CU-3P-9P1  Nine 3P-Osculating Conic Points

In projective geometry, given three distinct points Q,R,S one can define a special configuration of nine points that arise from the osculation of conics constrained by this triple.

See [77], page 137 (option 10.) and [80], pages 4-8.

Definition

Let Q,R,S be three fixed, non-collinear points. The Nine Osculating Conic Points associated with the triple (Q,R,S) are defined as the set of points P1,…,P9 such that:

• Each point Pi lies on a unique conic Ci that osculates a given cubic CU at Pi to order ≥ 5 (to the extent that it exists). Each point Pi lies on a unique conic Ci that has contact of order at least 5 with a given cubic CU at Pi; that is, the conic matches the cubic’s tangent direction, curvature, and higher derivatives up to fifth order at that point.
• Each conic Ci passes through the fixed triple Q,R,S, i.e., Q,R,S ∈ Ci.
• The conic Ci has maximal contact with CU at Pi under the constraint that it passes through Q,R,S.

 Geometric Interpretation

• These nine points represent the locus of maximal constrained osculation: they are the points where conics through Q,R,S “kiss” the curve CU most tightly.
• The configuration depends on both the geometry of CU and the choice of the triple (Q,R,S), but the number of such points is always nine under generic conditions.

 Properties

• The nine points are distinct and algebraically determined by the intersection of the osculating condition and the conic constraint.
• They form a projectively invariant configuration: any projective transformation preserving Q,R,S will map the nine points accordingly.

CU-3P-9P1 3P-Tritangent Points-01.fig

CU Point Validation

• The equation 3X+Q+R+S=2N has nine solutions.

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