Indicatrix

Definitions

  • Webster's Revised Unabridged Dictionary
    • n Indicatrix (Geom. of Three Dimensions) A certain conic section supposed to be drawn in the tangent plane to any surface, and used to determine the accidents of curvature of the surface at the point of contact. The curve is similar to the intersection of the surface with a parallel to the tangent plane and indefinitely near it. It is an ellipse when the curvature is synclastic, and an hyperbola when the curvature is anticlastic.
    • ***
Century Dictionary and Cyclopedia
    • n indicatrix In geometry, the curve of intersection of any surface with a plane indefinitely near and parallel to the tangent-plane at any point. The indicatrix is a hyperbola, a pair of parallel lines, or an ellipse, according as the surface is anticlastic, cylindrical, or synclastic, at the point of tangency.
    • n indicatrix In the theory of equations, a curve which exhibits the joint effect of the two middle criteria of Newton's rule, in the case of an equation of the fifth degree having all its roots imaginary.
    • n indicatrix In crystallography, a surface, in general (for a biaxial crystal) an ellipsoid having axes proportional to the principal refractive indexes, whose geometrical characters serve to exhibit the optical relations of the crystal: for a uniaxial crystal the surface becomes a spheroid and for an isotropic crystal a sphere. The indicatrix bears a simple relation to Fresnel's ellipsoid the axes of which are proportional to the reciprocals of the refractive indexes, that is, directly preportional to the light-velocities in the given axial direction. L. Fletcher, The Optical indicatrix, London, 1892.
    • ***

Etymology

Webster's Revised Unabridged Dictionary
NL

Usage


In science:

We may use geodesic-arc angle to construct the geodesic-arc sector S{x} (l1 , l2 ) ⊂ TxM , where l1 , l2 ∈ TxM are two unit vectors issued from the origin 0 ∈ TxM (and pointed to the indicatrix).
Finsleroid gives rise to the angle-preserving connection
RN and h2(x) is the value of curvature of the indicatrix supported by point x (see (2.6)).
Finsleroid gives rise to the angle-preserving connection
At each point x ∈ M , the ratio AREAFinsleroid Indicatrix /V OLU M EFinsleroid proves to be of the universal value in each dimension N (is independent of h), so that the ratio is exactly the same as it holds in the Riemannian limit (that is, when h = 1).
Finsleroid gives rise to the angle-preserving connection
However, there exists a simple possibility to elucidate the global structure of the F F P D -Finsleroid indicatrix.
Finsleroid gives rise to the angle-preserving connection
The equality (3.7) manifests that the transformation (6.26) maps -Finsleroid indicatrix in regions of the sphere S {ζ } regions of the F F P D {x} .
Finsleroid gives rise to the angle-preserving connection
S {−} {x} may be used to yield two covering charts for the indicatrix; they can be characterized by the angle ranges indicated in (6.22).
Finsleroid gives rise to the angle-preserving connection
Similarly to the Riemannian case proper, we need two charts to cover the indicatrix.
Finsleroid gives rise to the angle-preserving connection
Under the conditions formulated in the preceding theorem, the indicatrix supported by a point x ∈ M is of the constant curvature h2 (x), such that RFinsleroid Indicatrix = C (x) with C (x) = h2 (x).
Finsleroid gives rise to the angle-preserving connection
Therefore, since we adhere at measuring the angle α by the geodesic arc-length on the indicatrix, from the above formulas (2.1)-(2.6) we are entitled to conclude the following.
Finsleroid gives rise to the angle-preserving connection
To surely recognize the validity of this theorem, it is sufficient to take a glance on the equality (3.9) which represents infinitesimally the squared length of the geodesic arc on the indicatrix.
Finsleroid gives rise to the angle-preserving connection
If the indicatrix supported by a point x ∈ M is a space of constant curvature, then the conformal property (2.1) holds at the point x.
Finsleroid gives rise to the angle-preserving connection
The final condition is (Z5): The indicatrix is closed and regular.
Finsleroid gives rise to the angle-preserving connection
Let the Finsleroid indicatrix IF {x} supported by a fixed point x ∈ M be parameterized by means of a convenient variable set ua (for instance, we can take ua = ζ a/ζ N in regions with ζ N 6= 0, or ua = y a/yN , whenever yN 6= 0.
Finsleroid gives rise to the angle-preserving connection
We have taken into account the fact that the conformal factor κ , having been proposed by (2.5), equals 1/h(x) on the indicatrix.
Finsleroid gives rise to the angle-preserving connection
From this standpoint, it is easy to make the search into the curvature of the indicatrix.
Finsleroid gives rise to the angle-preserving connection
***