If I have two lines with equations;$$x=0$$ $$y=0$$ $$z=t$$ and $$x=t$$ $$y=10$$ $$z=t$$ are there any parabolas that cross through the two lines and in which the parabola matches the slope of the lines at the points of intersection?
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These are skew lines (not coplanar). So what do you mean by matching the slope at the point of intersection? – almagest Mar 23 '16 at 19:44
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No. All of the tangent lines of a parabola lie in a plane (in fact, the entire parabola lies in the same plane). The two lines you gave are of course their own tangent lines, so the parabola has to lie in some translation of the plane spanned by $(0,0,1)$ and $(1,0,1)$ (which is the $xz$-plane), so the the parabola has a constant $y$ value. It cannot be both 10 and 0.
Plutoro
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Are there any parametric equations that include parabolas as a subset that satisfy the conditions I specified considering that no parabolas that satisfy the conditions I specified? – Anders Gustafson Mar 24 '16 at 05:21
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I am unsure what you mean. Certainly there exist curves which have both of the lines you give as tangent lines. You take the union of all such curves and all parabolas. Does this sound like what you are asking for? – Plutoro Mar 24 '16 at 18:10