geometry ellipsoid

Describes the entry ellipsoid at version 0.1.0, registered by mime.scene._solids.Ellipsoid.

At a glance

• geometry ellipsoid, version 0.1.0, namespace mime

• Derivative: exact

• Validity: none declared

What it models, and when not to use it

ellipsoid

A solid ellipsoid given by its three semi-axes, along its part frame’s x, y and z, and centred on its centroid: an egg-shaped or flattened particle, a cell, or a smooth body whose three widths differ. As a body’s part it gives the body its volume and inertia and its surface samples, its one patch surface, for a solver that integrates over the surface. Its region solid has no registered shape (an ellipsoid’s distance has no closed form), so nothing evaluates it at run time, and a body whose hull it is declares its extent where a law reads one.

When not to use it. For a body with a feature the ellipsoid smooths away (a thread, a fin, an edge), or where a law needs the distance to its surface at run time.

Its derivative is exact: its volume, inertia, area, samples and reach are smooth functions of its semi-axes.

Its sampling. At a spacing, for a solver whose unknowns sit at the samples (the MRS rung), it samples its surface quasi-uniformly: as many samples as its quadrature (Gauss-Legendre in the cosine of the polar angle about z, the trapezoid rule round it) has at that spacing, chosen from that quadrature at a finer spacing by farthest points and Lloyd’s relaxation (mime.scene.quasi_uniform), each a quadrature sample with its cell’s area as its weight. Its hull query, its reach along d, is the length of (a d_x, b d_y, c d_z).

What its sampling costs. As on the sphere, the MRS rung’s drag at a given count is further from the closed form (Oberbeck’s and Edwardes’s, as Kim and Karrila give it) on the quasi-uniform sampling than on its quadrature’s grid, in translation and in rotation, on the two triaxial ellipsoids measured: the grid misses by three fifths to three quarters as much (the numbers are MIME-ANO-126’s). The ellipsoid samples quasi-uniformly all the same, as every kind does (a labelled default the maintainer may overturn); the separated quadrature chooses the defaults again.

Regions, surfaces, directions and frames. What a kind names and how a law or a stage reads it: the scene note.

Versions. 0.1.0, its first version, gained its quadrature, its sampling and its regions before any release, at the same version: before it the MRS rung refused it, so no result of it moved.

Settings

Setting	Type	Default	Dimension	Role	Default’s kind
semi_axes	Annotated[Quantity, Length]	{"required": true}	length (m)	physical	none

Each role and default’s kind is the entry’s setting_roles (mime.registry.SettingRole); a setting that declares none is required.

Capabilities

Generated from which of the optional methods of mime.registry.Geometry the kind implements; a consumer that needs one it lacks is refused when it plans, naming both.

Capability	What it gives	Used by	This kind
samples	surface samples by named patch, at a resolution the consumer chooses	the MRS; immersed-boundary markers	yes
area	the area of each patch	the samples’ weights	yes
mass_properties	volume and inertia at a density	the part resolver	yes
radial_reach	the hull’s radial reach about its axis	a body’s extent; contact on the hull	no
family_parameters	the parameters of the family it is	a surrogate trained on one family	no
tangent	a tangent it declares at a surface point	a law anisotropic along it	no
native	its native form, one of USD’s own primitives	the stage writer	no
drawing	a drawing, tessellated at the writer’s resolution	the stage writer, for a kind with no native form	yes
region_primitive	its regions as OpenMaterica’s primitives: exactly, or (a region with no end along an axis) clipped at the kind’s own length, or refused	OpenMaterica’s manifest and stage	no
reach	reach	not stated	yes
sampling	sampling	not stated	yes
support	support	not stated	yes

native: none, so the stage writes it in MIME’s schema with its drawing.

placement: none: its own frame is the frame its settings are written in.

Run-time capabilities (signed distance, outward normal, nearest point) are its regions’ and surfaces’ shapes, listed below.

Nodes it builds

Builds no node of its own.

Gradient status

The derivative through this entry is exact, the derivative of what it computes.

Validity

none declared

References

[@WeissteinEllipsoid]

Other declarations

Class variable	Type	Value
directions	ClassVar[tuple[str, ...]]	[]
moving	ClassVar[bool]	false
patches	ClassVar[Mapping[str, str]]	{"surface": "solid"}
reach_envelope	ClassVar[float]	0.0
region_shapes	ClassVar[Mapping[str, str]]	{}
regions	ClassVar[tuple[str, ...]]	["solid"]
support_basis	ClassVar[str]	closed form
support_error	ClassVar[float]	0.0
surfaces	ClassVar[Mapping[str, str]]	{}