buffalo_wings.airfoil.CstAirfoil

class buffalo_wings.airfoil.CstAirfoil(*, upper, lower, trailing_edge_thickness=0.0)[source]

Bases: CstGeometry

Canonical CST airfoil with fixed airfoil class exponents.

Notes

This runtime represents the airfoil-specific CST case with n1 = 0.5 and n2 = 1.0 on both sides. Unlike CstGeometry, the curve parameterization uses the canonical side parameter x = s**2 with full-airfoil curve parameter s = |u|. This keeps canonical curve-parameter derivatives finite at the leading edge while preserving the same geometric shape as the corresponding general CST geometry under the mapped parameter tau = sign(u) * u**2.

Methods

`arc_length`(u_s, u_e)	Calculate the arc-length distance between two points on surface.
`arc_length_breakpoints`()	Return the breakpoint locations in arc-length coordinates.
`breakpoint_parameter_limits`(*, index)	Return parameter limits for one breakpoint.
`breakpoints`()	Return the boundary and leading-edge breakpoints.
`camber_curve`(*[, num_points, spacing])	Return a camber-curve representation for this airfoil.
`chord`()	Return the airfoil chord length.
`curvature_from_xi`(xi, *, surface)	Return one-surface curvature values at surface-local `xi` locations.
`d2ydx2`(u)	Return the second surface derivative at curve parameter locations.
`demote_degree`(*[, count, continuity])	Lower the Bezier shape degree on both CST sides.
`dydx`(u)	Return the surface slope at curve parameter locations.
`k`(u)	Calculate the curvature at parameter location.
`leading_edge`()	Return the leading-edge location.
`normal`(u)	Calculate the unit normal at parameter location.
`promote_degree`(*[, count])	Raise the Bezier shape degree on both CST sides.
`slope_from_xi`(xi, *, surface)	Return one-surface slope values at surface-local `xi` locations.
`tangent`(u)	Calculate the unit tangent at parameter location.
`to_spec`()	Return the schema definition needed to recreate this airfoil.
`trailing_edge`()	Return the midpoint of the trailing-edge points.
`u_from_s`(s)	Return curve parameters that correspond to arc length.
`u_from_x`(x, *, surface)	Return curve parameters that correspond to `x`.
`u_from_xi`(xi, *, surface)	Convert surface-local `xi` coordinates to native parameters.
`xi_from_u`(u)	Convert native parameters to surface-local `xi` coordinates.
`xy_from_s`(s)	Return curve coordinates at arc-length locations.
`xy_from_u`(u)	Calculate canonical CST airfoil coordinates.
`xy_from_xi`(xi, *, surface)	Return one-surface coordinates at surface-local `xi` locations.
`xy_s`(s)	Calculate first derivatives at arc-length location.
`xy_s_breakpoint`(*, index)	Return one-sided arc-length derivatives at one breakpoint index.
`xy_ss`(s)	Calculate second derivatives at arc-length location.
`xy_ss_breakpoint`(*, index)	Return one-sided arc-length second derivatives at one breakpoint.
`xy_u`(u)	Calculate first derivatives with respect to the canonical parameter.
`xy_u_breakpoint`(*, index)	Return one-sided first derivatives at one breakpoint index.
`xy_uu`(u)	Calculate second derivatives with respect to the canonical parameter.
`xy_uu_breakpoint`(*, index)	Return one-sided second derivatives at one breakpoint index.

Attributes

`length`	Return the full airfoil surface length.
`lower`	Return the lower-side canonical CST geometry.
`spec`	Return the schema definition used to create this airfoil.
`trailing_edge_thickness`	Return the explicit trailing-edge thickness.
`upper`	Return the upper-side canonical CST geometry.

property upper: CstAirfoilSide

Return the upper-side canonical CST geometry.

This property exposes the upper-side canonical CST definition.

property lower: CstAirfoilSide

Return the lower-side canonical CST geometry.

This property exposes the lower-side canonical CST definition.

property spec: CstAirfoilSpec

Return the schema definition used to create this airfoil.

The returned schema reproduces the current runtime coefficients and trailing-edge thickness.

xy_from_u(u)[source]

Calculate canonical CST airfoil coordinates.

Parameters:: u (buffalo_core.typing.FloatInput) – Signed airfoil parameter values in [-1, 1].
Returns:: Tuple (x, y) of float64 arrays.
Return type:: tuple[FloatArray, FloatArray]

Notes

This uses x = u**2 on each surface branch rather than the general CST geometry parameterization x = |u|.

xy_u(u)[source]

Calculate first derivatives with respect to the canonical parameter.

Parameters:: u (buffalo_core.typing.FloatInput) – Signed airfoil parameter values in [-1, 1].
Returns:: Tuple (dx/du, dy/du) of float64 arrays.
Return type:: tuple[FloatArray, FloatArray]

Notes

At listed breakpoints, this method returns the minus-side derivative so array-valued evaluations remain single-valued.

xy_uu(u)[source]

Calculate second derivatives with respect to the canonical parameter.

Parameters:: u (buffalo_core.typing.FloatInput) – Signed airfoil parameter values in [-1, 1].
Returns:: Tuple (d2x/du2, d2y/du2) of float64 arrays.
Return type:: tuple[FloatArray, FloatArray]

Notes

At listed breakpoints, this method returns the minus-side second derivative so array-valued evaluations remain single-valued.

xy_u_breakpoint(*, index)[source]

Return one-sided first derivatives at one breakpoint index.

Notes

Endpoint breakpoints return the same exact boundary value for both entries. The interior leading-edge breakpoint returns finite canonical one-sided derivatives in the x = s**2 parameterization.

Return type:: tuple[tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)], tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)]]

xy_uu_breakpoint(*, index)[source]

Return one-sided second derivatives at one breakpoint index.

Notes

Endpoint breakpoints return the same exact boundary value for both entries. The interior leading-edge breakpoint returns finite canonical one-sided second derivatives in the x = s**2 parameterization.

Return type:: tuple[tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)], tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)]]

arc_length(u_s, u_e)

Calculate the arc-length distance between two points on surface.

Parameters:

u_s (buffalo_core.typing.FloatScalar) – Start point of distance calculation.
u_e (buffalo_core.typing.FloatInput) – End point of distance calculation.

Returns:

Distance from start point to end point.

Return type:

buffalo_core.typing.FloatArray

arc_length_breakpoints()

Return the breakpoint locations in arc-length coordinates.

Returns:: Arc-length coordinates measured from the minimum native parameter.
Return type:: list[FloatScalar]

Notes

These values include the two curve endpoints as boundary markers. Interior breakpoints correspond to the native-parameter interior breakpoints returned by breakpoints().

breakpoint_parameter_limits(*, index)

Return parameter limits for one breakpoint.

Notes

Endpoint breakpoints return the exact boundary parameter. Interior breakpoints return nearby one-sided parameters chosen within the neighboring breakpoint interval for the current generic breakpoint-side implementation. These limits exist to support the sampled fallback in the generic *_breakpoint methods and should not be treated as the primary source of truth when a subclass can provide exact one-sided values directly.

Return type:: tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)]

breakpoints()

Return the boundary and leading-edge breakpoints.

Returns:: Ordered parameter locations where surface branches meet or derivative one-sided limits may differ.
Return type:: list[float]

camber_curve(*, num_points=81, spacing='cosine')

Return a camber-curve representation for this airfoil.

Parameters:

num_points (int, default 81) – Number of shared surface samples to use when an approximate camber line must be derived from the airfoil geometry.
spacing ({"uniform", "cosine"}, default "cosine") – Spacing rule used for the shared surface-local sample locations in the approximate extraction path.

Returns:

Exact or approximate camber-curve result for this airfoil.

Return type:

AirfoilCamberResult

Raises:

ValueError – If num_points or spacing is invalid for the approximate extraction path.

chord()

Return the airfoil chord length.

Returns:: Distance between the leading-edge reference and trailing-edge midpoint reference.
Return type:: buffalo_core.typing.FloatScalar

curvature_from_xi(xi, *, surface)

Return one-surface curvature values at surface-local xi locations.

Parameters:

xi (buffalo_core.typing.FloatInput) – Surface-local coordinates in [0, 1] measured from the leading edge to the trailing edge.
surface ({"lower", "upper"}) – Surface to evaluate.

Returns:

Surface-oriented curvature values on the selected surface.

Return type:

buffalo_core.typing.FloatArray

d2ydx2(u)

Return the second surface derivative at curve parameter locations.

Parameters:: u (buffalo_core.typing.FloatInput) – Airfoil parameters.
Returns:: Second derivative values d^2y/dx^2 evaluated at u.
Return type:: buffalo_core.typing.FloatArray

demote_degree(*, count=1, continuity='NOT_CONNECTED')

Lower the Bezier shape degree on both CST sides.

Parameters:

count (int, default 1) – Number of Bezier degree-reduction steps applied to each side shape curve.
continuity ({"NOT_CONNECTED", "C0", "C1", "C2"},) – default=”NOT_CONNECTED” Symmetric endpoint continuity preserved during each side demotion step.

Returns:

Rebuilt CST airfoil with reduced-degree side shape curves.

Return type:

CstGeometry

Notes

This operation is intentionally approximate unless the side shape curves are exactly reducible to the requested lower degree.

dydx(u)

Return the surface slope at curve parameter locations.

Parameters:: u (buffalo_core.typing.FloatInput) – Airfoil parameters.
Returns:: Surface slope values dy/dx evaluated at u.
Return type:: buffalo_core.typing.FloatArray

k(u)

Calculate the curvature at parameter location.

Parameters:: u (buffalo_core.typing.FloatInput) – Parameter for desired locations.
Returns:: Curvature of surface matching the normalized shape of u.
Return type:: buffalo_core.typing.FloatArray

leading_edge()

Return the leading-edge location.

Returns:: (x, y) location of the leading-edge reference point.
Return type:: tuple[FloatScalar, FloatScalar]

property length: buffalo_core.typing.FloatScalar

Return the full airfoil surface length.

Returns:: Total airfoil surface length measured from the lower trailing edge to the upper trailing edge.
Return type:: buffalo_core.typing.FloatScalar

normal(u)

Calculate the unit normal at parameter location.

Parameters:: u (buffalo_core.typing.FloatInput) – Parameter for desired locations.
Returns:: Tuple (n_x, n_y) of float64 arrays matching the normalized shape of u.
Return type:: tuple[FloatArray, FloatArray]

promote_degree(*, count=1)

Raise the Bezier shape degree on both CST sides.

Parameters:: count (int, default 1) – Number of Bezier degree-elevation steps applied to each side shape curve.
Returns:: Rebuilt CST airfoil with exact elevated side shape curves.
Return type:: CstGeometry

slope_from_xi(xi, *, surface)

Return one-surface slope values at surface-local xi locations.

Parameters:

xi (buffalo_core.typing.FloatInput) – Surface-local coordinates in [0, 1] measured from the leading edge to the trailing edge.
surface ({"lower", "upper"}) – Surface to evaluate.

Returns:

Surface slope values dy/dx on the selected surface.

Return type:

buffalo_core.typing.FloatArray

tangent(u)

Calculate the unit tangent at parameter location.

Parameters:: u (buffalo_core.typing.FloatInput) – Parameter for desired locations.
Returns:: Tuple (t_x, t_y) of float64 arrays matching the normalized shape of u.
Return type:: tuple[FloatArray, FloatArray]

to_spec()

Return the schema definition needed to recreate this airfoil.

Returns:: Serialized airfoil definition that can recreate this runtime object.
Return type:: AirfoilDefinitionSpec

Notes

For runtime families covered by the current schema round-trip contract, this returns the same schema content as spec.

trailing_edge()

Return the midpoint of the trailing-edge points.

Returns:: (x, y) location of the trailing-edge midpoint reference.
Return type:: tuple[FloatScalar, FloatScalar]

property trailing_edge_thickness: buffalo_core.typing.FloatScalar

Return the explicit trailing-edge thickness.

This property reports the explicit trailing-edge gap as a fraction of chord.

u_from_s(s)

Return curve parameters that correspond to arc length.

Parameters:: s (buffalo_core.typing.FloatInput) – Arc lengths measured from the lower trailing edge.
Returns:: Curve parameters corresponding to s.
Return type:: buffalo_core.typing.FloatArray
Raises:: ValueError – When arc-length provided is larger than airfoil surface length.

u_from_x(x, *, surface)

Return curve parameters that correspond to x.

Parameters:

x (buffalo_core.typing.FloatInput) – Chordwise coordinates in the normalized airfoil frame.
surface ({"lower", "upper"}) – Surface to solve on.

Returns:

Curve parameters on the requested surface.

Return type:

buffalo_core.typing.FloatArray

Raises:

ValueError – If any requested chordwise coordinate lies outside the reachable x-range of the selected surface.

u_from_xi(xi, *, surface)

Convert surface-local xi coordinates to native parameters.

Parameters:

xi (buffalo_core.typing.FloatInput) – Surface-local coordinates in [0, 1] measured from the leading edge to the trailing edge.
surface ({"lower", "upper"}) – Surface to evaluate.

Returns:

Signed native CST airfoil parameters matching xi on the selected surface.

Return type:

buffalo_core.typing.FloatArray

Notes

General CST geometry uses the linear mapping u = +/- xi, with the sign determined by surface.

xi_from_u(u)

Convert native parameters to surface-local xi coordinates.

Parameters:: u (buffalo_core.typing.FloatInput) – Signed native CST airfoil parameters in [-1, 1].
Returns:: Surface-local xi values and upper-surface membership flags.
Return type:: SurfaceMappedValues

Notes

General CST geometry uses the linear mapping xi = |u|.

xy_from_s(s)

Return curve coordinates at arc-length locations.

Parameters:: s (buffalo_core.typing.FloatInput) – Arc length location of point.
Returns:: (x, y) coordinates matching the normalized shape of s.
Return type:: tuple[FloatArray, FloatArray]

xy_from_xi(xi, *, surface)

Return one-surface coordinates at surface-local xi locations.

Parameters:

xi (buffalo_core.typing.FloatInput) – Surface-local coordinates in [0, 1] measured from the leading edge to the trailing edge.
surface ({"lower", "upper"}) – Surface to evaluate.

Returns:

Tuple (x, y) of float64 arrays matching the normalized shape of xi.

Return type:

tuple[FloatArray, FloatArray]

xy_s(s)

Calculate first derivatives at arc-length location.

Parameters:: s (buffalo_core.typing.FloatInput) – Arc length location of point.
Returns:: (dx/ds, dy/ds) coordinates matching the normalized shape of s.
Return type:: tuple[FloatArray, FloatArray]

Notes

If s matches one of arc_length_breakpoints() exactly, this method returns the minus-side derivative limit. Subclasses should override xy_s_breakpoint() when exact one-sided breakpoint derivatives are available analytically.

xy_s_breakpoint(*, index)

Return one-sided arc-length derivatives at one breakpoint index.

Notes

This method composes the exact arc-length tangent values from the exact native breakpoint derivatives returned by xy_u_breakpoint().

Return type:: tuple[tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)], tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)]]

xy_ss(s)

Calculate second derivatives at arc-length location.

Parameters:: s (buffalo_core.typing.FloatInput) – Arc length location of point.
Returns:: (d^2x/ds^2, d^2y/ds^2) coordinates matching the normalized shape of s.
Return type:: tuple[FloatArray, FloatArray]

Notes

If s matches one of arc_length_breakpoints() exactly, this method returns the minus-side derivative limit. Subclasses should override xy_ss_breakpoint() when exact one-sided breakpoint second derivatives are available analytically.

xy_ss_breakpoint(*, index)

Return one-sided arc-length second derivatives at one breakpoint.

Notes

This method composes the exact arc-length curvature-vector values from the exact native breakpoint derivatives returned by xy_u_breakpoint() and xy_uu_breakpoint().

Return type:: tuple[tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)], tuple[TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’), TypeAliasForwardRef(‘buffalo_core.typing.FloatScalar’)]]