Draft:Gaussianoid conical frustum

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Gaussianoid Conical Frustum (GCF) (R=6 mm, θ=8°) shown via three cross-sections with Gaussian distributions. The blue line denotes the GCF axis; the red circle marks the radius enclosing 50% of the volume under each curve.

Gaussianoid Conical Frustum

A Gaussianoid Conical Frustum (GCF) is a four-dimensional geometric construct in which a traditional three-dimensional conical frustum is extended by associating a scalar field as a fourth dimension. The scalar quantity—such as temperature, concentration, electromagnetic intensity, probability, or other measurable parameters—varies across the spatial domain of the frustum. At any given cross-section along the frustum's principal axis, the scalar field distribution follows a 2D gaussian function, centered about the axis.

The GCF can be conceptualized as a 3D spatial volume whose radial cross-sections are embedded with scalar values forming a continuous field, thereby introducing an additional dimension of analysis. This modeling paradigm can be applied in scientific fields for describing spatial variations of physical, chemical, or statistical quantities in space and time.

The accompanying image visualizes a Gaussianoid Conical Frustum, described as $R=6\,{\text{mm}}$, $\theta =8^{\circ }$ through three horizontal cross-sections at progressive depths (e.g., $0\,{\text{mm}}$, $20\,{\text{mm}}$, and $40\,{\text{mm}}$) along the frustum's longitudinal axis. Key features illustrated include:

• The blue vertical line, representing the central axis of the GCF;
• The X and Y axes, defining the cross-sectional plane;
• The Z axis, depicted by consecutive slices along the spatial dimension;
• The ρ axis represents the scalar field;
• The red circle, described by mean radius, encloses a specified proportion, in this case 50%, of the scalar field's total volume under the curve each depth.

As the depth increases along the Z axis, the width of the Gaussian distribution in each cross-section also increases, following the increasing mean radius defined by the conical frustum angle.

Mathematical Representation

The concept of the Gaussianoid Conical Frustum (GCF) extends from classical geometry, beginning with the mathematical description of a conical frustum. A conical frustum may be defined as the portion of a cone that lies between two parallel planes cutting it, with the planes perpendicular to the axis. This results in a finite frustum bounded between two circular cross-sections of radii $r_{1}$ (bottom) and $r_{2}$ (top) and a height $h$. The volume $V$ of a finite conical frustum is:

$V={\frac {1}{3}}\pi h\left(r_{1}^{2}+r_{1}r_{2}+r_{2}^{2}\right)$

Alternatively, an infinite conical frustum (or conical surface) may be parameterized by a base radius $r_{1}$ and an angle $\theta$, corresponding to the generatrix angle of the cone, where the radius at any height $z$ is given by

$r(z)=r_{1}+z\tan \theta$ for $z\geq 0$.

Distinct from the standard geometric frustum, the GCF integrates a scalar quantity at each horizontal cross-section along the frustum’s axis. This scalar value—for example, temperature, electromagnetic intensity, or probability density—is is a two-dimensional Gaussian-like (gaussianoid) distribution centered on the frustum axis. The general form of the two-dimensional Gaussian function is:

$f(x,y)=A\exp \left(-\left[{\frac {(x-x_{0})^{2}}{2\sigma _{x}^{2}}}+{\frac {(y-y_{0})^{2}}{2\sigma _{y}^{2}}}\right]\right)$

Within the GCF framework, the parameters $\sigma _{x}$ and $\sigma _{y}$ may vary as functions of the longitudinal axis ($z$), describing the expanding or contracting spread of the distribution as one moves through the frustum. This embedding results in a four-dimensional object, where the spatial boundary evolves according to both the evolving radius of the frustum and the spatial characteristics of the scalar field at each cross-section.

The boundary at each $z$-position may be further defined by a quantile threshold (e.g., the radius enclosing 50% of the scalar field magnitude), which forms the characteristic conical frustum-like profile observed across the frustum’s length.

Conceptual Analogies

The Gaussianoid Conical Frustum represents just one example of a broader class of multi-dimensional constructs, wherein a traditional spatial solid is parameterized by an additional, often non-spatial, scalar field. While the fourth dimension is frequently associated with probability density—mirroring concepts prevalent in statistical mechanics and quantum physics—this framework is fully generalizable to any scalar property, such as temperature, chemical concentration, or intensity.

Such objects may be referred to as 4D parameterized solids or scalar-embedded manifolds.
Some generalized examples can be found in the following concepts:

• A parameterized spatial solid, where the 4th dimension embodies a continuously varying physical quantity such as temperature, density, radiopacity (CT scan) or water density (MRI scan).
• A 3D volume with an associated scalar field, a fundamental concept in fields such as vector calculus and continuum mechanics.
• A probability-embedded 3D shape, in the case that the scalar represents statistical density, echoing the use of probability fields in the study of quantum state distributions or diffusion processes.

This concept is analogous to describing a probability (or scalar) field over a domain in three-dimensional space, such as when visualizing electron cloud densities in chemistry or temperature distributions within a physical body. The resulting visualization is a 3D geometry whose cross-sections correspond to the spatial variation of the scalar field—here, governed by a Gaussian profile whose spatial extent changes along the axis, ultimately forming a conical or frustum-like shape within this 4D framework.

Applications

Depiction of a laser beam's pointing error as a Gaussianoid Conical Frustum (GCF). The probability of the beam passing through a given point forms a Gaussian profile that widens and decreases with distance from the source.

The Gaussianoid Conical Frustum (GCF) model serves as a useful abstraction in several scientific and technological domains where spatial uncertainty or field intensity varies systematically along an axis. One notable application is in the analysis of laser beam paths and beam divergence, where pointing errors are naturally modeled by a conical frustum spread, with a Gaussian distribution of probability profile, allowing accurate characterization of the spatial uncertainty or irradiance at varying distances from the source. Effectively describing the probability that the laser beam will pass through a given point in space by accounting for both the beam’s pointing error and the source’s positional uncertainty.

Gaussianoid conical frustum (GCF) K-wire position probability over 4 planar sections along the Z axis. Nerve strike risk estimation is displayed in the first column. The nerve is represented in yellow; mean radius in red.

Multiple K-wires are inserted into the left humerus, with the ulnar nerve and surrounding anatomical structures clearly visible. The GCF probability distribution is visible as the k-wire bundle spreading.

The GCF further finds application in medical and engineering contexts involving the placement and navigation of rod-like objects. For example, in navigated surgery or construction, the position and orientation uncertainties of K-wires, screws or drill bits are defined by a frustum-shaped region with Gaussian-like uncertainty or confidence levels at different depths. In the represented image a direct application of the GCF is represented: a K-wire placement is affected by a radial entrance point error and and angular positioning error, its positioning error, or final placement probability, is therfore described by a GCF probability.

A further example of the GCF concept occurs in ballistics and targeting applications. When a projectile is fired (in the absence of gravity or at sufficiently short ranges), the uncertainty in its trajectory due to aiming or mechanical tolerances forms a conical frustum region that expands with distance. The cross-sections perpendicular to the trajectory are characterized by a Gaussian or normal distribution of possible impact points, a foundational concept in the analysis of circular error probable (CEP) and precision of weapon systems.

See Also

• Frustum
• Gaussian curvature
• Differential geometry
• Circular error probable

Category:Probability Category:Geometric shapes Category:Differential geometry