Conic constant
The conic constant, denoted as K, is a dimensionless parameter in optical engineering that defines the shape of aspheric surfaces by specifying their deviation from a sphere, particularly when approximating conic sections such as ellipses, parabolas, or hyperbolas.[1] It appears in the standard equation for the sagitta (axial displacement) of a rotationally symmetric surface:z = \frac{c r^2}{1 + \sqrt{1 - (1 + K) c^2 r^2}} + \sum_{n=4}^{\infty} A_n r^n,
where c = 1/R is the curvature (with R as the vertex radius of curvature), r is the radial distance from the optical axis, and the higher-order terms A_n account for additional asphericity beyond the conic base.[1] This formulation allows precise control over surface geometry to optimize light focusing and minimize optical aberrations.[2] The value of K determines the specific conic section: K = 0 yields a sphere, -1 < K < 0 a prolate ellipsoid, K = -1 a paraboloid, K < -1 a hyperboloid, and K > 0 an oblate spheroid.[1] Also known as the Schwarzschild constant, K relates to the eccentricity e of the conic as K = -e^2, providing a mathematical link to classical conic geometry.[2] In optical design, the conic constant is essential for fabricating mirrors and lenses that reduce spherical aberration and other distortions, enabling high-performance systems in telescopes, camera objectives, and medical devices.[2] For instance, parabolic mirrors (K = -1) focus parallel rays to a single point without aberration, a principle applied in astronomical reflectors.[1] Hyperbolic conics (K < -1) are used in Cassegrain telescope secondaries to correct off-axis aberrations, as seen in space-based instruments like the Hubble Space Telescope.[3]
Fundamentals
Definition
The conic constant, denoted as k (or sometimes \kappa), is a dimensionless parameter that appears in the general equation of a conic section to characterize its shape and curvature. It is defined as k = -e^2, where e is the eccentricity of the conic. The term originates from the geometry of conic sections, the curves formed by intersecting a plane with a double cone, and has been adopted in optics for describing aspheric surfaces; it is also known as the Schwarzschild constant in honor of Karl Schwarzschild's contributions to optical theory.[4] The eccentricity e is a fundamental property of a conic section, defined as the constant ratio of the distance from any point on the conic to a fixed point called the focus and the distance from that point to a fixed line called the directrix. This ratio quantifies how much the conic deviates from a circle, with e = 0 for a circle and increasing values indicating greater elongation. In optics, the value of the conic constant k determines the type of conic surface as follows: k = 0 for a sphere, -1 < k < 0 for a prolate ellipsoid, k = -1 for a paraboloid, k < -1 for a hyperboloid, and k > 0 for an oblate spheroid. These classifications arise from the range of the eccentricity, since e = 0 yields k = 0 for a sphere (special case of ellipse), $0 < e < 1 gives -1 < k < 0 for prolate ellipsoids, e = 1 results in k = -1 for paraboloids, and e > 1 yields k < -1 for hyperboloids.[4]Mathematical Formulation
The conic constant k appears in the standard equation describing the sagitta z of a rotationally symmetric conic surface as a function of the radial distance r from the optical axis and the vertex radius of curvature R: z = \frac{r^2 / R}{1 + \sqrt{1 - (1 + k) (r / R)^2}}. This formula provides the axial displacement z (sag) for points at radial distance r, where the surface is tangent to the xy-plane at the vertex (0, 0, 0), and R > 0 defines the curvature at the vertex.[5][1] This sag equation derives from the general Cartesian equation of a conic section of revolution about the z-axis. For a prolate ellipsoid, the equation is \frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2} = 1, where a and b are semi-axes with b > a > 0. Substituting r^2 = x^2 + y^2 and solving the resulting quadratic for z yields the positive root matching the sag form, with parameters related to R and k via R = \frac{a^2}{b} and k = \frac{a^2}{b^2} - 1. This approach extends to other conics: a sphere corresponds to a = b (k = 0); a paraboloid emerges in the limit b \to \infty (k = -1); and hyperboloids use \frac{r^2}{a^2} - \frac{z^2}{b^2} = 1 with appropriate signs for one sheet (k < -1). Equivalently, the implicit quadratic form r^2 = 2 R z - (1 + k) z^2 encapsulates all cases, from which the explicit sag solves the quadratic equation for z.[5] The conic constant k relates directly to the eccentricity e of the underlying conic section by k = -e^2. For prolate ellipsoids ($0 < e < 1), k ranges from -1 to $0; for paraboloids (e = 1), k = -1; and for hyperboloids (e > 1), k < -1. Oblate spheroids, which flatten along the axis of rotation, correspond to k > 0, effectively extending the ellipse case with imaginary eccentricity.[6][7][4] As a dimensionless quantity, k has no units and fully characterizes the deviation from sphericity independent of scale. For example, k = -0.5 describes a prolate ellipsoid with e = \sqrt{0.5} \approx 0.707, while k = 0.5 yields an oblate spheroid.[7][1]Classification
Conic Types by Value
The conic constant k, a dimensionless parameter in the aspheric surface sag equation, classifies rotational surfaces of revolution based on its value, determining the underlying conic section type used in optical design. These classifications arise from the geometric properties encoded in the equation, where k relates to the eccentricity e of the conic via k = -e^2. For k > 0, the surface describes an oblate ellipsoid, characterized by a flattened shape along the axis of rotation, resembling a sphere squashed at the poles; this configuration is less common in optics but can model certain diverging or broad-beam applications.[1][8] At k = 0, the surface reduces to a sphere, the simplest case with zero eccentricity (e = 0), where all cross-sections are circular and the curvature is uniform, serving as the baseline for traditional spherical optics but introducing aberrations for large apertures.[5] For -1 < k < 0, the surface forms a prolate ellipsoid (or ellipse of revolution), with $0 < e < 1, featuring an elongated shape along the axis; this type supports converging foci between two finite points, ideal for imaging systems requiring aberration reduction without extreme deviations from sphericity.[1][8] The boundary at k = -1 marks the transition to a paraboloid (e = 1), where the surface has a single focus at infinity, enabling perfect collimation of parallel rays or focusing of point sources at infinity without spherical aberration on-axis; this limiting case bridges ellipsoids and hyperboloids.[5] For k < -1, the surface becomes a hyperboloid (e > 1), consisting of two sheets with real foci; the convex sheet diverges rays from one focus toward the other, while the concave sheet converges, often used in reflective systems for off-axis performance.[1][8] The transition at k = 0 separates oblate forms from spherical symmetry, while k = -1 delineates bounded (ellipsoidal) from unbounded (parabolic and hyperbolic) geometries, influencing focal properties and aberration profiles in design.[5] The following table summarizes the conic types, their k ranges, corresponding eccentricities, and key geometric traits for reference:| Conic Constant k | Eccentricity e | Conic Type | Key Properties |
|---|---|---|---|
| k > 0 | Imaginary (e^2 = -k < 0) | Oblate ellipsoid | Flattened along rotation axis; broad divergence |
| k = 0 | e = 0 | Sphere (spheroid) | Uniform circular cross-sections; symmetric curvature |
| -1 < k < 0 | $0 < e < 1 | Prolate ellipsoid (ellipse) | Elongated along axis; two finite converging foci |
| k = -1 | e = 1 | Paraboloid | Single focus at infinity; aberration-free on-axis collimation |
| k < -1 | e > 1 | Hyperboloid | Two sheets; diverging/ converging foci depending on branch |