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Intersection of Quadratic Surface and Plane

Quadratic surface

Quadratic surface is a second-order algebraic surface given by the general equation \[ Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0.\] Generally, there are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone, and hyperboloids of one sheet and two sheets. The intersection of a quadric surface and a plane is usually a quadric curve.
For hyperboloids, there are two kinds of cases: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\] or \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\] Both surfaces are asymptotic to the cone of the equation \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0.\] In the first case (+1 in the right-hand side of the equation), it is a one-sheet hyperboloid, which is also called a hyperbolic hyperboloid. The one-sheet hyperboloid is a connected surface, which has a negative Gaussian curvature at every point.
In the second case (−1 in the right-hand side of the equation), it is a two-sheet hyperboloid, which is also called an elliptic hyperboloid. The surface has two connected components and a positive Gaussian curvature at every point.
Let’s consider the intersection of a one-sheet hyperboloid of one sheet and a plane. An arbitrary plane is defined as: \[Ax+By+Cz+D=0\] The intersection of a one-sheet hyperboloid sheet and a plane is a quadric curve defined by the following system of equations: \[\left\{\begin{split} \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\\ Ax+By+Cz+D=0\\ \end{split}\right.\]

An example of plane sections with a hyperboloid

For simplicity the plane sections of the unit one-sheet hyperboloid are considered. Because a hyperboloid in general position is an affine transform of the unit hyperboloid, the result applies to the general case. The function of the unit one-sheet hyperboloid is defined as \[{x^2+y^2-z^2}=1\] Cartesian coordinates for the unit one-sheet hyperboloid can be defined by hyperbolic trigonometric functions: \[\left\{\begin{split} x=\cosh v\cos \theta \\ y=\cosh v\sin \theta \\ z=\sinh v\\ \end{split}\right.\] where \(\theta \in [-\pi,\pi]\), \(v \in (−\infty, \infty)\).
The plane sections of the unit one-sheet hyperboloid can have the following cases:
  • A plane with a slope with respect the axis less than 1 (1 is the slope of the lines on the hyperboloid) intersects the unit one-sheet hyperboloid in an ellipse；
  • A plane with a slope equal to 1 containing the origin intersects the unit one-sheet hyperboloid in a pair of parallel lines；
  • A plane with a slope equal 1 not containing the origin intersects the unit one-sheet hyperboloid in a parabola；
  • A tangential plane intersects the unit one-sheet hyperboloid in a pair of intersecting lines；
  • A non-tangential plane with a slope greater than 1 intersects the unit one-sheet hyperboloid in a hyperbola.
The plane we chose has an inclination of 30° to the x axis and a bias as 1 \[z=\frac{\sqrt{3}}{3}x +1\]
The intersection is defined by the following system of equations: \[\left\{\begin{split} x^2+y^2-z^2-1=0 \quad (1) \\ \frac{\sqrt{3}}{3}x -z+1=0 \quad (2)\\ \end{split}\right.\] We can also substitute \(z=\frac{\sqrt{3}}{3}x +1\) in to the equation of the unit one-sheet hyperboloid, then we get the the projection of the intersections on the \(XY\) plane: \[\frac{2}{3}x^2-\frac{2\sqrt{3}}{3}x+y^2=0,\] which is an ellipse.

Why to use this example

This example of the intersection of a hyperboloid of one sheet and a plane was chosen for the following reasons:
  1. The function of the hyperboloid of one sheet is a specific form of quadric surfaces, which has some important properties. For example, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid, and a hyperboloid can be constructed by using straight lines which makes the towers very strong at low cost, as shown in the following two pictures:
  2. The intersection of a hyperboloid of one sheet and a plane may have different forms of quadric curves. Usually the hyperboloid intersects many planes into hyperbolas, while the quadric curve of the intersection in our example is an ellipse.
For the 3D print, we also add a base for the upper or lower pieces of the 3D print, in order to make the printing process more stable.
In the above example, we investigate the intersection of a hyperboloid of one sheet and a plane. This helps us to better understand the concept of quadric surfaces.

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