Skip to main content

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.←

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more