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We classify all the relative positions between an ellipsoid and an ellipticparaboloid when the ellipsoid is small in comparison with the paraboloid ({\itsmall} meaning that the ellipsoid cannot be tangent to the paraboloid at twopoints simultaneously). This provides an easy way to detect contact between thetwo surfaces by a direct analysis of the coefficients of a fourth degreepolynomial.

Classification of the relative positions between an ellipsoid and an elliptic paraboloid