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.
Questions tagged [cross-sections]
72 questions
22
votes
6 answers
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 other 3D (or any dimensional) object?
I have a…
Beasted1010
- 391
12
votes
3 answers
What is the maximum volume of $N$-D slice of an $M$-D hypercube?
Consider a unit hypercube of M dimensions. We wish to make a cut of dimension N through it. What is the largest N-D size (length, area, volume, ...) we can achieve, $S(M, N)$, and what cut gives it?
Examples:
$S(M=2, N=1) = \sqrt2$
Linear cut…
Eric
- 653
8
votes
1 answer
How to find the $2d$ cross-sections of the $n$-hypercube that are regular $2n$-gons?
I'm currently trying to do something that sounds doable, but on which I've now been stuck for a while.
The idea is that, if you consider an $n$-hypercube, the polygon found by doing a $2d$ cross-section of it can have between 3 and $2n$ edges, with…
Dak
- 83
8
votes
2 answers
Construct the intersection of a cube by a plane through $3$ points on its edges, no pair of which is on the same face
So this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by only using the means of construction.
Problem.…
hayk
- 273
7
votes
1 answer
What is the maximum area of a 2D slice of a unit tesseract? Is it 2?
(This question is a special case of this old question, in the hopes of making progress on a more tractable piece of the puzzle.)
Let $H = [0,1]^4\subseteq \mathbb R^4$ be the unit 4-hypercube, and let $P$ be some plane, specified e.g. by the set of…
RavenclawPrefect
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5
votes
1 answer
Area of a circle from the edge to a point offset from the center
I am trying to come up with a way to calculate the cross-sectional area of the shape shown in the figure below. My first method would be to subtract the circle from the rectangle like this: $$(Y)\left(\frac{OD-ID}{2}\right)-A_{circle}$$, however, I…
LSUEngineer
- 53
4
votes
1 answer
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 the shape relative to the direction of motion.
I…
James Storey
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4
votes
0 answers
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 "nice" if its intersection with any line consists…
sitiposit
- 366
3
votes
2 answers
Volume of a solid with a circular base
Find the volume of the solid with a circular base of radius 9 and the cross sections perpendicular to the y-axis are squares.
I've never solved a problem like this before. How can I go about setting up the integral for this particular problem?
Roman Arefov
- 123
3
votes
1 answer
Can the cross section of parallelepiped be a regular pentagon
Came across this question in a children's recreational mathematics book. Apparently, the cross section of a cube cannot be a regular pentagon. It could be a irregular pentagon though.
But if we generalize this problem, can the cross section of…
Helen
- 235
3
votes
1 answer
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 keeps those vertices on the same side of the…
Yitzchak Shmalo
- 165
2
votes
2 answers
Equation of a section plane in hyperbolic paraboloid
Find the equation of a plane passing through $Ox$ and intersecting a hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z$ $(p>0, q>0)$ along a hyperbola with equal semi-axes.
My attempt: The equation of a plane passing through $Ox$ is $By+Cz=0$.…
Alice P.
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2
votes
1 answer
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 furthermore calculate the vertical curvature of $c$ at…
Rebellos
- 21,666
2
votes
0 answers
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 of subset, namely one that is the product of its…
Evan Aad
- 11,818
2
votes
0 answers
How To Find Volume Using Integrals Under Many Curves and Shapes(Specifically a skull)
For a calculus project, I need to find the theorized volume of a shape made from curves and objects but the problem is I have so many shapes and different types of graphs that I have no clue how to find the volume of my skull.
This is my graph…
adsf
- 147