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In $\mathbb R^3$ , the intersection of a plane and a sphere (e.g. $x^2 + y^2 + z^2 = 1$) is either empty, a single point, or a circle. All isometries of those circles are realized by isometries of the full sphere. In contrast, every plane intersects a cone (e.g. $x^2 + y^2 - z^2 = 0$) in a conic section which has reflection symmetries along one, two, or more axes. The one reflection is realized by an isometry of the cone, but the second, in general, is not.

What surfaces in $\mathbb R^3$ , or general subsets of $\mathbb R^3$ , are such that all non-empty planar intersections are either 1 point or have nontrivial symmetry? When are those symmetries not extensible to a symmetry of the full surface?

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    I think in the sphere example you need reflections of the sphere as well as rotations.2012-07-20
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    Do all conic sections have two reflection symmetries? Seems like the parabolas do not.2012-07-20
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    A point has trivial symmetry? What do you mean by nontrivial?2012-07-20
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    @tomasz He wrote "such that all non-empty planar intersections are either 1 point or have nontrivial symmetry?"2012-07-20
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    @ThomasAndrews: yes, making it sound like 1 point does not have nontrivial symmetry, hence the question.2012-07-20
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    @tomasz You are wrong, there are no double parabolas that can be gotten via conic sections. The parabolas are limit points of the ellipses. The hyperbolas are the intersections of a plane with both directions of the cone.2012-07-20
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    @ThomasAndrews: Right, my mistake. I thought about the same example as you and got confused. What do you mean by limit points though?2012-07-20
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    I'll edit the question to reflect the clarifications here. Thanks.2012-07-20
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    @tomasz There's a sense in projective geometry that all conic sections are "the same" once you add the points at infinity. For example, a parabola is an ellipse with a single point "at infinity." A hyperbola is an ellipse with two points at infinity.2012-07-20

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