Interface Conic

  • All Superinterfaces:
    CurveSegment, GenericCurve

    @UML(identifier="GM_Conic",
         specification=ISO_19107)
    public interface Conic
    extends CurveSegment
    Any general conic curve. Any of the conic section curves can be canonically represented in polar co-ordinates (ρ, φ) as:
    Conic curve

    where "P" is semi-latus rectum and "e" is the eccentricity. This gives a conic with focus at the pole (origin), and the vertex on the conic nearest this focus in the direction of the polar axis, φ=0.

    For e=0, this is a circle. For 0 < e < 1, this is an ellipse. For e=1, this is a parabola. For e>1, this is one branch of a hyperbola.

    These generic conics can be viewed in a two-dimensional Cartesian parameter space (uv) given by the usual coordinate conversions u=ρcos(φ) and v=ρsin(φ). We can then convert this to a 3D coordinate reference system by using an affine transformation, (uv) → (xyz) which is defined by:

    (TODO: paste the matrix here, same as AffinePlacement)

    This gives us φ as the constructive parameter. The direct position given by (x₀, y₀, z₀) is the image of the origin in the local coordinate space (u, v) Alternatively, the origin may be shifted to the vertex of the conic as

    u' = ρcos(φ) - P/(1 + e)   and   v' = ρsin(φ)

    and v can be used as the constructive parameter. In general, conics with small eccentricity and small P, use the first or "central" representation. Those with large eccentricity or large P tend to use the second or "linear" representation.

    Since:
    GeoAPI 1.0