Solids With Known Cross Section

Solids with Known Cross Section

Cross sections help us visualize how multiple 2D planes can make up a 3D object. For example, a camping tent has cross sections with the shape of a triangle or rectangle (depending on if you are looking at the parallel or perpendicular angle) or a soccer ball has cross sections in the shape of a circle. The Egyptian Pyramids were even built one cross section square at a time. By breaking down a 3D object into simple 2D shapes, we can easily visualize a 3D model and calculate its volume.

For me personally, one of my favorite foods is ice cream and it’s to die for out of a waffle cone! But every time I go to an ice cream parlor, I struggle with the million-dollar question, waffle cone or cup? Off the top of my head, I choose a scoop in a cup because I think I’ll get more ice cream (which is the main goal of course!) but I just never know. To help figure this out, the function I chose to focus on is an absolute value function when rotated on the y-axis forms a cone with ellipse cross sections.

F(x) = -|5x|+1

After calculating the volume, I'll be able to tell just how much ice cream I'm getting out of a cone.

(For the purposes of math and 3D printing, calculations are scaled down and do not represent the volume of an ice cream cone. However, the same approach would work with correct measurements.)

Volume

Approximate Volume

To approximate the volume, I broke down the function into 10 elliptical cross sections. The cone has a height of 1 inch; therefore, each cylinder will be 0.1 inch tall. The cross sections and base of the cylinder dimensions are width of the absolute value and 2.5 of the absolute value.
To find the volume of each cross section we used the following formulas.
$A = \prod ab$

$V = A * height$

The approximate volume of my model is 0.12089 in³.

Actual Volume

To find the actual volume I integrated the area of the cross section in terms of y.

$V = \frac{5}{2}\prod \int_{-0.2}^{0.2}(\frac{1}{5}(1-y)^{2})$

$V = 0.12733 in^{3}$

These are similar which proves this an accurate way to calculate and visualize the solid. To double check my work I also calculated the volume using the volume of a cone equation. However, this will not be as accurate because this is for a more uniform shape. But I wanted to include it in this post for comparison reasons.

$V = \frac{1}{3}\prod abh$

$V = \frac{1}{3}\prod(0.2)(0.5)(1))$

$V = 0.1047 in^{3}$

Comments

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...

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...

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...

MA 391 Composition and Communication

Search This Blog