative to a specific object.
• Image-Centered: Pixel or image coordinate system.
• Normalized: Scaled or adjusted frame for analysis or invariance.
• Task-Centered: Defined relative to task-specific landmarks or goals.

Note that equal numeric coordinates in different frames do not represent the same physical location without transforming between frames.

Coordinate values differ from geometric objects. For instance, a coordinate triple (x, y, z) can represent a location (point), a displacement (vector), an axis direction, a rotation vector, or an arbitrary feature triple depending on the declared schema. Therefore, coordinate names, array shapes, or numeric values alone do not establish geometric semantics. The geometric object type, coordinate frame, units, and transformation behavior must be explicitly declared.

A rigid change of coordinate frame is mathematically expressed as:

Here, p^A are the coordinates of one physical point expressed in frame A, p^B are the coordinates of the same point expressed in frame B, R_A^B is the declared rotation matrix mapping coordinate components from frame A into frame B, and t_A^B is the translation vector from the origin of frame A expressed in frame B. Note that frame-direction conventions vary across disciplines and software; thus, the transform direction and matrix multiplication convention form part of the representation identity.

Dimensionality and coordinate domain also matter. Pixel or image coordinates, normalized image coordinates, planar physical coordinates, three-dimensional Cartesian coordinates, geographic or curvilinear coordinates, joint-angle coordinates, and constrained manifold coordinates are not interchangeable simply because they share two or three numeric components. Geometric operations such as Euclidean distance or averaging require justification based on the coordinate domain and underlying geometry.

Landmarks, Poses, Skeletons, and Configurations

Landmark representations encode ordered or labeled collections of spatial points whose landmark identity and correspondence are integral to the schema. Preserved metadata includes landmark names or IDs, entity side or role, dimensionality, coordinate frame, units, confidence or validity states, and whether coordinates are measured, reconstructed, tracked, inferred, or normalized. Permuting landmark rows without updating their IDs results in a different representation, even if all coordinates remain the same.

Skeletal representations augment landmark configurations with declared anatomical or model connectivity, segment identity, parent–child structure, or a kinematic-tree-like schema where applicable. Fixed skeletal connectivity is part of the geometric schema but does not transform the representation into a general relational or graph analysis object. A skeletal model is an abstraction representing anatomical or kinematic structure, not a complete representation of the body.

Pose, posture, and pose estimate are distinct concepts. A geometric pose denotes either the position and orientation of a rigid body or a configuration of landmarks or joints under a declared convention. Posture is a behavioral or anatomical interpretation of body configuration. A pose estimate is derived evidence produced by an estimation procedure. One geometric configuration does not automatically imply a unique behavioral posture label.

Body-centered and root-relative configurations translate landmarks relative to an anchor such as the pelvis, torso, head, object, centroid, or other reference. This removes selected global-position information and exposes internal configuration. Root-relative coordinates answer a different representational question than absolute coordinates and inherit the anchor’s uncertainty.

A generic centered-and-scaled landmark coordinate is expressed as:

Here, p_j is the coordinate of landmark j in the original declared frame, o is the declared origin or anchor used for centering, s > 0 is the declared scale reference, and p̃_j is the resulting centered-and-scaled landmark coordinate. Changing the anchor or scale reference alters the representation, and this normalization intentionally removes selected translation and scale information.

Orientation and Rigid-Pose Parameterizations

Orientation defines the rotational relation of a directed frame, segment, rigid body, or object relative to another declared frame. Orientation differs from position and must be distinguished between absolute orientation (relative to a global or fixed frame) and relative orientation (between segments or bodies). Preserved metadata includes source and target frames, handedness, angle units, active/passive rotation convention, and whether the orientation was directly measured or estimated.

Common orientation parameterizations include:

• Euler/Cardan Angles: Three sequential rotations about defined axes; depend on axis order and can suffer from singularities (gimbal lock).
• Rotation Matrices: 3×3 orthogonal matrices representing rotation; redundant coordinates with orthogonality constraints.
• Unit Quaternions: Four-dimensional vectors with unit norm; represent rotation without singularities but have sign ambiguity (±q represent the same rotation).
• Axis–Angle/Rotation Vectors: Vector along axis of rotation with magnitude equal to rotation angle; periodicity and sign conventions apply.
• Direction Vectors: Unit vectors indicating specific directions, lacking full 3D orientation information.

Equivalent physical orientations can have non-equivalent raw coordinate representations due to parameterization characteristics.

A rigid pose combines position/translation and orientation relative to declared frames. It can be represented by translation plus quaternion, homogeneous transform matrices, dual quaternions, translation plus Euler angles, or other parameterizations. Component-wise averaging or interpolation on pose parameters is not automatically valid because orientations lie on a nonlinear, constrained rotation manifold.

Wraparound, singularity, and representation discontinuity occur in orientation parameterizations. Angular coordinates near −π and π can denote nearby orientations despite large numeric differences. Euler angles lose uniqueness near gimbal-lock configurations, and quaternion sign flips occur without physical discontinuity. Numerical continuity of stored parameters must be distinguished from physical orientation continuity.

Trajectories, Shapes, Regions, and Spatial Structures

Trajectory representations encode ordered geometric positions or poses with declared point correspondence, spatial frame, and sequence or time parameterization when timing is represented. Trajectories preserve paths or pose evolution, but kinematic quantities like speed and acceleration are descriptors derived from temporal change, not inherent in the trajectory representation itself.

Point-set and point-cloud representations focus on membership and geometry, with row order possibly meaningless. Point identity and correspondence must be preserved when known, along with units, frame, sampling density, normals or attributes if present, and ordering. Treating unordered point clouds as ordered landmark sets falsely implies correspondence.

Region and occupancy representations include masks, polygons, bounding regions, occupancy grids, volumetric cells, signed or unsigned distance fields, or other spatial-support encodings. They distinguish occupied, observable, valid, unknown, occluded, and outside-domain states. A bounding box or convex hull approximates or summarizes a region and should not be substituted silently for the region itself.

Surfaces and meshes combine vertex geometry with declared connectivity/topology, faces, normals, textures, or boundaries. Vertex coordinates alone do not define the same object as a mesh because topology specifies how vertices form a surface. Mesh connectivity here is geometric schema; graph-theoretic relation modeling requires additional semantics beyond adjacency.

Shape correspondence and reparameterization are critical when comparing configurations or shapes. Known landmark correspondence or explicit alignment procedures are necessary for pointwise comparison. Different parameterizations of the same curve sampled densely can produce different arrays representing the same geometric path; array index should not be mistaken for homologous geometric identity.

Projection, Viewpoint, Reconstruction, and Occlusion

Physical three-dimensional geometry differs fundamentally from its two-dimensional projection. Image-plane points encode where projected evidence appears under a camera or viewpoint model. Depth information is lost or confounded with projection unless additional evidence constrains it. Distinct 3D configurations can produce similar or identical 2D projections; therefore, one 2D pose should not be interpreted automatically as a unique 3D configuration.

A generic pinhole camera projection is expressed as:

where (X, Y, Z) are coordinates of one 3D point in the declared source/world frame, (u, v) are its image-plane coordinates, R and t are the declared extrinsic rotation and translation mapping into the camera frame, K is the declared intrinsic camera matrix under the pinhole model, λ is a nonzero projective scale proportional to depth, and superscript T denotes vector transpose. This ideal relation omits lens distortion and does not produce a unique inverse 2D-to-3D mapping without additional depth, geometric, multiview, or model constraints.

Viewpoint and projection state form part of the representation identity. Camera-centered, orthographic, perspective, normalized-device, screen/image, depth-map, reconstructed-world, and body-centered representations preserve different spatial information. Apparent distance, size, angle, and shape in an image can change with viewpoint even when physical geometry remains unchanged.

Reconstruction provenance matters. Three-dimensional landmarks, poses, surfaces, or trajectories can be triangulated, depth-derived, model-fitted, inertially integrated, multi-view reconstructed, or inferred by learned estimators. Reconstructed coordinates are derived evidence whose calibration, model assumptions, priors, initialization, occlusion handling, and uncertainty remain part of the representation semantics.

Occlusion and visibility states must be distinguished carefully. A landmark may be physically present but not visible due to being outside the sensing region, self-occluded, externally occluded, inferred despite invisibility, or truly absent from the represented model. Observed-versus-inferred status must be preserved; occlusion should not be encoded as zero coordinate, disappearance of body parts, or behavioral absence.

Alignment, Normalization, Invariance, and Geometric Equivalence

Alignment and normalization operations intentionally remove or standardize selected geometric variation. These include translation alignment, scale normalization, rigid alignment, similarity alignment, reflection handling, orientation normalization, root centering, Procrustes-like alignment, and task-defined registration. The transform class and fitted reference are part of the representation identity; alignment is not scientifically neutral because it determines which variation is suppressed.

Geometric equivalence differs from numerical equality. Two coordinate arrays represent the same underlying shape up to translation, rotation, uniform scale, permutation, reparameterization, or reflection only when the scientific representation explicitly declares the corresponding equivalence relation. Conversely, absolute location, facing direction, body size, handedness, left–right anatomy, or asymmetry can be target-relevant, making such transformations scientifically non-equivalent.

Chirality (handedness) must be explained explicitly. Mirroring preserves many pairwise distances while reversing handedness and left–right geometry; allowing reflections in alignment can erase behaviorally relevant asymmetry or anatomical laterality. Reflection admissibility must be stated rather than assumed.

Missingness, Uncertainty, Comparability, and Provenance

Missing and partial geometric representations arise from various causes: missing coordinates, occluded landmarks, invalid or low-confidence estimates, truncated fields of view, absent model elements, untracked entities, unknown correspondence, interpolated points, or structurally inapplicable components. Masks, confidence values, covariance or uncertainty objects, partial structures, or explicit validity states can preserve these distinctions. Imputed or inferred geometry is not observed geometry.

Uncertainty and sensitivity in geometric representations arise from coordinate error, calibration error, frame-registration error, landmark localization uncertainty, depth uncertainty, orientation uncertainty, correspondence ambiguity, occlusion, projection-model mismatch, fitted alignment, normalization anchor/scale uncertainty, and reconstruction assumptions. Uncertainty can be anisotropic and correlated across landmarks or coordinates, so a single scalar confidence value is often insufficient for demanding geometric comparisons.

A worked comparison example of a short body-movement episode with several labeled landmarks might include: raw image-plane landmarks; calibrated/reconstructed 3D landmarks; a world-frame configuration; a root-relative body-centered configuration; a body-size-normalized configuration; one rigid-body orientation represented both as Euler angles and as a quaternion; and a short spatial trajectory. This example would demonstrate coordinate-frame transformation without physical movement, information loss by 3D-to-2D projection, nonuniqueness of orientation coordinates, the effect of root/scale normalization, representation of one occluded landmark as unavailable rather than zero, and a correspondence failure caused by swapping left/right landmark IDs. It would also include a boundary example showing that speed or joint-angle statistics are descriptors derived from the geometric representation rather than the representation itself.

Spatial and Geometric Behavioral Representation provenance includes all information needed to reconstruct and scientifically interpret both the reusable representation definition and each instance. This includes representation definition/version, geometric object type, entity/landmark/segment IDs and correspondence, dimensionality, coordinate frame origin/axis order/handedness, units, calibration/reference state, orientation and pose parameterization, skeleton/mesh connectivity, trajectory parameterization, region/occupancy semantics, projection model and viewpoint, observed-versus-reconstructed status, normalization/alignment transform and admissible equivalence class, reflection policy, masks/occlusion/invalid states, uncertainty/confidence, source evidence and support, implementation/version, comparability conditions, and sensitivity findings. A defensible geometric representation explicitly states what spatial object is encoded, in which coordinate geometry, under which correspondences and transforms, which spatial distinctions are preserved or removed, and which components are measured, reconstructed, inferred, or unavailable.