Office Action 1: Notice of Allowance (2025-05-30)

CLAIMS:
What is claimed is:
1\. An atmospheric re-entry vehicle, comprising: a main body defining a forward end of the vehicle; a base defining an aft end of the vehicle; and a heat shield at the base, the heat shield including a heat shield outer surface; wherein the main body and the heat shield are configured such that a centerline of the heat shield is offset relative to a centerline of the main body; and wherein at least a portion of the heat shield outer surface is at least substantially axisymmetric relative to the centerline of the heat shield.
2\. The atmospheric re-entry vehicle of claim 1, wherein the main body includes a main body outer surface that is at least substantially axisymmetric relative to the centerline of the main body.
3\. The atmospheric re-entry vehicle of claim 1, wherein the centerline of the main body extends in a direction between the forward end of the vehicle and an aft end of the main body; and wherein the centerline of the heat shield extends in a direction between a forward end of the heat shield and an aft end of the heat shield.
4\. The atmospheric re-entry vehicle of claim 1, wherein the centerline of the main body is perpendicular to a tangent of a forwardmost point of the main body; and wherein the centerline of the heat shield is perpendicular to a tangent of an aftmost point of the heat shield.
5\. The atmospheric re-entry vehicle of claim 1, wherein the centerline of the heat shield is offset relative to the centerline of the main body by an offset angle within a range of 1° to 10°.
6\. The atmospheric re-entry vehicle of claim 1, wherein the heat shield outer surface is configured such that net aerodynamic forces acting on the heat shield outer surface during atmospheric re-entry are at an angle relative to the centerline of the main body.
7\. The atmospheric re-entry vehicle of claim 1, wherein the heat shield outer surface is configured such that net aerodynamic forces acting on the heat shield outer surface generate lift during atmospheric re-entry at a zero angle of attack.
8\. The atmospheric re-entry vehicle of claim 1, wherein the heat shield is actively cooled.
9\. The atmospheric re-entry vehicle of claim 1, wherein the vehicle is a rocket.
10\. The atmospheric re-entry vehicle of claim 1, wherein the vehicle is a reusable upper stage rocket of a multi-stage rocket system.
11\. The atmospheric re-entry vehicle of claim 1, further comprising a nozzle including a throat and a centerbody extending aft of the throat; wherein the centerbody defines the at least a portion of the heat shield outer surface that is at least substantially axisymmetric relative to the centerline of the heat shield.
12\. The atmospheric re-entry vehicle of claim 11, wherein the nozzle is an aerospike nozzle.
13\. The atmospheric re-entry vehicle of claim 1, further comprising: a high pressure chamber; and a nozzle that exhausts gas from the high pressure chamber, the nozzle including a throat and a centerbody extending aft of the throat; wherein the centerbody defines the at least a portion of the heat shield outer surface that is at least substantially axisymmetric relative to the centerline of the heat shield.
14\. The atmospheric re-entry vehicle of claim 13, wherein the centerbody is a truncated toroidal aerospike.
15\. The atmospheric re-entry vehicle of claim 1, further comprising: a high pressure chamber; and a nozzle that exhausts gas from the high pressure chamber, the nozzle including a throat and a centerbody extending aft of the throat; wherein the centerbody includes a centerbody sidewall defining an expansion surface and a centerbody base extending aft of the centerbody sidewall.
16\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody is a truncated toroidal aerospike.
17\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody base defines the at least a portion of the heat shield outer surface that is at least substantially axisymmetric relative to the centerline of the heat shield.
18\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody sidewall is non-axisymmetric relative to the centerline of the heat shield.
19\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody sidewall and the centerbody base collectively form a blunt body.
20\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody sidewall includes a rigid wall with a cone shape.
21\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody sidewall has a truncated and oblique cone shape.
22\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody base has a semi-spherical shape.
23\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody base has a frustoconical shape.
24\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody base has a multi-conic shape.
25\. The atmospheric re-entry vehicle of claim 15, wherein the centerbody base has an ellipsoidal shape.
26\. The atmospheric re-entry vehicle of claim 15, wherein at least one of the centerbody sidewall and the centerbody base is actively cooled.
27\. The atmospheric re-entry vehicle of claim 1, wherein the heat shield includes a first heat shield outer surface and a second heat shield outer surface; wherein the first heat shield outer surface is at least substantially axisymmetric relative to the centerline of the heat shield; and wherein the second heat shield outer surface is non-axisymmetric relative to the centerline of the heat shield.
28\. The atmospheric re-entry vehicle of claim 1, further comprising a plurality of thrust cans spaced relative to one another; wherein each thrust can is disposed relative to one of a plurality of openings extending through the heat shield.
29\. The atmospheric re-entry vehicle of claim 28, wherein each thrust can includes a high pressure chamber and an initial nozzle portion.
30\. The atmospheric re-entry vehicle of claim 28, wherein the plurality of openings extending through the heat shield are circumferentially spaced relative to one another.
31\. The atmospheric re-entry vehicle of claim 1, wherein the centerline of the main body is perpendicular to a tangent of a forwardmost point of the main body; wherein the centerline of the heat shield is perpendicular to a tangent of an aftmost point of the heat shield; wherein the main body includes a main body outer surface that is at least substantially axisymmetric relative to the centerline of the main body.
32\. The atmospheric re-entry vehicle of claim 31, wherein the centerline of the heat shield is offset relative to the centerline of the main body by an offset angle within a range of 1° to 10°.
33\. The atmospheric re-entry vehicle of claim 31, wherein the heat shield outer surface is configured such that net aerodynamic forces acting on the heat shield outer surface during atmospheric re-entry are at an angle relative to the centerline of the main body.
34\. The atmospheric re-entry vehicle of claim 31, wherein the heat shield outer surface is configured such that net aerodynamic forces acting on the heat shield outer surface generate lift during atmospheric re-entry at a zero angle of attack.
35\. The atmospheric re-entry vehicle of claim 31, wherein the heat shield includes a first heat shield outer surface and a second heat shield outer surface; wherein the first heat shield outer surface is at least substantially axisymmetric relative to the centerline of the heat shield; and wherein the second heat shield outer surface is non-axisymmetric relative to the centerline of the heat shield.
36\. The atmospheric re-entry vehicle of claim 31, further comprising a plurality of thrust cans spaced relative to one another; wherein each thrust can is disposed relative to one of a plurality of openings extending through the heat shield.

OFFICE ACTION:
Notice of Pre-AIA  or AIA  Status
The present application, filed on or after March 16, 2013, is being examined under the first inventor to file provisions of the AIA .
Reasons for Allowance
The following is an examiner’s statement of reasons for allowance: claims 1 – 36 are allowable over the prior art of record. The prior art of record fails to teach the main body and the heat shield are configured such that a centerline of the heat shield is offset relative to a centerline of the main body; and wherein at least a portion of the heat shield outer surface is at least substantially axisymmetric relative to the centerline of the heat shield.
Any comments considered necessary by applicant must be submitted no later than the payment of the issue fee and, to avoid processing delays, should preferably accompany the issue fee.  Such submissions should be clearly labeled “Comments on Statement of Reasons for Allowance.”
Conclusion



CITED REFERENCE SUMMARIES: Ref US20240151194A1: Claims:
An apparatus, comprising: a nozzle configured to control a flow of thrust from the apparatus; and an ablative material inside of or attached to the nozzle, configured to deflect the thrust at a predefined angle, wherein the ablative material is configured to ablate-away, leaving un-deflected thrust for a majority of the burn, or a fugacious material configured to ablate-away, melt-away, or be removed by ablating or melting a portion of a supporting structure of the fugacious material.
a nozzle configured to control a flow of thrust from the apparatus; and
an ablative material inside of or attached to the nozzle, configured to deflect the thrust at a predefined angle, wherein
the ablative material is configured to ablate-away, leaving un-deflected thrust for a majority of the burn, or
a fugacious material configured to ablate-away, melt-away, or be removed by ablating or melting a portion of a supporting structure of the fugacious material.
The apparatus of claim 1 , wherein the fugacious material is configured to deflect the thrust at a predefined angle.
The apparatus of claim 1 , wherein the fugacious material is configured to ablate away, leaving substantially un-deflected thrust for a majority of the burn.
The apparatus of claim 1 , wherein the fugacious material includes a plurality of elements, the plurality of elements being fabricated as a single part or placed within one or more other parts.
The apparatus of claim 1 , wherein the ablative material or the fugacious material is in the form of an adhesive.
The apparatus of claim 5 , wherein the adhesive is filled with additives to modify an ablation rate.
The apparatus of claim 1 , wherein the ablative material comprises a hole in the center and is positioned in an axially to a direction of the nozzle, the ablative material together with the hole are configured to act as a timing feature, increasing a flow of thrust from the hole as the hole ablates away.
the ablative material together with the hole are configured to act as a timing feature, increasing a flow of thrust from the hole as the hole ablates away.
The apparatus of claim 1 , further comprising: a meltable deflector attached to the inside of the nozzle using a meltable interface material, wherein the meltable deflector is composed of a ceramic material or a metal, and the meltable interface material is composed of polymers, elastomers, metals, or alloys.
a meltable deflector attached to the inside of the nozzle using a meltable interface material, wherein
the meltable deflector is composed of a ceramic material or a metal, and
the meltable interface material is composed of polymers, elastomers, metals, or alloys.
The apparatus of claim 1 , further comprising: one or more fasteners configured to couple the ablative material or the fugacious material to the interior of the nozzle.
one or more fasteners configured to couple the ablative material or the fugacious material to the interior of the nozzle.
The apparatus of claim 1 , further comprising: epoxy material configured to couple the ablative material or fugacious material to the interior of the nozzle.
epoxy material configured to couple the ablative material or fugacious material to the interior of the nozzle.
The apparatus of claim 1 , wherein the nozzle with a plurality of integrated fugacious elements are configured to redirect the rocket exhaust plume, creating a torque before being removed by ablation or melting from the plume.
The apparatus of claim 10 , wherein the nozzle is manufactured by way of machining, additive manufacturing, three dimensional (3D) printing, or casting or injection molding.
An apparatus, comprising: a nozzle configured to control a flow of thrust from the apparatus; and an ablative material inside of or attached to the nozzle, configured to deflect the thrust at a predefined angle, wherein the ablative material being configured to ablate-away, leaving un-deflected thrust for a majority of the burn; a fugacious material configured to ablate-away, melt-away, or be removed by ablating or melting a portion of a supporting structure of the fugacious material; or a meltable deflector attached to the inside of the nozzle using a meltable interface material, wherein the meltable deflector is composed of a ceramic material or a metal, and the meltable interface material is composed of polymers, elastomers, metals, or alloys.
an ablative material inside of or attached to the nozzle, configured to deflect the thrust at a predefined angle, wherein the ablative material being configured to ablate-away, leaving un-deflected thrust for a majority of the burn;
a fugacious material configured to ablate-away, melt-away, or be removed by ablating or melting a portion of a supporting structure of the fugacious material; or
The apparatus of claim 13 , wherein the fugacious material is configured to deflect the thrust at a predefined angle.
The apparatus of claim 13 , wherein the fugacious material is configured to ablate away, leaving substantially un-deflected thrust for a majority of the burn.
The apparatus of claim 13 , wherein the fugacious material includes a plurality of elements, the plurality of elements being fabricated as a single part or placed within one or more other parts.
The apparatus of claim 13 , wherein the ablative material or the fugacious material is in the form of an adhesive.
The apparatus of claim 17 , wherein the adhesive is âfilledâ with additives to modify an ablation rate.
The apparatus of claim 13 , wherein the ablative material comprises a hole in the center and is positioned in an axially to a direction of the nozzle, the ablative material together with the hole are configured to act as a timing feature, increasing a flow of thrust from the hole as the hole ablates away.
The apparatus of claim 13 , further comprising: one or more fasteners configured to couple the ablative material or the fugacious material to the interior of the nozzle.
The apparatus of claim 13 , further comprising: epoxy material configured to couple an ablative material or a fugacious material to the interior of the nozzle.
epoxy material configured to couple an ablative material or a fugacious material to the interior of the nozzle.
The apparatus of claim 13 , wherein the nozzle with a plurality of integrated fugacious elements are configured to redirect the rocket exhaust plume, creating a torque before being removed by ablation or melting from the plume.
The apparatus of claim 22 , wherein the nozzle is manufactured by way of machining, additive manufacturing, three dimensional (3D) printing, or casting or injection molding.
Ref US7431242B1: Claims:
A nose cone structure for a body entering an atmosphere, that provides enhanced aerodynamic and aerothermal performance, the structure comprising: a convex shell having a longitudinally oriented x-axis and having a non-symmetric transverse cross sectional shape, when projected onto a plane parallel to an xz-plane of an (x,y,z)) Cartesian coordinate, the shell comprising a fore-body and an aft-body joined to the fore-body, the aft-body having a projection on a yz-plane comprising: first and second linear segments, spaced apart from each other at a first end of each segment by a selected distance of substantially 2R m and located on opposite sides of the x-axis, with each of the first and second segments being oriented substantially at a selected non-zero angle Î¸ c relative to the x-axis and having a second segment end having a separation distance of substantially R M from the x-axis; a third linear segment, oriented substantially perpendicular to the x-axis adjacent to the second end of each of the first and second linear segments; first and second curvilinear segments, tangent to and joining the first and third linear segments, and tangent to and joining the second and third linear segments, respectively; and a third curvilinear segment, having first and second ends, positioned tangent at the first end of the third curvilinear segment to the second end of the first linear segment; a fourth curvilinear segment, having first and second ends, positioned tangent at the first end of the fourth curvilinear segment to the second end of the second linear segment; and the fore-body having a projection on the yz-plane comprising: a fifth curvilinear segment, having first and second ends, positioned tangent to a second end of the first linear segment at a fifth curvilinear segment first end; a sixth curvilinear segment, having first and second ends, positioned tangent to a second end of the second linear segment at a sixth curvilinear segment first end; and a fourth linear segment, having first and second ends, positioned tangent to the fifth curvilinear segment and tangent to the sixth curvilinear segment at the respective first and second ends of the fourth linear segment.
a convex shell having a longitudinally oriented x-axis and having a non-symmetric transverse cross sectional shape, when projected onto a plane parallel to an xz-plane of an (x,y,z)) Cartesian coordinate, the shell comprising a fore-body and an aft-body joined to the fore-body, the aft-body having a projection on a yz-plane comprising: first and second linear segments, spaced apart from each other at a first end of each segment by a selected distance of substantially 2R m and located on opposite sides of the x-axis, with each of the first and second segments being oriented substantially at a selected non-zero angle Î¸ c relative to the x-axis and having a second segment end having a separation distance of substantially R M from the x-axis; a third linear segment, oriented substantially perpendicular to the x-axis adjacent to the second end of each of the first and second linear segments; first and second curvilinear segments, tangent to and joining the first and third linear segments, and tangent to and joining the second and third linear segments, respectively; and a third curvilinear segment, having first and second ends, positioned tangent at the first end of the third curvilinear segment to the second end of the first linear segment; a fourth curvilinear segment, having first and second ends, positioned tangent at the first end of the fourth curvilinear segment to the second end of the second linear segment; and
first and second linear segments, spaced apart from each other at a first end of each segment by a selected distance of substantially 2R m and located on opposite sides of the x-axis, with each of the first and second segments being oriented substantially at a selected non-zero angle Î¸ c relative to the x-axis and having a second segment end having a separation distance of substantially R M from the x-axis;
a third linear segment, oriented substantially perpendicular to the x-axis adjacent to the second end of each of the first and second linear segments;
first and second curvilinear segments, tangent to and joining the first and third linear segments, and tangent to and joining the second and third linear segments, respectively; and
a third curvilinear segment, having first and second ends, positioned tangent at the first end of the third curvilinear segment to the second end of the first linear segment;
a fourth curvilinear segment, having first and second ends, positioned tangent at the first end of the fourth curvilinear segment to the second end of the second linear segment; and
the fore-body having a projection on the yz-plane comprising: a fifth curvilinear segment, having first and second ends, positioned tangent to a second end of the first linear segment at a fifth curvilinear segment first end; a sixth curvilinear segment, having first and second ends, positioned tangent to a second end of the second linear segment at a sixth curvilinear segment first end; and a fourth linear segment, having first and second ends, positioned tangent to the fifth curvilinear segment and tangent to the sixth curvilinear segment at the respective first and second ends of the fourth linear segment.
a fifth curvilinear segment, having first and second ends, positioned tangent to a second end of the first linear segment at a fifth curvilinear segment first end;
a sixth curvilinear segment, having first and second ends, positioned tangent to a second end of the second linear segment at a sixth curvilinear segment first end; and
a fourth linear segment, having first and second ends, positioned tangent to the fifth curvilinear segment and tangent to the sixth curvilinear segment at the respective first and second ends of the fourth linear segment.
The structure of claim 1 , wherein said first and second curvilinear segments are sectors of first and second circles having substantially equal radii.
The structure of claim 1 , wherein said third and fourth curvilinear segments are sectors of third and fourth circles having substantially equal radii.
The structure of claim 1 , wherein said fifth curvilinear segment comprises a section of a first ellipse, having a selected first major axis length and a selected first minor axis length.
The structure of claim 4 , wherein said sixth curvilinear segment comprises a section of a second ellipse, having a selected second major axis length and a selected second minor axis length.
The structure of claim 4 , wherein at least one of said angle Î¸ c , said distance 2R M , said first major axis, said first minor axis, said second major axis and said second minor axis are chosen so that an aerodynamic lift force associated with said structure during re-entry of said structure into a selected atmosphere is at least equal to a selected threshold lift force.
The structure of claim 1 , wherein; said fifth curvilinear segment, projected on an xz-plane y=0, is substantially a portion of a first ellipse that is described by relations x 2 /(H 1 ) 2 +(zâz 5 ) 2 /(H 2 ) 2 =1, where H 1 and H 2 are selected lengths of a first semi-major axis and a first semi-minor axis of the first ellipse; and said sixth curvilinear segment, projected on an xz-plane is substantially a portion of a second ellipse that is described by relations x 2 /(H 3 ) 2 +(zâz 6 ) 2 /(H 4 ) 2 =1, where H 3 and H 4 are selected lengths of a second semi-major axis and a second semi-minor axis of the second ellipse.
said fifth curvilinear segment, projected on an xz-plane y=0, is substantially a portion of a first ellipse that is described by relations x 2 /(H 1 ) 2 +(zâz 5 ) 2 /(H 2 ) 2 =1, where H 1 and H 2 are selected lengths of a first semi-major axis and a first semi-minor axis of the first ellipse; and
said sixth curvilinear segment, projected on an xz-plane is substantially a portion of a second ellipse that is described by relations x 2 /(H 3 ) 2 +(zâz 6 ) 2 /(H 4 ) 2 =1, where H 3 and H 4 are selected lengths of a second semi-major axis and a second semi-minor axis of the second ellipse.
The structure of claim 1 , wherein said fore-body shape is defined by relations
x 2 / a â¡ ( z ) 2 + ( y 2 + z 2 ) / R M 2 = 1 , â¢ a â¡ ( z ) = H 1 â¢ { 1 - ( z - z 5 ) 2 / ( H 2 ) 2 } 1 / 2 â¢ ( - R M â¢ < _ â¢ z â¢ < _ â¢ z a ) , = A â¡ ( z - z 1 ) + B â¡ ( z a â¢ < _ â¢ z â¢ < _ â¢ z b ) = H 3 â¢ { 1 - ( z - z 6 ) 2 / ( H 4 ) 2 } 1 / 2 â¢ ( z b â¢ < _ â¢ z â¢ < _ â¢ R M ) , â¢ z 5 = - R M + H 2 , z 6 = R M - H 4 ,
where R M is a selected positive radius value, H 1 and H 2 are respective selected lengths of a semi-major axis and a semi-minor axis of a first ellipse, H 3 and H 4 are respective selected lengths of a semi-major axis and a semi-minor axis of a second ellipse, each coordinate pair, (z a ,x a ) and (z b ,x b ), lies on said fourth linear segment, and A and B are selected constant values for said fourth linear segment.
The structure of claim 8 , wherein A=m=(x b âx a )/(z b âz a ) is a slope of said fourth linear segment with respect to said z-axis and B is chosen to be said value x a and z 1 =z a .
The structure of claim 8 , wherein A=m=(x b âx a )/(z b âz a ) is a slope of said fourth linear segment with respect to said z-axis and B is chosen to be said value x b and z 1 =z b .
The structure of claim 1 , wherein said aft-body comprises a substantially cylindrically symmetric, having a maximum diameter of substantially 2R M , a length x L , and being defined substantially by the relations y 2 +z 2 ={R M âx Â·tan Î¸ c } 2 , (0â¦xâ¦x L ) tan Î¸ c =( R M âR m )/x L .
y 2 +z 2 ={R M âx Â·tan Î¸ c } 2 , (0â¦xâ¦x L )
tan Î¸ c =( R M âR m )/x L .
The structure of claim 1 , wherein, for âR M â¦zâ¦R M and âR z â¦yâ¦R z with R z ={R M 2 âZ 2 } 1/2 , said fore-body has an extent in said x-coordinate direction given by 0â¦xâ¦a(z){1â(y/R z ) 2 } 1/2 .
The structure of claim 1 , wherein, for âR M â¦zâ¦R M and âR z â¦y<R z with R z ={R M 2 âZ 2 } 1/2 , said fore-body has an extent in said x-coordinate direction given by x=w(y,z)Â·a(R yz )+(1âw(y,z))a(âR yz ), with R yz {y 2 +z 2 } 1/2 , where w(y,z) is a selected weight function having a domain of âR M â¦zâ¦R M and âR z â¦yâ¦R z and a range of [0,1], and a(z) is the extent of said fore-body at y=0 in the x-coordinate direction.
The structure of claim 1 , wherein at least one of said fourth linear segment, said fifth curvilinear segment and said sixth curvilinear segment is replaced by a function P(z) that is at least one of a polynomial and a trigonometric function satisfying the following conditions: ( dP/dz ) z=zâ²a =( H 1 /H 2 ) zâ² a /{1 âzâ² a 2 } 1/2 , ( dP/dz ) z=zâ²b =( H 3 /H 4 ) zâ² b /{1 âzâ² b 2 } 1/2 . P ( zâ² a )= H 1 {1â zâ² a 2 )} 1/2 , P ( zâ² b )= H 1 {1â zâ² b 2 )} 1/2 , zâ² a =( z a âz 5 )? H 2 , zâ² b =( z b âz 6 )? H 4 , d 2 P/dz 2 <0( zâ² A â¦zâ¦zâ² B ),
( dP/dz ) z=zâ²a =( H 1 /H 2 ) zâ² a /{1 âzâ² a 2 } 1/2 ,
( dP/dz ) z=zâ²b =( H 3 /H 4 ) zâ² b /{1 âzâ² b 2 } 1/2 .
P ( zâ² a )= H 1 {1â zâ² a 2 )} 1/2 ,
P ( zâ² b )= H 1 {1â zâ² b 2 )} 1/2 ,
zâ² a =( z a âz 5 )? H 2 ,
zâ² b =( z b âz 6 )? H 4 ,
d 2 P/dz 2 <0( zâ² A â¦zâ¦zâ² B ),
where z a and z b are selected values of said coordinate z.
The structure of claim 1 , wherein said aft-body projection on said yz-plane is determined by a weighted sum S(z)(âR M â¦zâ¦R M ) of (i) a combination of said fifth curvilinear segment C 5 , said fourth linear segment L 4 and said sixth curvilinear segment C 6 and (ii) a selected polynomial or trigonometric function P(z), having respective weights w and 1âw, where the sum S(z) is tangent to said second end of said third curvilinear segment and is tangent to said second end of said fourth curvilinear segment, respectively.
A nose cone structure for a body entering an atmosphere, that provides enhanced aerodynamic and aerothermal performance, the structure comprising: a convex shell having a longitudinally oriented x-axis and having a non-symmetric transverse cross sectional shape, when projected onto a plane parallel to an xz-plane of an (x,y,z)) Cartesian coordinate, the shell comprising a fore-body and an aft-body joined to the fore-body, the aft-body having a projection on a yz-plane comprising: first and second linear segments, spaced apart from each other at a first end of each segment by a selected distance of substantially 2R m and located on opposite sides of the x-axis, with each of the first and second segments being oriented substantially at a selected non-zero angle Î¸ c relative to the x-axis and having a second segment end having a separation distance R M from the x-axis; a third linear segment, oriented substantially perpendicular to the x-axis adjacent to the second end of each of the first and second linear segments; first and second curvilinear segments, tangent to and joining the first and third linear segments, and tangent to and joining the second and third linear segments, respectively; and a third curvilinear segment, having first and second ends, positioned tangent at the first end of the third
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