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5
votes
1answer
197 views

Volume of a sphere with three holes drilled in it.

Suppose that the sphere $x^2+y^2+z^2=9$ has three holes of radius $1$ drilled through it. One down the $z$-axis, one along the $x$-axis, and one along the $y$-axis. What is the volume of the resulting ...
2
votes
0answers
75 views

Angle between two planes, finding the second plane

I've got a plane given by $x-y+z=0$ and a line given by $r=(0,0,a)+t(1,1,b)$. How do I find another plane that contains the line and intersects the other plane at an angle of $45^\circ$? Is ...
0
votes
1answer
86 views

Eigenvectors for the equation of the second degree and right-hand rule

I'm trying to find the Eigenvectors for the equation of the second degree (for example Elliptic cone). The estimated values $V_1$, $V_2$ and $V_3$ must satisfy the right-hand rule. How can we verify ...
4
votes
1answer
202 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
1
vote
1answer
234 views

Find the volume of the solid about y-axis, i.e. 1/(x^2+3x+2), bounds x=0, x=1

find intersection points: 1/(x^2+3x+2)=1, so x=(sqrt(5)-3)/2, x= -(sqrt(5)+3)/2 thus we have: V(volume)= Pi*integral((1)^2-(1/(x^2+3x+2)^2) from x= -(sqrt(5)+3)/2, x= (sqrt(5)-3)/2 ...
1
vote
1answer
84 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
0
votes
1answer
84 views

Correcting plane parameters with the fixed azimuth angles

I am trying to reconstruct specific 3d objects such as cubes, pyramids and so on. For this, i am using point cloud data and then fitting planar surfaces for the segmented point patches. Planes ...
1
vote
0answers
34 views

How to fit a cuboid into a polyhedra?

I have multiple points which create a solid (polyhedra). And now I want to place a cuboid inside this solid in a way that it uses the maximum amout of space inside. Are there any solutions for this ...
2
votes
1answer
199 views

solid angle of polyhedron

I have an interesting thought when I draw polygon and 3D polyhedron. My question is: Can I know the number of Face, Edge, and Vertex from a given 'space angle constraint'? For example, a vertex a cube ...
1
vote
3answers
165 views

Surface area comparison of a solid cut in half

This was just a thought I had while driving this morning, no particular application. If you make a straight line cut through a solid object such that the resulting two pieces have the same volume, ...
2
votes
1answer
223 views

Prove that $S$ is a sphere.

Let $S\subset {\mathbb{R}}^3$ with the following properties: $1.$ For any line $l$: $|l\cap S|=2$ or $|l\cap S|=1$ or $|l\cap S|=0$ $2.$ For any plane $P$ $P\cap S=\text{circle}$ or $|P\cap S|=1$ ...
0
votes
1answer
201 views

Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is \begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation} and it is easy to estimate the curvature numerically for given ...
0
votes
2answers
218 views

get a rotation matrix from an oriented vector quicker than Euler

I'm in $R^3$ and I have a solid 3d object and a vector, I would like to rotate and orient the solid according to this vector. I found that the simplest way to do that is to use euler angles, the ...
1
vote
1answer
238 views

Surface of a sphere inside a concentric cube

Given a sphere of radius r, and a cube of unit length with the same center as the sphere, what is the surface area of the sphere that is inside the cube, when $\sqrt{2}/2 <r < \sqrt{3}/2$?
3
votes
0answers
137 views

Maximum length of pencil in a pencil case

What is the maximum length of an unsharpened, cylindrical pencil inside an empty rectangular pencil box? Or, in a rectangular cuboid of dimensions $x \times y \times z$, what is the maximum possible ...
2
votes
1answer
46 views

What's the name of each pseudo-rectangle in a spherical surface?

Consider the common surface of a spherical segment crossed with a spherical wedge. This produces a pseudo-rectangle in the sphere surface, and a perfect rectangle in a mercator projection. What's the ...
0
votes
1answer
144 views

Volume of a cylinder on a slope

A cylindrical water tank which is 35 feet in diameter and 105 feet in length is placed temporarily on an 18.5 degree slope. The filler is located flush with the top of the tank at midpoint. What is ...
4
votes
0answers
103 views

Icosahedral symmetry

Is there a clear and correct reference for full icosahedral symmetry that includes a presentation and its action on vertices, edges, and faces? The Wikipedia article for icoshedral symmetry gives ...
0
votes
0answers
144 views

how to get optimal vector, which is parallel to intersection line of many plane (Least Square way)

My idea is to construct the best optimal 3D line representing the intersection of many 3D planes. (As we know, due to fitting errors or data errors, the fitted planes might not intersect exactly ...
2
votes
0answers
143 views

Faces on a rotating cube

This one is probably a softball question, but... Fact: A cube has 6 faces Fact (unless my math is wrong): A cube that is cut in half center horizontal can be reoriented to have 18 unique faces. ...
1
vote
0answers
146 views

Creating polyhedra with playing cards

In here, George Hart gives some exciting examples on how to create polyhedra by cutting playing cards and sliding them inside one another. I was wondering if such an approach can be generalized to ...
0
votes
1answer
169 views

producing tetrahedra within a cube

A common math problem involves dividing a cube into regular and irregular tetrahedra, where the points of the cube must also be the points of the tetrahedra. A problem I'm working on seems to be ...
1
vote
3answers
379 views

What is the formula for a 3D line?

Just like we have the formula $y=mx+b$ for $\mathbb{R}^{2}$, what would be a formula for $\mathbb{R}^{3}$? Thanks.
1
vote
1answer
50 views

Help with school solid geometry

In the triangular pyramid $MABC$ all side edges equals $1$, $\angle AMB = \angle BMC = 60 ^\circ$, $\angle AMC = 45 ^\circ$. Find: 1) square of the $\triangle ABC$; 2) dihedral angle on the $AB$ ...
1
vote
0answers
62 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
0
votes
1answer
166 views

least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
0
votes
2answers
266 views

Analytic geometry section of cone and sphere

How to show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circle ? I understand that the two equations taken together represent the circle. but how to go about finding the ...
-1
votes
1answer
260 views

Sphere to ellipsoid [closed]

If there is a sphere diameter 0.72mm and I was wanting to calculate the angle of the z axis that would convert the sphere to an ellipsoid with a major axis of 0.82 mm What is the equation that I can ...
2
votes
1answer
111 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
0
votes
1answer
113 views

What is the minimum number of blocks to build this?

A rectangular solid is built using $N$ cubes of a side length of 1cm. When viewed from such an angle such that only 3 of the sides of the rectangular solid are visible, there are 231 $cm^2$ of area ...
2
votes
1answer
112 views

3D Geometry Proof by Contradiction /Contrapositive (high school)

Could someone evaluate my work? A plane perpendicular to one of 2 parallel lines is perpendicular to the other line also. My two column proof so far: Let AB || CD and AB be perpendicular to plane ...
0
votes
1answer
108 views

Steepest slope gradient of a vertical plane

I know the steepest slope gradient (Azimuth) of a 3D plane can be obtained by projecting normal vector onto XY Plane. So, when the plane is slant, the steepest gradient will be a some value. ...
2
votes
1answer
90 views

Is the intersection of a bunch of cylinders a sphere?

Suppose we have a 3-D shape $S$ with a center $C$, so that a point $p$ is in $S$ if and only if for any direction $\vec d$, $p$ is contained within a cylinder of radius $1$, extending infinitely both ...
5
votes
2answers
408 views

Automorphism group and congruences of the cube

I want to prove that the automorphism group of the cube is $\mathbb{Z}_2 \times S_4$, by using information about the congruences of a cube. By the cube, I mean the graph of the platonic solid, i.e. ...
0
votes
1answer
81 views

Tetrahedralize Mesh

Say I have a triangle mesh which forms the shell of an object which may not be convex. For every triangle, I have the vertices and a normal. I want to turn this mesh into a solid. I want to break up ...
1
vote
2answers
1k views

How do I find the nearest point on a sphere?

Say I have a sphere of radius $6$ centered at $(3, 4, 5)$. What's the nearest point on the surface of the sphere to point $(1, 2, 3)$, which is within the sphere? I feel that this is a minimization ...
0
votes
2answers
147 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
1
vote
1answer
59 views

Coordinates for vertices of the “silver” rhombohedron.

The "silver" rhombohedron (a.k.a the trigonal trapezohedron) is a three-dimensional object with six faces composed of congruent rhombi. You can see it visualised here. I am interested in replicating ...
7
votes
1answer
333 views

What tesselated three-dimensional shape gives the maximum volume with the minimum surface area?

I recently read an article on the future of buildings. I have long been interested in architecture and it seems to me that this article makes some very good points. It got me thinking about the ...
0
votes
1answer
1k views

Volume of a trapezoidal prism

A pile of ore has a rectangular base, 60 feet wide and 500 feet long. If the sides of the pile are all inclined 45degrees to the horizontal, and the ore weighs 110 lb. per cu.ft. Find the number of ...
0
votes
2answers
145 views

Sixth Platonic solid

A Sixth Platonic solid? [1] Wouldn't gluing a tetrahedron's one triangle to a another tetrahedron's triangle make a platonic solid ? See the picture to see clearly what I mean. Tetrahedron stacked ...
3
votes
1answer
1k views

Volume from revolving $e^x$ around $y$-axis

In school, I had a problem something like this: A region R is bounded by $x$-axis, $y$-axis, $x = 3$, and $y = e^x$. What is the volume of the solid produced by revolving it around the $y$-axis. ...
1
vote
1answer
58 views

What kind of solid has a face adjacency graph whose spanning trees are not feasible nets

Was reading an introductory graph theory book, and it says that nets of solids can be represented using adjacency graphs, and new nets can be discovered by searching for all the spanning trees of the ...
1
vote
1answer
318 views

Surface area of revolution about $y$-axis in terms of $f(x)$

I need to find a formula for the surface area of a solid of revolution rotated around the $y$-axis. The curve is $f(x)=x^2$ on $[0,1]$. However, my answer must be in terms of $f$, not $f^{-1}$.
0
votes
1answer
140 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Do angles $CAD$ and $CBD$ equals in general tetrahedron?
1
vote
1answer
184 views

Find volume of region using change of variables

I want to find the volume of the region $R$ that lies between $$z= x^2 + y^2, \quad z= 4(x^2 + y^2), \quad z = 1, \quad z = 4$$ Using the transformation \begin{align} x &= ...
23
votes
1answer
567 views

A problem of J. E. Littlewood

Many years ago I picked up a little book by J. E. Littlewood and was baffled by part of a question he posed: "Is it possible in 3-space for seven infinite circular cylinders of unit radius each to ...
1
vote
2answers
138 views

name of this shape [3d solid]

what is the name of this 3d solid please? "faces" of 3 and 4 sides. Thanks!
12
votes
1answer
263 views

What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?

I tried for a while, not very hard, to construct a polyhedron with exactly six faces, whose areas were respectively 1, 2, 3, 4, 5, and 6 units. I did not meet with any success. Still, it seems that ...
0
votes
1answer
868 views

Parametric and implicit representation of a cone

http://mathworld.wolfram.com/Cone.html shows the parametric and implicit representation of a cone, I am wondering what the equation would look like if we also consider the bottom circle face for the ...