The solid-geometry tag has no wiki summary.
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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 $AB$ ...
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27 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 ...
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1answer
36 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 ...
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30 views
Mymultiple image geometry
I have to work with multiple aerial images. the objective is to reconstruct 3d features.
For a particular object, i want to find the images which are giving good viewing geometry than others. so ...
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2answers
28 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 ...
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1answer
48 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 ...
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1answer
76 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 ...
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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 ...
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1answer
30 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.
...
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1answer
56 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 ...
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2answers
107 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. ...
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1answer
30 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 ...
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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 ...
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2answers
80 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 ...
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1answer
39 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 ...
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1answer
131 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 ...
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1answer
529 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 ...
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83 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 ...
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1answer
283 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.
...
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1answer
47 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 ...
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1answer
84 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}$.
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equation of a disc in the 3D space with unit normal vector and center point
I create disc equations in 3D space with have dip, dip
direction(aspect),(or the unit normal vector of the disk)and disk
center point.How i create the disc? for me dip direction is
...
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1answer
86 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?
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1answer
106 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 &= ...
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1answer
440 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 ...
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2answers
84 views
name of this shape [3d solid]
what is the name of this 3d solid please?
"faces" of 3 and 4 sides.
Thanks!
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1answer
215 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 ...
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1answer
205 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 ...
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84 views
Using Laser Distance Meter to calculate surface area of roof with minimum measurements required
Equipped with a Laser Distance Meter I am trying to calculate the total area of roofs (flat surfaces, not necessarily rectangular), and I am wondering how much data I need to collect in order to do ...
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2answers
230 views
Intersection of a Cone and Sphere
Show that a the cone $xy + yz + xz = 0$ cuts the sphere $x^2 + y^2 + z^2 = r^2$ into two equal circles and find their area.
I have been trying to substitute one of the variables, say $z$, from the ...
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2answers
138 views
Surface of Cone
A cone is cut by a plane ($z=d$). ($0<d<c$)
The top point of the cone is $A(a,b,c)$
Button of the Cone is a circle on $z=0$ plane and center of it is $O(0,0,0)$.
If $a=b=0$ then
...
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1answer
133 views
Surfaces of constant projected area
Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area.
Question: If $A$ is a (bounded) subset of $\mathbb ...
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1answer
212 views
Volume and surface area of a sphere by polyhedral approximation
Exposition:
In two dimensions, there is a (are many) straightforward explanation(s) of the fact that the perimeter (i.e. circumference) and area of a circle relate to the radius by $2\pi r$ and $\pi ...
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2answers
159 views
What is the volume of this 3d shape?
I'm wondering if there is an equation that represents the volume of an arbitrary 3d primitive
matching this description:
1.) Point at center of sphere
2.) Each edge is the length of the radius
3.) 3 ...
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1answer
449 views
To find the volume of tetrahedron by using all surfaces areas?
I am looking for a formula: $V=f(S_1,S_2,S_3,S_4)$, where $S_1$, $S_2$, $S_3$, and $S_4$ are the areas of the four faces.
We know ...
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1answer
149 views
Find volume of a revolved solid by integrating wedges.
So, lets say that I wanted to find the volume of the solid formed by rotating the area between
$f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
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1answer
251 views
How would you make a (physical) dodecahedron with edges instead of faces?
Call this a "math problem disguised as a woodworking problem" or vice versa.
Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of ...
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About the Platonic Solids in all dimensions
I am asking about the Platonic solids in all dimension, some reference about the proofs of many of the statement made in here.
I would like to here about how to think about higher dimensions mainly ...
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3answers
280 views
Platonic Solids
It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
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4answers
686 views
How to cut a cube into an icosahedron?
Edit: Originally I asked this about a using a cube, but it is not a requirement to start with a cube, just how to end up with an icosahedron as on of the answers showed how to make dodecahedron a ...
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203 views
How to prove there are exactly eight convex deltahedra?
A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
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1answer
133 views
Polyhedron with $11$ faces
Show that there is no polyhedron with exactly $11$ faces such that each face is a polygon having an odd number of sides.
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1answer
1k views
Volume of a parabolic solid
We intend to find the volume of a solid described as follows:
The $X$, $Y$ and $Z$ axes are such that the base of the solid is in the $XY$-plane and the vertical direction is parallel to the ...
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2answers
161 views
Shape/volume of this solid (don't know the name)
Here is the construction of the solid. Take an ellipse, make a copy of it, and put it
on top of the original ellipse. Now turn the top ellipse by $90^\circ$ (quarter turn).
Glue the two boundaries. I ...
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1answer
798 views
Ordering vertices in counter-clockwise manner in 3D space.
This is my first question in math and if I cannot get it right for the first time, please forgive me. I'm working on a simulation and I need to order vertices of a triangle in counter-clockwise ...
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1answer
166 views
Convert a 'world' point to a 'local' point relative to a plane
I'm trying to find a way of converting a point relative to the world into a point relative to an arbitrary plane (and a way to convert back). This will be coded into C++ so I'll have to able to write ...
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1answer
370 views
Volume of the torus with an integral
I am stuck on the following question:
Find the volume of the torus obtained by rotating the circle $(x-a)^2+y^2=b^2$, where $a>b$, around the y-axis.
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1answer
466 views
Solid of revolution about a slanted line
I just thought about this idea and I decided to work on it.
After taking on a general case, which proved to be too difficult, I tried a specific case. Something simple like the curve $y_1 = x^2$ ...
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1answer
311 views
Intersection of squares/cubes/hypercubes.
One can form a polygon of $4 n$ sides by intersecting $n$ congruent squares (treated as closed sets, i.e., filled squares):
Q1. For which of the ...
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Icosahedral symmetry as permutation group
Hopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the ...
