A plane cutting through a paraboloid

Geometry Level 5

A paraboloid is given by

z = 0.25 x 2 + 0.1 y 2 z = 0.25 x^2 + 0.1 y^2

It is cut by a plane passing through ( 10 , 10 , 20 ) (10, 10, 20) with a unit normal of ( 1 2 , 1 2 , − 1 2 ) ( \frac{1}{2}, \frac{1}{2}, -\frac{1}{\sqrt{2}} ) . The resulting cut is an ellipse as the animation below shows. Find the sum of the semi-minor and semi-major axes of this ellipse.


The answer is 17.2.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Yuriy Kazakov
Feb 16, 2020

I use Mathcad and approach Ellipse in Space - 2

I find the parametric equations for ellipse line.

The curvature is given by κ = ∥ f ′ × f ′ ′ ∥ ∥ f ′ ∥ 3 \kappa = \dfrac{\lVert f' \times f'' \rVert}{\lVert f' \rVert^3}

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