Skip to main content

Quadric surfaces

Introduction
Quadric surfaces are of the form \(Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\), where the sum of the powers of any term is less than or equal to 2. However, we don't usually see them in this form. Translations combine the \(x^2\) and \(x\) terms; we only see linear terms if they appear without the cooresponding squared term. Additionally, the \(xy\), \(xz\), and \(yz\) terms are eliminated with rotations. We can use translations and rotations to turn the given equation into a more intuitive form because shapes are the same regardless of where they're located or how they're oriented.
There are seven standard quadric surfaces to consider.
1. Ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\)
2. Cone \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0\)
3. Hyperboloid of One Sheet \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\)
4. Hyperboloid of Two Sheets \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1\)
5. Elliptic Paraboloid \(z=\frac{x^2}{a^2} + \frac{y^2}{b^2}\)
6. Hyperbolic Paraboloid \(z=\frac{x^2}{a^2} - \frac{y^2}{b^2}\)
7. Cylinder \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = r\)
When intersected by a plane, we see conic sections! These can be hyperbolas, parabolas, ellipses, circles, or a combination of these depending on how many times the surface is sliced. Clearly, some of these appear more intriguing than others. A slice that reveals only a line, point, or pair of lines isn't all that interesting. (Source: https://www.youtube.com/watch?v=VAz-X6uaSs4)
The surface and plane I chose
Consider a cone centered at the origin given by \(\frac{z^2}{144}=\frac{x^2}{25}+\frac{y^2}{4}\).
I chose this function because cones show every conic section and are most interesting to slice with a plane! I picked values for a, b, and c such that the opening of the cone is an ellipse and not a circle, just to spice things up.
Now, let's slice it. I want to show a hyperbola because these are the coolest conic sections in my opinion. To do this, I want a vertical plane like the one pictured. (Source: https://math.libretexts.org)
The equation of a plane is given by \(Ax+By+Cz+D=0\). To find the equation of a plane, all you need is a point and a normal vector. For example, choose the plane perpendicular to (1, -1, 0). Points perpendicular to a vector \(\vec{v}=\left\lceil\begin{matrix} v_{1}\\v_{2}\\v_{3} \end{matrix}\right\rceil\) are the points \(\vec{x}=\left\lceil\begin{matrix} x_{1}\\x_{2}\\x_{3} \end{matrix}\right\rceil\) such that \(\vec{v}\cdot\vec{x}=0\). So, we do the following: \[1\cdot x+(-1)\cdot y+0\cdot z=0\] \[x-y=0\] \[x=y.\] Here's the cone sliced with the plane \(x=y\):
I like how it's cut so that the ellipse-like opening is at a slant.

The intersection
Now for the math! We are intersecting \(\frac{z^2}{144}=\frac{x^2}{25}+\frac{y^2}{4}\) with the plane \(x=y\). To do this, we can use a really simple substitution. Set \(x\) equal to \(y\) to obtain \[\frac{z^2}{144}=\frac{x^2}{25}+\frac{x^2}{4}\] \[z^2=144[x^2(\frac{1}{25}+\frac{1}{4})]\] \[z^2=144(0.29x^2)\] \[z^2=41.76x^2\] \[z=\pm 6.46x\] On the 3D print, we can see the hyperbola by pulling the two halves of the model apart. The model is just under 4 inches tall, so \(z\) is from around -2 to 2. It is about one inch at its widest, where \(y\) ranges from around -0.5 to 0.5.
Hopefully now you can see why cones are the coolest surface and hyperbolas are the most interesting cross-section. Our cone may look more like a squished hourglass than any cone we're used to seeing, but that's okay! If we only wanted to see one half of this hourglass shape, we would have chosen \[z^2=c^2(\frac{x^2}{a^2}+\frac{y^2}{b^2})=\frac{c^2}{a^2}x^2+\frac{c^2}{b^2}y^2=A^2x^2+B^2y^2\] \[z=\pm\sqrt{A^2x^2+B^2y^2}.\] Additionally, our cone opens up along the z-axis. If we wanted to change that, we would only need to change our equation slightly. For example, a cone opening along the x-axis would be given by \(\frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x^2}{a^2}.\) (Source: https://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx)

Word count: 522

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

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more