Geometry and images: API index

Find an operation by its purpose. Entries link to the complete signature and contract.

HMath.Geometry

API	Purpose
Point2	Cartesian two-dimensional point, also used for vectors.
P2	Declaration and type contract below.
Point3	Cartesian right-handed three-dimensional point/vector.
P3	Declaration and type contract below.
PointND	A point of dimension 1..16.
PointND	Declaration and type contract below.
point2	Given x,y, construct Point2 after bounded finite-coordinate validation.
point3	Given x,y,z, construct Point3 after bounded finite-coordinate validation.
pointND	Construct a checked ND point from 1..16 bounded finite coordinates.
Line	Parametric line origin+t*direction; direction must be nonzero.
Line	Declaration and type contract below.
Ray	Parametric ray origin+t*direction, t>=0.
Ray	Declaration and type contract below.
Segment	Closed segment with two endpoints; zero length is valid.
Segment	Declaration and type contract below.
Plane	Plane normal dot x = offset, with a nonzero normal.
Plane	Declaration and type contract below.
Triangle	Closed triangle; zero-area triangles are rejected by triangle operations.
Triangle	Declaration and type contract below.
Rectangle	Axis-aligned rectangle with ordered corners.
Rectangle	Declaration and type contract below.
Circle	Circle with nonnegative radius.
Circle	Declaration and type contract below.
Ellipse	Rotated ellipse: positive semiaxes, counterclockwise angle in radians.
Ellipse	Declaration and type contract below.
Polygon	Simple polygon; vertices are implicitly closed and no duplicate endpoint is needed.
Polygon	Declaration and type contract below.
Sphere	Closed sphere, including a zero radius.
Sphere	Declaration and type contract below.
AABB	Ordered, closed axis-aligned bounds in 1..16 dimensions.
AABB	Declaration and type contract below.
OBB	Oriented three-dimensional box: center, three orthonormal axes, half-extents.
OBB	Declaration and type contract below.
Capsule	Capsule around the closed axis segment.
Capsule	Declaration and type contract below.
Cylinder	Finite cylinder with endpoint centers and radius; endpoints must differ.
Cylinder	Declaration and type contract below.
Cone	Finite cone with apex, base center, and radius; axis must be nonzero.
Cone	Declaration and type contract below.
Frustum	Convex frustum represented by six inward half-spaces n dot x >= offset.
Frustum	Declaration and type contract below.
Orientation	Sign of an exact predicate on the input Double values.
Clockwise	Declaration and type contract below.
Collinear	Declaration and type contract below.
Counterclockwise	Declaration and type contract below.
Location	Point position with boundary reported separately.
Outside	Declaration and type contract below.
Boundary	Declaration and type contract below.
Inside	Declaration and type contract below.

HMath.Geometry.Collision

API	Purpose
gjk	Convex intersection by Gilbert-Johnson-Keerthi support/simplex iterations.
Penetration	Normal moves the second shape away from the first by penetrationDepth.
Penetration	Declaration and type contract below.
penetrationDepth	Estimated overlap depth along penetrationNormal, in coordinate units.
penetrationNormal	Separating direction of the returned EPA penetration estimate.
penetrationError	Remaining absolute support gap at the stopping iteration.
penetrationIterations	Completed expanding-polytope iterations.
epa	Expanding polytope penetration estimate.
separatingAxis	Separating-axis collision and minimum overlap for convex polygons.

HMath.Geometry.Internal

API	Purpose
Point2	Cartesian two-dimensional point, also used for vectors.
P2	Declaration and type contract below.
Point3	Cartesian right-handed three-dimensional point/vector.
P3	Declaration and type contract below.
PointND	A point of dimension 1..16.
PointND	Declaration and type contract below.
Line	Parametric line origin+t*direction; direction must be nonzero.
Line	Declaration and type contract below.
Ray	Parametric ray origin+t*direction, t>=0.
Ray	Declaration and type contract below.
Segment	Closed segment with two endpoints; zero length is valid.
Segment	Declaration and type contract below.
Plane	Plane normal dot x = offset, with a nonzero normal.
Plane	Declaration and type contract below.
Triangle	Closed triangle; zero-area triangles are rejected by triangle operations.
Triangle	Declaration and type contract below.
Rectangle	Axis-aligned rectangle with ordered corners.
Rectangle	Declaration and type contract below.
Circle	Circle with nonnegative radius.
Circle	Declaration and type contract below.
Ellipse	Rotated ellipse: positive semiaxes, counterclockwise angle in radians.
Ellipse	Declaration and type contract below.
Polygon	Simple polygon; vertices are implicitly closed and no duplicate endpoint is needed.
Polygon	Declaration and type contract below.
Sphere	Closed sphere, including a zero radius.
Sphere	Declaration and type contract below.
AABB	Ordered, closed axis-aligned bounds in 1..16 dimensions.
AABB	Declaration and type contract below.
OBB	Oriented three-dimensional box: center, three orthonormal axes, half-extents.
OBB	Declaration and type contract below.
Capsule	Capsule around the closed axis segment.
Capsule	Declaration and type contract below.
Cylinder	Finite cylinder with endpoint centers and radius; endpoints must differ.
Cylinder	Declaration and type contract below.
Cone	Finite cone with apex, base center, and radius; axis must be nonzero.
Cone	Declaration and type contract below.
Frustum	Convex frustum represented by six inward half-spaces n dot x >= offset.
Frustum	Declaration and type contract below.
Orientation	Sign of an exact predicate on the input Double values.
Clockwise	Declaration and type contract below.
Collinear	Declaration and type contract below.
Counterclockwise	Declaration and type contract below.
Location	Point position with boundary reported separately.
Outside	Declaration and type contract below.
Boundary	Declaration and type contract below.
Inside	Declaration and type contract below.
finite	Test whether a Double is neither NaN nor infinite.
require	Given predicate then diagnostic text, return Right () if satisfied or Left (DomainError text).
coord	Accept a finite geometry coordinate only when its absolute value is <=1e50, bounding downstream Double products.
coords2	Extract [x,y] from a Point2 without validation.
coords3	Extract [x,y,z] from a Point3 without validation.
coordsND	Extract ordered coordinate list from a PointND without validation.
check2	Validate both Point2 coordinates as finite with absolute magnitude <=1e50; invalid data returns DomainError.
check3	Validate all Point3 coordinates as finite with absolute magnitude <=1e50; invalid data returns DomainError.
checkND	Require 1..16 coordinates, all finite and bounded by 1e50 in magnitude.
checkPair	Validate both ND points and require equal dimensions.
checked	Return the supplied list if every Double is finite, otherwise Left NonFiniteResult.
point2	Given x,y, construct Point2 after bounded finite-coordinate validation.
point3	Given x,y,z, construct Point3 after bounded finite-coordinate validation.
pointND	Construct a checked ND point from 1..16 bounded finite coordinates.
dot	Unchecked sum of pairwise list products using zipWith; unequal lists truncate to the shorter length.
norm	Return scaled Euclidean list norm to reduce intermediate overflow; empty/all-zero list yields zero.
sub	Unchecked coordinatewise left-minus-right list subtraction, truncating to the shorter list on mismatch.
plus	Unchecked coordinatewise list addition, truncating to the shorter list on mismatch.
scale	Given Double scalar then coordinate list, return all coordinates multiplied by it without validation.
cross	Return right-handed cross product of two Point3 vectors.
add2	Add Point2 coordinates componentwise without validation.
sub2	Subtract right Point2 coordinates from left, without validation.
scale2	Given scalar then Point2, scale both coordinates without validation.
cross2	Return 2D determinant axby-aybx as Double without validation.
dot2	Return 2D dot product axbx+ayby without validation.
norm2	Return scaled Euclidean length of a Point2 without checked finite validation.
from3	Convert a list of exactly three Doubles to Point3; all other lengths return origin P3 0 0 0.
edges	Pair each polygon-list element with its cyclic successor, including the last-to-first edge.
orientExact	Given finite points a,b,c, compute exact Rational determinant of their represented Double coordinates.
orient	Compare orientExact determinant with zero, returning GT for counterclockwise, EQ for collinear, LT for clockwise.
onSegment	Given endpoints a,b then point p, test exact collinearity and inclusive coordinate bounds.
checkPolygon	Validate a simple polygon with 3..256 bounded finite vertices, no adjacent duplicates, nonzero area and no nonadjacent edge intersection.
signedArea	Return oriented shoelace area using exact Rational intermediate arithmetic on finite Double coordinates.
ccw	Return polygon vertices in counterclockwise order by reversing negative signed area; zero-area input remains unchanged.
locateUnchecked	Given valid simple polygon vertices then finite Point2, return Boundary for an edge point, otherwise Inside/Outside by ray parity.
firstOr	Given fallback then list, return its first element or the fallback for empty input.

HMath.Geometry.Polygon

API	Purpose
HullAlgorithm	Four distinct planar convex-hull construction algorithms.
GrahamScan	Declaration and type contract below.
MonotoneChain	Declaration and type contract below.
Quickhull	Declaration and type contract below.
GiftWrapping	Declaration and type contract below.
convexHull	Convex hull with at most 4096 input points.
polygonArea	Unsigned simple polygon area.
polygonCentroid	Area-weighted centroid; works for nonconvex simple polygons.
shoelace	Signed shoelace area; clockwise polygons have negative area.
BooleanOperation	Boolean set operation on closed polygon regions; ring boundaries have zero area.
Union	Declaration and type contract below.
Intersection	Declaration and type contract below.
Difference	Declaration and type contract below.
polygonBoolean	Exact edge-arrangement Boolean.
clipPolygon	Sutherland-Hodgman clipping against a convex CCW/clockwise clip polygon.
minimumBoundingCircle	Deterministic incremental minimum enclosing circle, O(n^3), at most 256 points.
minimumBoundingRectangle	Minimum-area oriented enclosing rectangle.
minkowskiSum	Convex Minkowski sum, via pairwise sums followed by a hull.

HMath.Geometry.Primitives

API	Purpose
Metric	Minkowski p must be finite and at least one.
Euclidean	Declaration and type contract below.
Manhattan	Declaration and type contract below.
Chebyshev	Declaration and type contract below.
Minkowski	Declaration and type contract below.
distance	Euclidean, L1, Linfinity, or scaled Lp distance.
cosineDistance	One minus cosine similarity; zero vectors are undefined.
hammingDistance	Number of unequal positions (not normalized).
jaccardDistance	Set Jaccard distance; two empty sets have distance zero.
hausdorffDistance	Symmetric Hausdorff distance on two nonempty finite clouds, at most 1024 each.
mahalanobisDistance	Mahalanobis distance using a checked symmetric positive-definite covariance.
pointLineDistance	Distance to an infinite line; a zero direction is an error.
pointSegmentDistance	Distance to a closed segment, including a zero-length segment.
pointPlaneDistance	Unsigned perpendicular plane distance.
orientation	Exact orientation of the represented finite Double coordinates.
incircle	Exact in-circle sign, orientation-independent: positive means inside.
polygonLocation	Simple polygon position, with boundary separate from interior.
triangleLocation	Triangle position using exact orientation signs.
sphereContains	Closed sphere containment.
boxContains	Closed axis-aligned box containment.
ellipseContains	Closed ellipse containment after rotation to its principal axes.
triangleBarycentric	Weights (a,b,c) summing to one; valid outside the triangle as well.
circumcenter	Circumcenter, obtained from exact rational perpendicular-bisector equations.
incenter	Incenter weighted by opposite side lengths.
Intersection2	Two-dimensional intersection, with a coincident line or overlapping segment explicit.
NoIntersection	Declaration and type contract below.
IntersectionPoint	Declaration and type contract below.
OverlapSegment	Declaration and type contract below.
CoincidentLine	Declaration and type contract below.
segmentIntersection	Closed segment intersection with exact degeneracy predicates.
lineIntersection	Infinite-line intersection from origins and nonzero direction vectors.
linePlaneIntersection	Lineplane result: Nothing paralleldisjoint, Just Nothing coincident, Just (Just p) a unique intersection.
planeIntersection	Plane intersection: same nested convention as linePlaneIntersection, with the unique intersection represented by a line.
rayTriangle	Moller-Trumbore ray intersection; coplanar rays return Nothing because their hit set is not a unique transverse surface crossing.
rayBox	Slab intersection interval [entry,exit], clipped to the ray t>=0.
raySphere	Ray-sphere near/far nonnegative parameters; tangent gives equal roots.
CircleIntersection	Circle intersections: coincidence is distinct from disjointness and tangency.
DisjointCircles	Declaration and type contract below.
CoincidentCircles	Declaration and type contract below.
CirclePoints	Declaration and type contract below.
circleIntersection	Given two circles, return DisjointCircles, CoincidentCircles, or CirclePoints for their intersection.
triangleIntersection	Closed triangle overlap polygon, including isolated tangencies as vertices.
sphereSDF	Sphere signed distance (negative inside).
boxSDF	Axis-aligned box exact signed Euclidean distance in 1..16 dimensions.
obbSDF	Oriented box distance after validating its orthonormal axes.
capsuleSDF	Capsule distance from its closed axis minus radius.
cylinderSDF	Exact capped finite-cylinder signed distance.
coneSDF	Exact finite-cone distance by reducing to its radial/axial cross-section.

HMath.Geometry.Spatial

API	Purpose
SpatialKind	Explicit index construction algorithm; hash width must be positive finite.
SpatialHash	Declaration and type contract below.
KDTree	Declaration and type contract below.
BallTree	Declaration and type contract below.
Quadtree	Declaration and type contract below.
Octree	Declaration and type contract below.
RTree	Declaration and type contract below.
BVH	Declaration and type contract below.
PointCloud	Validated homogeneous finite cloud; a nonempty cloud defines its dimension.
pointCloud	Construct a nonempty cloud of at most 65536 points, with consistent dimensions.
cloudPoints	Original cloud observations.
SpatialIndex	Opaque checked spatial-search tree/hash index.
buildSpatialIndex	Build the named structure.
rangeQuery	Closed range search, sorted original IDs.
radiusQuery	Exact closed radius query; uses the index range query then Euclidean filtering.
NeighborQuery	A budgeted nearest-neighbor answer; False means unexplored leaves may improve the answer.
NeighborQuery	Declaration and type contract below.
queryNeighbors	Nearest candidate IDs paired with Euclidean distances, in increasing distance order.
queryLeaves	Number of visited spatial-index leaves charged to the query budget.
queryExhaustive	True only when unexplored nodes cannot improve the returned neighbors; False marks a budget-limited approximation.
nearestQuery	Given neighbor count k, positive leaf budget, index, then query point, search for nearest neighbors.

HMath.Geometry.Transform

API	Purpose
affineTransform	Construct an affine matrix [linear,translation;0,1].
applyTransform	Apply an affine or projective matrix and divide by homogeneous w.
composeTransforms	Checked matrix composition; projective and affine matrices may be mixed.
inverseTransform	Gauss-Jordan inverse with partial pivoting, bounded to matrices of size 17.
perspective	Vertical field of view in (0,pi), positive aspect, 0<near<far.
orthographic	Orthographic bounds with left<right, bottom<top and 0<=near<far.
lookAt	World-to-camera transform from eye, target and nonparallel up direction.
extractFrustum	Extract six inward clipping planes from projection*view.
cullSphere	True means the closed sphere may be visible (conservative plane culling).
cullBox	True means the closed 3D AABB may be visible; corners need not be allocated.

HMath.Geometry.Triangulation

API	Purpose
IndexedTriangle	Counterclockwise indices into the caller's input point list.
delaunay	Incremental cavity replacement.
constrainedDelaunay	Insert noncrossing constrained edges by convex quadrilateral flips, then Lawson-flip every unconstrained illegal edge.
voronoi	Voronoi cells clipped to an ordered nonzero rectangle, one polygon ring per site (empty only if the site lies outside the clipping rectangle).
earClip	Ear clipping of a simple polygon.

HMath.Image

API	Purpose
Boundary	Image sampling policy: Constant fill,edge Clamp,half-sample symmetric Reflect,or periodic Wrap.
Constant	Declaration and type contract below.
Clamp	Declaration and type contract below.
Reflect	Declaration and type contract below.
Wrap	Declaration and type contract below.
Image	Opaque finite CPU image with row-major spatial axes and interleaved channels.
fromTensor	Tensor shape must be spatial dimensions followed by a channel axis.
toTensor	Copy an image to Tensor Double with shape spatial dimensions followed by channel count.
normalize	Global min/max scaling; a constant image maps to the requested lower bound.
gamma	Positive gamma, nonnegative inputs; computes x^gamma without clipping.
image	Validate all dimensions before allocating boxed immutable array storage.
imageShape	Return spatial dimensions only, excluding the last implicit channel axis.
imageChannels	Return channel count, between one and sixteen for validated images.
imageValues	Return flat row-major samples with channels interleaved at each spatial coordinate.
pixel	Given image then complete spatial coordinates, return all channel values in order.
channel	Given zero-based channel index then image, return a copied one-channel image of the same spatial shape.
combineChannels	Concatenate channels of a nonempty same-spatial-shape image list in input order.
mapImage	Output is validated, including arithmetic overflow inside the callback.
zipImage	Given binary sample function then equally shaped/channel-count images, return a new validated image.
brightness	Add a finite offset without implicit clipping.
contrast	Scale about a caller-defined pivot, without clipping.
gradient	Returns row derivative, column derivative, magnitude, and atan2(row,col) direction in radians.
laplacian	Given boundary then one-channel 2D image, apply the four-neighbor 3-by-3 Laplacian [0,1,0;1,-4,1;0,1,0].
convolve	Given boundary,kernel dimensions,flat weights,then image, reverse all kernel axes and correlate.
correlate	Correlation uses sample(ix+kernelOffset-center).
gaussianBlur	Separable Gaussian, truncated at an explicitly supplied integer radius.
boxBlur	Given boundary,radius,then image, average a centered full-dimensional box neighborhood independently per channel.
medianBlur	Given boundary,radius,then image, choose the median of each centered odd-sized neighborhood independently per channel.
bilateralBlur	Per-channel bilateral filtering: spatial and range Gaussian weights are multiplied, normalized locally, and computed relative to the center sample.
sharpen	Unsharp mask x+amount*(x-Gaussian(x)); no implicit range clipping.
GradientOperator	Image gradient stencil selector, choosing Sobel,Prewitt,or Scharr.
Sobel	Declaration and type contract below.
Prewitt	Declaration and type contract below.
Scharr	Declaration and type contract below.
canny	Canny: Gaussian smoothing, Sobel gradient, four-direction nonmaximum suppression, and 8-connected hysteresis. low/high are absolute normalized gradient magnitudes, not quantiles.
Morphology	Grayscale footprint operation selector: erosion,dilation,opening,or closing.
Erosion	Declaration and type contract below.
Dilation	Declaration and type contract below.
Opening	Declaration and type contract below.
Closing	Declaration and type contract below.
morphology	Flat grayscale morphology with an explicit odd-rank Boolean footprint.
morphologicalGradient	Given boundary,footprint dimensions,Bool footprint,then image, return dilation minus erosion.

HMath.Image.Color

API	Purpose
ColorSpace	Color coordinate convention,including sRGB,linear RGB,HSV,HSL,XYZ,Lab,and YCbCr. convertColor validates each source convention's ranges.
SRGB	Declaration and type contract below.
LinearRGB	Declaration and type contract below.
HSV	Declaration and type contract below.
HSL	Declaration and type contract below.
XYZ	Declaration and type contract below.
Lab	Declaration and type contract below.
YCbCr	Declaration and type contract below.
convertColor	Explicit source/destination spaces; three interleaved channels required.
grayscale	Rec.709 luminance, explicitly converting the stated source to linear RGB.
srgbToLinear	Transfer functions extend linearly below their positive breakpoints; finite negative/out-of-gamut channels are retained.
linearToSrgb	Encode a finite linear-light channel with the sRGB piecewise transfer function (linear branch through 0.0031308).

HMath.Image.Core

API	Purpose
Image	Opaque finite CPU image with row-major spatial axes and interleaved channels.
image	Validate all dimensions before allocating boxed immutable array storage.
fromTensor	Tensor shape must be spatial dimensions followed by a channel axis.
toTensor	Copy an image to Tensor Double with shape spatial dimensions followed by channel count.
imageShape	Return spatial dimensions only, excluding the last implicit channel axis.
imageChannels	Return channel count, between one and sixteen for validated images.
imageValues	Return flat row-major samples with channels interleaved at each spatial coordinate.
pixel	Given image then complete spatial coordinates, return all channel values in order.
channel	Given zero-based channel index then image, return a copied one-channel image of the same spatial shape.
combineChannels	Concatenate channels of a nonempty same-spatial-shape image list in input order.
mapImage	Output is validated, including arithmetic overflow inside the callback.
zipImage	Given binary sample function then equally shaped/channel-count images, return a new validated image.
normalize	Global min/max scaling; a constant image maps to the requested lower bound.
brightness	Add a finite offset without implicit clipping.
contrast	Scale about a caller-defined pivot, without clipping.
gamma	Positive gamma, nonnegative inputs; computes x^gamma without clipping.
Boundary	Image sampling policy: Constant fill,edge Clamp,half-sample symmetric Reflect,or periodic Wrap.
Constant	Declaration and type contract below.
Clamp	Declaration and type contract below.
Reflect	Declaration and type contract below.
Wrap	Declaration and type contract below.
coordinates	Enumerate zero-based row-major coordinates of a dimension list. [] yields [[]]; zero/negative extents yield no coordinates.
sample	Reflect is half-sample symmetric (... b a | a b c | c b ...).
build	Checked output allocation for internal coordinate-wise reference kernels.
at	Unchecked image access: given image,complete spatial coordinates,then channel, return one sample.
spatialSize	Return number of spatial pixels (shape product), excluding channel multiplicity.
finite	Test whether Double is neither NaN nor infinite.
require	Given predicate then diagnostic, return Right () or image-prefixed Left InvalidArgument.
checked	Return finite Double values unchanged, or Left NonFiniteResult when any sample is non-finite.

HMath.Image.Filter

API	Purpose
convolve	Given boundary,kernel dimensions,flat weights,then image, reverse all kernel axes and correlate.
correlate	Correlation uses sample(ix+kernelOffset-center).
gaussianBlur	Separable Gaussian, truncated at an explicitly supplied integer radius.
boxBlur	Given boundary,radius,then image, average a centered full-dimensional box neighborhood independently per channel.
medianBlur	Given boundary,radius,then image, choose the median of each centered odd-sized neighborhood independently per channel.
bilateralBlur	Per-channel bilateral filtering: spatial and range Gaussian weights are multiplied, normalized locally, and computed relative to the center sample.
sharpen	Unsharp mask x+amount*(x-Gaussian(x)); no implicit range clipping.
GradientOperator	Image gradient stencil selector, choosing Sobel,Prewitt,or Scharr.
Sobel	Declaration and type contract below.
Prewitt	Declaration and type contract below.
Scharr	Declaration and type contract below.
gradient	Returns row derivative, column derivative, magnitude, and atan2(row,col) direction in radians.
laplacian	Given boundary then one-channel 2D image, apply the four-neighbor 3-by-3 Laplacian [0,1,0;1,-4,1;0,1,0].
canny	Canny: Gaussian smoothing, Sobel gradient, four-direction nonmaximum suppression, and 8-connected hysteresis. low/high are absolute normalized gradient magnitudes, not quantiles.
Morphology	Grayscale footprint operation selector: erosion,dilation,opening,or closing.
Erosion	Declaration and type contract below.
Dilation	Declaration and type contract below.
Opening	Declaration and type contract below.
Closing	Declaration and type contract below.
morphology	Flat grayscale morphology with an explicit odd-rank Boolean footprint.
morphologicalGradient	Given boundary,footprint dimensions,Bool footprint,then image, return dilation minus erosion.
validateBoundary	Validate Constant boundary value as finite; Clamp,Reflect,Wrap are always valid.
scalar2D	Require a two-dimensional one-channel image and return (height,width).
windowOffsets	Given radius then image, enumerate centered neighborhood offsets.

HMath.Image.Histogram

API	Purpose
histogram	Given bin count in [2,4096] then image with every sample in [0,1], return one Int count list per channel.
equalizeHistogram	CDF-min equalization with quantized bin mapping.
clahe	2D adaptive equalization. clipFactor>=1 is relative to uniform tile occupancy; the cap is max(ceil(tileSamplesbins),floor(factor*tileSamplesbins)).
threshold	Strict comparison x>level; each channel is thresholded independently.
otsu	Otsu between-class variance maximum.
adaptiveThreshold	Local mean or Gaussian threshold: x > localMean-offset.

HMath.Image.Measure

API	Purpose
TemplateMetric	Template matching score: squared difference or normalized correlation.
SquaredDifference	Declaration and type contract below.
NormalizedCorrelation	Declaration and type contract below.
templateMatch	Valid 2D, one-channel matching.
rawMoment	Raw M_pq=sum(row^pcolumn^qintensity), orders 0..8 each.
centralMoment	Given row exponent p,column exponent q,then scalar 2D image, compute the unnormalized intensity-weighted central moment around its centroid.
imageCentroid	Intensity-weighted (row,column) centroid.
integralImage	Padded 2D summed-area image, per channel: output shape (height+1,width+1) has a leading zero row/column.
rectangleSum	Given (row,column) origin,(height,width) extent,then padded integralImage output, return per-channel sum of the half-open rectangle.
meanSquaredError	Given two images of equal spatial shape and channels, return average squared sample difference.
peakSignalToNoise	PSNR in decibels.
structuralSimilarity	Mean local SSIM with uniform valid windows and population covariance.

HMath.Image.Spatial

API	Purpose
Interpolation	Image resampling interpolation mode, choosing nearest,bilinear,bicubic,or Lanczos kernels.
Nearest	Declaration and type contract below.
Bilinear	Declaration and type contract below.
Bicubic	Declaration and type contract below.
Lanczos	Declaration and type contract below.
resize	2D resize uses pixel-center mapping (i+0.5)*old/new-0.5, independently per channel, without antialiasing.
warp	Forward 3x3 homogeneous transform acts on [x(column),y(row),1].
translate	Translation in pixel columns/rows; output dimensions equal input.
rotate	Angle in radians, clockwise on row-down image coordinates; rotate about the center pixel coordinate while preserving the input canvas dimensions.
Connectivity	Spatial connected-component adjacency: axis neighbors or all surrounding neighbors.
AxisConnected	Declaration and type contract below.
FullyConnected	Declaration and type contract below.
connectedComponents	Nonzero is foreground.
distanceTransform	Exact brute-force Euclidean distance to the nearest zero-valued spatial sample, with strictly positive per-axis spacing.

HMath.Quaternion

API	Purpose
Quaternion	A finite scalar-first Hamilton quaternion.
quaternion	Finite scalar-first constructor.
components	Scalar, x, y, z components.
norm	Scaled Euclidean norm (may overflow only when the true norm is unrepresentable).
normalize	Normalize without forming a squared norm; zero has no normalization.
inverse	Multiplicative inverse; uses component scaling to avoid norm overflow.
conjugate	Quaternion conjugation.
multiply	Hamilton product.
add	Component sum.
scale	Checked real scaling.
rotateVector	Rotate a finite three-vector; any nonzero quaternion is normalized first.
axisAngle	Axis-angle conversion.
toAxisAngle	Principal rotation angle in [0,pi]; identity uses the x axis.
fromEuler	Intrinsic ZYX rotation from (roll,pitch,yaw).
toEuler	Intrinsic ZYX angles, choosing roll zero at gimbal lock.
toMatrix	Active rotation as a row-major 3x3 matrix.
fromMatrix	Validate SO(3) within 1e-9, then use the largest-diagonal conversion near pi.
nlerp	Shortest-arc normalized linear interpolation on [0,1].
slerp	Shortest-arc spherical interpolation, with stable NLERP near equal rotations.
squad	Shoemake SQUAD through q1 to q2 using neighboring q0 and q3.
exponential	General quaternion exponential; sinc expansion stabilizes a tiny vector part.
logarithm	Principal quaternion logarithm; the negative real axis chooses positive x.
swingTwist	Decompose q = swing * twist around a nonzero axis.

HMath.Rotation

API	Purpose
SO3	Normalized representative of SO(3).
so3	Normalize a nonzero quaternion into SO(3).
rotationQuaternion	Underlying unit Hamilton quaternion.
identityRotation	Identity rotation.
composeRotation	Composition applying the right rotation first.
inverseRotation	Inverse of a unit rotation.
rotate	Apply a rotation to a finite three-vector.
expSO3	Exponential of a rotation vector; vector norm is the angle in radians.
logSO3	Principal logarithm as a rotation vector.
SE3	Rigid transformation x -> R*x+t.
se3	Construct a rigid transform from a rotation and finite translation.
rotationPart	Rotation component.
translationPart	Translation component.
identityTransform	Identity rigid transform.
compose	Compose transforms; b acts first.
inverse	Inverse rigid transform.
transformPoint	Apply a rigid transform.
expSE3	SE(3) exponential.
logSE3	Principal SE(3) logarithm, stable at identity and at rotations near pi.
toHomogeneous	Row-major homogeneous 4x4 matrix, bottom row [0,0,0,1].
fromHomogeneous	Validate a homogeneous rigid matrix (bottom row tolerance 1e-12, rotation 1e-9).
DualQuaternion	Unit real part and Study-orthogonal dual part.
dualQuaternion	Construct a rigid dual quaternion.
dualParts	Real and dual Hamilton parts.
toDualQuaternion	Encode translation as 0.5(0,t)r.
fromDualQuaternion	Decode translation from 2dconjugate(r).
multiplyDual	Rigid dual-quaternion product.
inverseDual	Inverse rigid dual quaternion.
interpolateDual	Shortest-rotation screw-linear interpolation on [0,1].