Intersection of two paraboloids

Question

Consider two paraboloids. The first one is given by $x^2 + y^2 = z+5$. So, it intersects the x-y plane in the circle $x^2+y^2=5$. The second paraboloid is exactly the same as the first one, only shifted in the x-y plane. It's equation becomes $(x-1)^2+(y-1)^2=z+5$. From the figure below, it seems clear that the two should intersect in a parabola.

However, when we actually try and solve the two equations simultaneously, we get from the second equation $$x^2 + y^2 -2x - 2y + 2 = z+5.$$ And substituting $x^2+y^2=z+5$ into this we get $$2x+2y=2.$$

Now this is a linear function. However, the picture seems to suggest that it should be a parabola which is non-linear. What am I missing here?

Answer 1

What you have obtained is the equation of the plane containing the intersection parabola.

The parabola in cartesian form is indeed defined by two different equations, as the following

• $x^2+y^2=z+5$

• $x+y=1$

You can parametrize it by

• $x=t$
• $y= 1-t$
• $z(t)=t^2+(1-t)^2-5=2t^2-2t-4$

Answer 2

Alternatively, exploiting symmetry profitably, you can simply express the equations as $$ \eqalign{ & x^{\,2} + y^{\,2} = \left( {x - 1} \right)^{\,2} + \left( {y - 1} \right)^{\,2} = z + 5 \cr & \left( {x - 1/2 + 1/2} \right)^{\,2} + \left( {y - 1/2 + 1/2} \right)^{\,2} = \left( {x - 1/2 - 1/2} \right)^{\,2} + \left( {y - 1/2 - 1/2} \right)^{\,2} = z + 5 \cr & \left\{ \matrix{ \left( {x - 1/2} \right)^{\,2} + 1/4 + \left( {x - 1/2} \right) + \left( {y - 1/2} \right)^{\,2} + 1/4 + \left( {y - 1/2} \right) = z + 5 \hfill \cr \left( {x - 1/2} \right)^{\,2} + 1/4 - \left( {x - 1/2} \right) + \left( {y - 1/2} \right)^{\,2} + 1/4 - \left( {y - 1/2} \right) = z + 5 \hfill \cr} \right. \cr & \left\{ \matrix{ \left( {x - 1/2} \right)^{\,2} + \left( {y - 1/2} \right)^{\,2} = z + 9/2 \hfill \cr \left( {x - 1/2} \right) + \left( {y - 1/2} \right) = 0 \hfill \cr} \right. \cr} $$

That is, a paraboloid with vertex in $(1/2,\, 1/2,\, -9/2)$ cut by the diametral plane through the vertex.

Answer 3

All sections of a paraboloid cut parallel to a plane containing the axis of symmetry is a parabola.. as is an intersection curve of two parabloids with parallel axes.