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Quadratic Surfaces: Cracking an Egg

Quadratic Surfaces: Cracking an Egg

A quadratic surface is the graph of an equation of three variables \(x\), \(y\), and \(z\); and a quadratic surface is the counterpart in 3D of conic sections in the plane. Conic sections are the name given to parabolas, ellipses, circles, and hyperbolas which result from intersecting a cone with a plane as shown below.

The graphs of the quadratic surfaces are the ellipsoid, elliptic paraboloid, hyperbolic paraboloid, cone, hyperboloid of one sheet, and hyperboloid of two sheets. The graphs of these surfaces are shown below.

Note that while a circle is a conic section a sphere is not a quadratic surface. That seems true but is not exactly accurate. A circle can be thought of as a special type of ellipse; the formula for an ellipse is \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] where \(a\) and \(b\) are constants that determine the width and breadth of our ellipse—i.e. how wide and long the ellipse is—and a circle is the special case where \(a=b\). An ellipsoid and sphere act in the exact same manner; where a sphere is a special case of an ellipsoid. The equation of an ellipsoid is \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\] where \(a\), \(b\), and \(c\) are constant the determine the width, breadth, and height of our ellipsoid, and in the same manner as an ellipse and sphere, if \(a=b=c\) then the ellipsoid is a sphere.

Now that we have thoroughly define what an ellipsoid is, we can begin actually observing a quadratic surface, a plane, and their intersections. It should not be surprising that the surface we will be looking is an ellipsoid. The actual formula of our ellipsoid is as follows \[ \frac{x^2}{35^2}+\frac{y^2}{35^2}+\frac{z^2}{50^2}=1\] this results in an ellipsoid that is reminiscent of the egg we studied when discussing solids of revolution and looks like the following image

If we intersect our ellipsoid with a flat plane in the horizontal direction our cross sections are circles, if we intersect it with a flat plane in the vertical direction the cross sections are ellipses, and is we intersect it with a flat plane at any other angle the cross sections are ellipses also. This seems relatively straight forward, so the plane we will actually observe intersecting our ellipsoid is the one defined by the equation \[ z=x\sin(y)\] which looks like the following

When we intersect the ellipsoid with this plane horizontally this makes it appear as though the "egg" has been cracked, which is interesting in itself and in that it results in some interesting cross sections. When viewed top-down it results in parabolas(red) as cross sections, and when viewed straight-on it results in a pseudo cross section(blue) that resembles the sine wave; shown below.

It is interesting that by changing our intersecting plane from a flat one to one that is more "interesting we can manage to get parabolic cross sections from an ellipsoid. A 3D print of the ellipsoid as well as its intersections via interesting and boring planes can be seen below.

→Image coming soon.←

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