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0
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2answers
55 views

Point of tangency between plane and sphere

How can one find the point of tangency between a plane and a sphere in $\mathbb{R}^3$? The equations of the plane and the sphere are $x + y + z - 5 = 0$ and $(x-1)^2 + (y-2)^2 + (z+1)^2 = 3$ ...
1
vote
1answer
79 views

Find the area of the Grayed triangle Given the following Figure

Can you help me find the area of the gray triangle in the given figure. I'm having a hard time finding the base value of the triangle, I've managed to find the sides for the big triangle but not ...
0
votes
1answer
456 views

Line Drawing Using Bresenham Algorithm

Indicate which raster locations would be chosen by Bersenham’s algorithm when scan converting a line from screen co-ordinates (1,1) to (8,5). First the straight values (initial values) must be found ...
0
votes
1answer
107 views

vectors in 3D space and Right-Hand Rule

Suppose we have three vectors in 3D space. My questions are: How we check if these vectors are satisfy the right-hand rule or not. I know that it's possible to make the three vectors satisfy the ...
2
votes
3answers
112 views

Generalization of angle bisector to tetrahedron

Let $I$ be the center of the inscribed sphere of a tetrahedron $ABCD$ and let $I_A$ be the length of the line passing through $I$ from the vertex $A$ to the opposite face. I am looking for a formula ...
2
votes
0answers
21 views

The exact type of my 3d model

I have reconstructed vertical features (hole like objects lie on a vertical face) lie on two connected faces. To understand the situation, I say I have 2 walls with many windows and doors on ...
0
votes
1answer
94 views

Perspective projection onto y/z plane?

On wikipedia there is an article on 3d perspective projection onto the x/y plane. http://en.wikipedia.org/wiki/3D_projection#Perspective_projection How do I project onto the y/z plane? If i have a ...
1
vote
1answer
118 views

Icosahedron and dual dodecahedron coordinates and rotations

I'm trying to build a 20 sided die in Actionscript 3, like one in the picture. I figure the best way would be to make it out of 20 equilateral triangles rotated in 3D space. The vertices of the ...
1
vote
1answer
22 views

calculate size of faces to have similar volume of platonic solids

I need to build some prototype physical shapes of the 5 platonic solids. But the problem is that I want to make them all about the same size, so how to calculate how big should the faces of my ...
2
votes
1answer
45 views

what will be the angle at the centre?

Taken a tetrahedron of same edges, a point is taken inside it which is equidistant from all $4$ vertices, i.e if a sphere is made taking it as a centre, all the vertices will be on the sphere, now ...
2
votes
1answer
132 views

Disk and washer problem

I need to find the volume of the solid obtained by rotating the region bounded by the following functions: $xy = 1, y = 0, x = 1, x = 2;$ about $x = -1$ I think the outer radius of the washer is ...
5
votes
3answers
137 views

Which solids are characterized by their orthographic projections?

If I know the orthographic projections of a given solid in Euclidean 3-space onto the $xy$, $xz$ and $yz$ planes, under which circumstances can I reconstruct the solid based on that information alone? ...
1
vote
0answers
50 views

Finding an angle in a straight pyramid

We have a straight pyramid with a square ABCD as its base and apex S. We're given the pyramid's height 8 and the angle 48 deg. between SA and SC. I've already managed to calculate the pyramid's volume ...
2
votes
0answers
107 views

Finding an angle in a pyramid

We have a straight pyramid with a square ABCD as its base and apex S. We're given the pyramid's height 8 and the angle 48 deg. between SA and SC. I've already managed to calculate the pyramid's volume ...
13
votes
1answer
758 views

Is there a dissection proof of the Pythagorean Theorem for tetrahedra?

Of the many nice proofs of the Pythagorean theorem, one large class is the "dissection" proofs, where the sum of the areas of the squares on the two legs is shown to be the same as the area of the ...
5
votes
1answer
215 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 ...
3
votes
0answers
95 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
96 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
226 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
298 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
91 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
98 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
37 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
239 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
209 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
224 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
283 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
238 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
266 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
164 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
50 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
175 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
108 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
145 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
147 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
160 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
214 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 ...
2
votes
3answers
635 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
65 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
201 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
328 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 ...
4
votes
1answer
133 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
116 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
134 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
129 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
96 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 ...
7
votes
2answers
549 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
106 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 ...