Pumpkin being cut

The quadric surface is derived from the quadratic function by adding an unknown number， We have all learned the quadratic function. It is just a two-dimensional plane transformed into a three-dimensional figure. If the quadric surface is intercepted by a parallel plane, its section is a quadratic curve. Usually, we call the surface represented by the quadratic equation of three variables as the quadric surface. The so-called three unknowns are the x, y, and z in the three-dimensional coordinate axis. Their powers are 1 or 2, which makes them form some fixed graphics, such as Ellipsoid, Hyperboloid of one sheet, Hyperbolic cylinder, and so on. In order to show it better, an ellipse closer to life would be a good example. For example, it’s almost Halloween and you decide to buy a pumpkin. If you find a great pumpkin, it can have an equation to show its appearance:

This equation is the coordinates of the circle of the plane plus the new unknown z, where a, b and c can be constants, So this equation is the equation of all ellipses, and different ellipses can be obtained by changing a, b, and c. You can also subtract or add some constants after x, y, z to change the position of the ellipse so that you can get the ellipses in different positions.

So in that case, for the sake of brevity, the position of the reunion will not be changed. We will bring in the constants, for example, a=b=4, c=3. Then our equation is obvious, then we can get a new equation and "pumpkin"

It can be seen that this is obviously not a ball, it should be flattened. Okay, now we have found a perfect "pumpkin". You bought it and decided to use it to make a pumpkin lantern after taking it back, so you have to open a flat mouth from above. If a=b, then all cuts parallel to the XY plane should be a circle because it does not depend on z at this time. oop, due to not being careful enough, the knife did not cut the pumpkin in parallel, but was crooked a lot, which resulted in the cutting plane not being parallel to the XY plane. Then suppose the equation of this knife is x+√3z=3, and this plane divides the previous model into two.

The incision is an ellipse, which is formed by the intersection of two equations. It can be seen that its tangent plane is also an ellipse. Then their intersection point is this ellipse, and the coordinates of the intersection point are:

This is one of the tangent planes, and for an ellipse with a=b (in the equation). Except for the tangent plane parallel to the XY plane, in other cases, all tangent planes should be an ellipse, all of which depend on the position of the plane.

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...

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...

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