CONIC SECTIONS BY RATIONAL BEZIER FUNCTIONS
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Conic sections can be considered a subset of rational quadratic Bézier curves. These curves are defined based on three control points and three weights associated with the three control points. First and last control points (endpoints) belong to the curve while the middle control point (midpoint) is provided at the intersection of the tangents passing at the endpoints. This paper shows that: 1) the weights associated with the endpoints can always be arbitrarily selected, 2) closed-form solutions are provided for the midpoint weight, 3) by changing sign to the midpoint weight, the complementary part of the conic section is described, and 4) the closed-form expressions of the midpoint weight is an only function of the variation of the eccentric/hyperbolic anomaly of the endpoints. These results are mathematically demonstrated and are provided for all three different conic sections: ellipse, parabola, and hyperbola. Numerical examples are also given to show these new findings on conic sections, which make potential applications in astrodynamics possible.
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