Differentiable Ray–Ellipsoid Intersection

• Differentiable ray–ellipsoid intersection is defined by algebraic and geometric conditions that yield smooth intersection points and local tangent geometry.
• Closed-form representations and discriminant-based criteria enable efficient computation and robust intersection tests in practical applications.
• This approach is vital in computer graphics, simulation, and optimization, providing computational efficiency and numerical stability through precise algebraic analysis.

A differentiable ray–ellipsoid intersection concerns both the algebraic-geometric and computational aspects of determining and characterizing the points at which a ray intersects an ellipsoid in such a way that the intersection location, existence, and local tangent geometry are smoothly determined functions of all parameters. This is foundational in numerous applied and theoretical fields, including computer graphics, physical simulation, optimization, and geometric modeling. The research record provides rigorous closed-form representations and necessary and sufficient algebraic criteria for when and how these intersections occur and how they can be computed efficiently and robustly.

1. Algebraic Formulation and Parameterization

The ellipsoid in $\mathbb{R}^n$ is most rigorously described via a quadratic form:

$${Q = \{ x \in \mathbb{R}^n : (x - c)^\top \left[ I - \varepsilon^2 v v^\top \right] (x - c) = a^2 - c^2 \},}$$

where $c$ is the ellipsoid center, $v$ a unit “major axis” direction, $\varepsilon$ an eccentricity parameter ($\varepsilon<1$ for ellipsoids), and $a > c$ scalar parameters. This generalizes to arbitrary orientations and centers via $v$ and $c$ (Dearing, 2017).

A ray is typically given as $x(t) = x_0 + t d$ with $d$ a unit direction and $x_0$ the origin point.

Substituting the ray into the ellipsoid’s equation leads to a (generally quadratic) equation in $t$:

$$(x_0 + t d - c)^\top [I - \varepsilon^2 v v^\top] (x_0 + t d - c) = a^2 - c^2.$$

This expands to $A t^2 + 2 B t + C = 0$ where

• $A = d^\top [I - \varepsilon^2 v v^\top] d$,
• $B = (x_0 - c)^\top [I - \varepsilon^2 v v^\top] d$,
• $$C = (x_0 - c)^\top [I - \varepsilon^2 v v^\top] (x_0 - c) - (a^2 - c^2)$$.

The discriminant $\Delta = B^2 - A C$ encodes the intersection character. $\Delta>0$ yields two real intersection points, $\Delta=0$ a tangent (grazing) contact, and $\Delta<0$ no real intersection.

This framework generalizes to intersections of higher-dimensional hyperplanes with ellipsoids, which yield $(n-1)$-dimensional ellipsoids, characterized with closed-form expressions for center, axes, and orientations (Dearing, 2017).

2. Characteristic Polynomial and Discriminant Criteria

For the more general setting of quadric–quadric intersections, or ray–ellipsoid contact, the key tool is the characteristic polynomial $\mathfrak{P}(\lambda) = \det(\lambda E + Q)$, where $E$ and $Q$ are symmetric matrices encoding the ellipsoid and other quadric (Brozos-Vázquez et al., 2021). In the ray–ellipsoid setting, the quadratic-in-$t$ described above is a low-degree analog of this polynomial.

The analysis of $\mathfrak{P}(\lambda)$’s roots reveals contact characteristics:

• A pair of complex conjugate roots signals transversal (non-tangent) intersection.
• All roots real indicates either strict separation or tangency.

For degree-four polynomials (arising in general quadric–quadric cases), discriminants $\Delta_4$ and $\Delta_3$ suffice: transversal contact exists iff $\Delta_4 < 0$ or ($\Delta_4 = 0$ and $\Delta_3 < 0$). These conditions can be checked without explicit root-finding.

Within the quadratic case relevant for ray–ellipsoid intersections, the sign of the discriminant of the $t$-quadratic directly encodes the contact type. This general principle enables robust, differentiable determination of intersection points and their local geometry.

3. Symmetries, Orientation, and Orthogonal Transformation

The ellipsoid’s symmetry about its major axis is structurally encoded via the quadratic form. The matrix $[I - \varepsilon^2 v v^\top]$ has eigenvalue $(1-\varepsilon^2)$ along $v$, and eigenvalue $1$ in the orthogonal complement, giving precise control of shape and orientation.

When a hyperplane or ray is introduced, an orthonormal change of basis—constructed by aligning axes with $v$ and the hyperplane normal—reduces the problem to canonical form (Dearing, 2017). The coordinate system is rotated such that the intersection (e.g., with a hyperplane) is characterized simply, and the parameters for the resulting section (center, axes) are given in closed form in this new system.

After computation, inversion of the transformation returns parameters to the original space, ensuring that the solution accommodates any orientation and preserves the underlying differentiability.

4. Efficiency and Computational Considerations

Closed-form expressions derived for intersection center, axis lengths, and orientations allow efficient computation once the required inner products and transformations are assembled (Dearing, 2017). For ray–ellipsoid problems, the main costs are the evaluation of the relevant quadratic (or, in higher-order cases, quartic) polynomials and, if needed, the computation of discriminants—not explicit root-solving unless an intersection is confirmed.

In scenarios involving repeated intersection checks (as in collision detection, ray tracing, or simulation), this approach offers both high computational efficiency and numerical robustness.

For more complicated objects, zone-based or hierarchical algorithms are suggested (Brozos-Vázquez et al., 2021). These may first cull non-interacting regions using separating planes, then apply the discriminant-based test, and finally compute intersection points and normals only as needed.

5. Necessary and Sufficient Geometric Conditions

Transversal, hence differentiable, intersection between a ray and an ellipsoid is guaranteed if the analysis outlined above (for the characteristic polynomial or the ray–ellipsoid quadratic) yields the appropriate sign structure of discriminants and roots.

A supplemental requirement is geometric “smallness”: For two quadrics, e.g., if an ellipsoid is to be intersected by another quadric (or approximated by a moving or swept ray), the largest semi-axis of the smaller ellipsoid must not exceed the smallest semi-axis of the other, and maximum/minimum principal curvatures must satisfy:

$$\kappa_\text{max}^Q \leq \kappa_\text{min}^\mathcal{E}.$$

This assures the absence of degenerate or multiple curves and that algebraic discriminant tests correspond directly to smooth intersection in $\mathbb{R}^n$ (Brozos-Vázquez et al., 2021).

In ray–ellipsoid cases, the intersection location, normal, and character (transversal or tangent) remain smooth functions of the ray and ellipsoid parameters as long as the discriminant does not vanish, certifying differentiability.

6. Applications and Algorithmic Implications

Differentiable ray–ellipsoid intersection is essential in:

• Rendering (e.g., smooth shading and shadowing of ellipsoidal surfaces),
• Collision detection in rigid body simulation,
• Geometric optimization such as minimal covering ball problems,
• Signal source localization via conic/time-difference-of-arrival models.

The closed-form and discriminant-based criteria enable algorithms to (a) efficiently test for intersection, (b) compute intersection points and normals, and (c) ensure the output varies smoothly with input geometry. In simulation or optimization loops, this is crucial for stability and convergence.

A plausible implication is that these methods, due to their algebraic and geometric generality, serve as a backbone for robust, differentiable geometric computation across domains.

7. Extensions and Theoretical Significance

The principles underlying differentiable ray–ellipsoid intersection, including quadratic form parameterization, discriminant-root analysis, and the use of orthogonal transformations, provide a unified framework applicable to arbitrary $n$ and to other conic sections (hyperboloids, paraboloids, etc.) (Dearing, 2017). The approach readily generalizes to time-dependent intersection problems, moving objects, or higher-dimensional simulation.

The algebraic-geometric perspective, especially as codified by characteristic polynomials and their discriminants, bridges computational geometry, classical algebra, and practical algorithmics, offering both rigorous foundations and efficient implementations (Brozos-Vázquez et al., 2021).