Noted at this time, the effort described below is limited to the unique reflective qualities of the Parabola or Paraboloid regardless of reflected wave form. It assumes the following unless specified otherwise: a constrained nominal parabola vertex (origin) and focus; planar wave front normal to the axis of rotation; a point source focus emanating a spherical wave front; a reflected spherical wave front from any discrete point on the Paraboloid surface. Therefore, best case scenarios are represented (i.e., an “as measured” surface may possess worse performance characteristics, but not better). Additionally noted, the use of Parabola (2D) and Paraboloid (3D) interchangeably.
To begin… some context. The Parabola and the Circle, as defined by conic section, exists as singular entities. This, unlike the Ellipse (special exception for the Circle) and Hyperbola which are infinite in shape variation. Our ability to manipulate the Parabola and Circle is limited strictly to dilation. However, while the Circle can be offset with a “parallel” path creating another Circle, the Parabola cannot. A locus of points offset “parallel” to the Parabola does not, and cannot, constitute a new Parabola. Thus, a distinguishable uniqueness of the Parabola. Simply put, any point not on the Parabola does not possess its qualities of reflection. As a result, real world efforts in Paraboloid fabrication most certainly result in an overall Hyperboloid, Ellipsoid or combination of the two but likely never the Paraboloid.
Please reference prior posts: ANGLE of REFLECTION as AFFECTED by DEVIATIONS from PARABOLA ECCENTRICITY; OFFSET or PARALLEL PATH to the PARABOLA; THREE-DIMENSIONAL COMPENSATION to the PARABOLOID
As to manufacturing, the geometric characteristic of the nominal Paraboloid surface, while easily defined mathematically in its most basic two-dimensional form as y=x2, is often controlled by GD&T (Geometric Dimensioning & Tolerancing) using “Profile of a Surface” (⌓) to establish compliance within an allowable deviation. Additionally, a common metrology metric used to evaluate Paraboloid surface conformance is RMS (Root Mean Square). Both view the nominal paraboloid surface within a “parallel” tolerance “band” which by definition is not paraboloid in shape. Adequate? Yes. Correct? This seems debatable to myself. I would offer alternative methodologies with the assertion that we define and measure conformance based on the specific reflective properties of the Parabola.
To that end, the use of eccentricity seems most appropriate. It describes the essence of Parabola uniqueness as function of the controlling vertex, focus and directrix. The eccentricity (e) of the Parabola is 1, given as the ratio of the distance from focus (f) to a point (P) on the Parabola to the distance from same point (P) to its normal intercept of the directrix. Using eccentricity as conformance criteria necessitates the delineation of a tolerance “band” as a “non-parallel” region, straddling the nominal Parabola and tapering to convergence at the vertex. This bilateral “band” then contains all allowable vertex constrained paraboloids. This can be thought of as analogous to GD&T “Circularity” (◯) yet instead the Circle, the Parabola. An example might appear as a given Paraboloid surface equation y=(√(x2+z2))2 with an eccentricity (e) tolerance of 1 ± .0002 or 1.0002/.9998 (unitless). Eccentricity is easily calculated for discrete points (P) measured on the Paraboloid surface and evaluated against the defined tolerance. Deviation in the extreme would re-define the geometry into something other than a Parabola.
Another similar approach to the same is the utilization of an exponent solution. Using y=axn for a deviated point (P), this is the natural logarithm solution for n where a point lying on the Parabola has a value n of 2.
Yet another, albeit different method, is the application of the 2f rule (stated below). It allows for the calculation of reflected spherical wave front deviation (∆W) where Tx is the normal intercept with the latus rectum and Rx is the intercept with the axis of rotation at X0,Yf,Z0. The result is the in-phase (at 2f) intercept over or under-shoot of the nominal focus.
2f Rule: For the Parabola, the total path length of any incoming signal, parallel to the Axis of Symmetry, originating at the Latus Rectum and reflected to termination at the Focus is twice the distance from Vertex to Focus.
Lastly, applying y=ax2 to the deviated point (P) while constraining the vertex allows Parabola dilation and a resultant focus as origin of Tx and terminus of Rx for a reflected spherical wave front. Focal length deviation (∆f) from nominal (X0,Yf,Z0) to resultant (X0,Yfn,Z0) can then be determined offering a catacaustic curve (in this case linear) representation.
For the later examples, these calculations can yield density graphs. A histogram to depict a normalized probability density, with frequency specific bin width equal to allowable phase shift (i.e., 1/16th λ). And, a KDE (Kernel Density Estimation) plot to show a normalized statistical density. In each case, graphs are centered about the nominal Parabola focus. For the KDE plot of wave front deviation (2f rule), the area under the phase shift curve represents over-shot (ellipsoidal data) and under-shot (hyperboloidal data) densities relative to nominal focus. In the KDE plot of focal length deviation (∆f), the area under the phase shift curve represents data densities along the optical axis relative to nominal focus with the highest peak indicating optimal focus. In-phase data in both cases is represented directly underneath any specific point along the phase shift curves.
n < 2 or e > 1 [Hyperbolic]
n = 2 or e = 1 [Parabolic]
n > 2 or 0 < e < 1 [Elliptical]
D [Aperture Diameter (m)]
O [Axis Offset Displacement (in)]
f = ( D .1016 ) – ( O / 2 ) [Focal Length (in)]
a = 1 / ( 4f ) [Dilation Coefficient]
d = -f [Directrix (in)]
X [Paraboloid Metrology Data, Vertical Axis (in)]
Y [Paraboloid Metrology Data, Optical Axis (in)]
Z [Paraboloid Metrology Data, Horizontal Axis (in)]
x = ( √ ( X2 + Z2 ) ) – O [Resolved for Point P (in)]
y = Y [Resolved for Point P (in)]
n = ( ln ( y / a )) / ln x [Exponent Solution]
e = ( √ ( x2 + ( f – y )2 )) / ( y – d ) [Eccentricity]
∆W† = ( f + y ) – ( √ ( x2 + ( f – y ) 2 ) ) [Wave Front Deviation, 2f Rule (in)]
∆fn‡ = ( ( 1 / 4 ) / ( y / x 2) ) – f [Focal Length Deviation (in)]
∆P* = ( 1 / (8a3x2)) (4a2x2(√(4a2x2 + 1)) + (√(4a2x2 + 1)) – (√((4a2x2 + 1) (16a3x2 y + 8a2x2 + 1)))) [Data Point Deviation (in)]
∆PRMS = √ ( ( P12 + P22 … Pn2 ) / n ) [Root Mean Square (in)]
ρ = n / ( ∆fnmax + | ∆fnmin | ) [Focal Density Along Optical Axis]
* Deviation normal to nominal surface
† Best case reflected wave front (Tx – normal intercept with lr plane, Rx – intercept with optical axis at resolved 0,f,0)
‡ fn is resolved origin for Tx, and terminus for Rx, of dilated paraboloid (vertex constrained) calculated from deviated point, Best Case
Nominal Focus (f) Tx, Focal Axis Parallelism for Deviated Point Px,y
+ value converges with focal axis
– value diverges from focal axis
Δ∠° = (tan-1 ((f – y) / x)) – (tan-1 ((( x2 / 4y) – y) / x))
