Questions tagged [cross-sections]

A cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections.

43 questions
22 views

Vertical cross-section of $S : r(u,v) = (u+v,uv,u^2v)$.

Exercise : Consider the surface $S : r(u,v) = (u+v,uv,u^2v), \; (u,v) \in \mathbb R^2$. Express the vertical cross section $c$ of the surface at the point $(2,1,1)$ with direction $(2,1)$ and ...
18 views

Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
3k views

Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Conic sections seem to get special attention in early math classes. My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some ...
22 views

Do all functions represent a section of an n dimensional object by another object of n-1 dimension?

If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what ...
36 views

I have no idea how to solve this problem using areas of known cross section

The problem involving cross sections I am so confused on how to find volume using known cross sections. I've never understood it. This problem that I've encountered is very difficult, and I tried ...
22 views

What is the terminology for a subset of a product of sets that is the product of its cross-sections?

Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind ...
107 views

cross section of torus

Is there any particular name for the plane revolved about an external axis to form a torus? I was thinking of "cross section," but that could be taken as a vertical plane cutting the whole torus in ...
71 views

In what sense are “projecting” and “taking sections” of polytopes dual operations?

It seems to be folklore that projecting and taking sections of polytopes are somehow "dual operations" (e.g. explicitly noted in the abstract of this paper, or suggested by this answer to an MO ...
132 views

cross section for y=1

My class has just started Multivariable, and I'd just gotten back my quiz results. However, I don't understand why for B), the correct graph is apparently in the other direction I thought it would be —...
29 views

Cross sections of randomly distributed defects

A $10 \times 10 \times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how ...
19 views

Cylindric sections

When a cylinder intersects with a plane, what are the resultant shapes and curves? I think that the curves are hyperbola, parabola and line, and the shapes are circle, ellipse, rectangle and trapezium....
133 views

Finding the cross section of a prolate spheroid at a given rotation [theta (x,y), theta (x,z), theta (y,z)] on a central plane

I currently have an assignment in which I have to model the drag forces acting on a rugby ball as it rotates through the air. One of the variables in the drag force equation is the cross section of ...
92 views

Volume of a solid formed by a triangle base with square cross sections parallel to a line

Reviewing for a test, I was given this problem. "The base of a solid is the region in the first quadrant bounded by the line x = -2y + 6 and the coordinate axes. What is the volume of the solid if ...
233 views

Cross sections of a cube

Suppose we take the set of all cross sections of a cube and construct from them a set $A$ whose elements are sets of vertices of the cube as follows. If there exists a cross section of the cube which ...
222 views

Finding the volume of a solid s using cross sections

I am a given a problem that reads "The base of $S$ is a region enclosed by $y = 2-x^2$ and the $x$-axis. Cross-sections perpendicular to the $y$-axis are quarter circles." The instructions are "Find ...
33 views

42 views

Cross-Sections of Solids

I know already that conic sections (or conics) have been widely explored and many things about them are already known. I was wondering if this sort of exploration has taken place for any other sorts ...
66 views

3D Ellipsoid Average Value Integrals

An ellipsoidal meteor is careening down from outer space. The comet's exterior takes the shape of $\frac{(x-30)^2}{36}+\frac{(y-70)^2}{9}+\frac{(z-40)^2}{25}=1$. When the meteor hits the Martian ...
72 views

Why are the middle cross sections of dual polyhedrea the same?

A tetrahedron is self-dual, so it is no surprise that in both the tetrahedron and its dual the middle cross section is the same shape (a square). A cube and an octahedron are dual, and the middle ...
81 views

Single-Variable Integration Volume Problem. “When lengths change linearly, areas change quadratically”?

Let $C$ be a cone with base any shape (not necessarily a disk or an ellipse) having area $A$, and height $h$ (that is, the distance from the apex to the plane containing the base is $h$). Note ...
88 views

An inverse problem for cross sections

I apologize beforehand for the vague title and the length of the description I am using to setup my question; I can't seem to be more concise without sacrificing clarity. Call a region in the plane "...
29 views

Island Sketching

"The Island Euleria sank over the course of a year. A helicopter captured 11 photographs of the top of the island. It took a picture every time the island had sunk three feet, and one of the island ...
78 views

2D Cross-section of data points

I have a set of points with x and y values. I would like to draw a straight line through these points and have a cross section through these points. Said another way, I would like to rotate these ...