The (3d) tag is for things related to 3-dimensions. For geometry of 3-dimensional solids, please use instead (solid-geometry). For geometry that is not on a plane, but otherwise agnostic of dimensions, perhaps (euclidean-geometry) or (analytic-geometry) should also be considered.

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Minimal definition of a Platonic Solid

I searched in Google for the definition of Platonic solid, and I found: A platonic solid is a polyhedron all of whose faces are congruent regular polygons, and where the same number of faces ...
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If the projections of $OA$ and $OB$ on the plane $z=0$ make angles $\phi_1$ and $\phi_2$,respectively,with the $x-$axis

$OA,OB,OC,$ with $O$ as origin, are three mutually perpendicular lines whose direction cosines are $l_1,m_1,n_1;l_2,m_2,n_2$ and $l_3,m_3,n_3$ respectively. If the projections of $OA$ and $OB$ on the ...
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34 views

How to draw the 3d planes?

The equations for planes I have are $x=0$, $x=1$, $y=0$, $y=1$, $z=0$, $z=1$ but these to me look like equation of the lines. Apparently, they form a cube in 3d coordinates. Is it possible use some ...
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Prove that for all values of $\lambda$ and $\mu,$ the two planes intersect on the same line.

Prove that for all values of $\lambda$ and $\mu,$ the planes $\frac{2x}{a}+\frac{y}{b}+\frac{2z}{c}-1+\lambda(\frac{x}{a}-\frac{2y}{b}-\frac{z}{c}-2)=0$ and ...
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Show that there is a common line of intersection of the three given planes.

Let $x-y\sin\alpha-z\sin\beta=0,x\sin\alpha-y+z\sin\gamma=0$ and $x\sin\beta+y\sin\gamma-z=0$ be the equations of the planes such that $\alpha+\beta+\gamma=\frac{\pi}{2}$,(where ...
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36 views

The new position of $O,$ when triangle is rotated about side $AB$ by $90^\circ$ can be

Consider the triangle $AOB$ in the $xy$-plane where $A\equiv(1,0,0);B\equiv(0,2,0);$ and $O(0,0,0)$.The new position of $O,$ when triangle is rotated about side $AB$ by $90^\circ$ can be ...
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32 views

3d collsion rig math

I'm in quite the bind. The end result I'm looking for is to create a finger rig with collision detection. This will be done in a 3d app. The diagrams are acting as a basic visual display. B1 is ...
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18 views

Method of finding Arc length parameterization of a 3d curve

r(t) = cos^3 t i + sin^3 t j; 0 < t < pi/2. r'(t) = -3cos^2 t sin t + 3 sin^2 t cos t ||r'(t)|| = 3 sin t cos t Now to find the arc length parameterization, we need S = integration from t0 ...
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15 views

A parallelopiped is formed by planes drawn through the points $(1,2,3)$ and $(9,8,5)$ parallel to the coordinate planes

A parallelopiped is formed by planes drawn through the points $(1,2,3)$ and $(9,8,5)$ parallel to the coordinate planes then which of the following is not the length of an edge of this rectangular ...
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21 views

A rod of length $2$ units whose one end is $(1,0,-1)$ and the other end touches the plane $x-2y+2z+4=0$

A rod of length $2$ units whose one end is $(1,0,-1)$ and the other end touches the plane $x-2y+2z+4=0,$ then find the center of the region which the rod traces on the plane. The rod sweeps out the ...
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52 views

Find the equation of the plane which bisects that angle between the given planes which is acute and which contains the origin.

Consider the planes $3x-6y+2z+5=0$ and $4x-12y+3z=3$.Find the equation of the plane which bisects that angle between the given planes which is acute and which contains the origin. The two bisectors ...
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43 views

If the shortest distance between the skew lines $AB$ and $CD$ is $8$ ,find the volume of the tetrahedron.

Given a tetrahedron $ABCD$ with $D$ at the top and $AB=12,CD=6$.If the shortest distance between the skew lines $AB$ and $CD$ is $8$ and the angle between them is $\frac{\pi}{6}$,then find the volume ...
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23 views

Segments joining each vertex with the centroid of the opposite face are concurrent at the point $P$.Find the position vector of $P$

$OABC$ is a tetrahedron where $O$ is the origin.Position vectors of its angular points $A,B$ and $C$ are $\vec{a},\vec{b}$ and $\vec{c}$ respectively. Segments joining each vertex with the centroid of ...
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30 views

Prove that there are infinite planes containing both the lines $L_1,L_2$ and that there is a unique plane containing both the lines $L_3,L_4.$

Let $L_1:\frac{x-1}{1}=\frac{y-0}{1}=\frac{z-2}{-5}$ $L_2:\frac{x-2}{2}=\frac{y-1}{2}=\frac{z+3}{-10}$ and let $L_3:\frac{x-0}{-6}=\frac{y-1}{9}=\frac{z-0}{-3}$ ...
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Find the equation of the plane perpendicular to the plane $P$ and containing line $L_1$.

Consider a plane $P$ passing through $A(\lambda,3,\mu)$,$B(-1,3,2)$ and $C(7,5,10)$ and a straight line $L$ with positive direction cosines passing through $A$,bisecting $BC$ and makes equal angles ...
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If $P$ and $Q$ are points on two skew lines.If $P$ and $Q$ are nearest to each other.Find $\vec{OP},\vec{OQ}$

Let $P$ and $Q$ be points on the lines $L_1:\vec{r}=6\hat{i}+7\hat{j}+4\hat{k}+\lambda(3\hat{i}-1\hat{j}+\hat{k})$ and $L_2:\vec{r}=-9\hat{j}+2\hat{k}+\mu(-3\hat{i}+2\hat{j}+4\hat{k})$ respectively ...
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51 views

How to interpret a 3D plot

I am trying to interpret 3D plots but its driving me nuts. Lets say I want to see the relationship between 3 variables and I have data as following: ...
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37 views

Rotate around point whilst maintaining forward facing direction

I have a stationary point, the origin (O) at (0, 0, 0) and a moving Point (P) that I wish to have rotating around O with a fixed radius (d). However, rather than just rotating around I always want to ...
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Finding rotation and translation of a planar object in 3d

I have a planar object, say a polygon $A_1A_2\ldots A_n$ in the 3-dimensional Euclidean space. It is translated by a vector $v$ and rotated by a rotation matrix $R$, and the resulting image is ...
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Curl and Divergence definitions - Is this definition mathematically correct?

I've been using Stewart's Calculus: Early Transcendentals in my Calculus class, and one of the definitions of curl offered by the book was $$\text{curl } \vec F = \nabla\times \vec F$$ where ...
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72 views

Equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at an angle of $\frac{\pi}{3}$

Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at an angle of $\frac{\pi}{3}$. Let the direction ratios of the two ...
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The equation of the plane which passes through the point of intersection of two space lines and at greatest distance from the point $(0,0,0)$

The equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance ...
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32 views

The value of $m$ for which straight line $3x-2y+z+3=0=4x-3y+4z+1$ is parallel to the plane $2x-y+mz-2=0$ is

The value of $m$ for which straight line $3x-2y+z+3=0=4x-3y+4z+1$ is parallel to the plane $2x-y+mz-2=0$ is $(A)-2\hspace{1cm}(B)8\hspace{1cm}(C)-18\hspace{1cm}(D)11$ My Attempt:The plane ...
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relative sizes of noncoplanar objects

Say that there are two objects in the distance, one of a known size, the other of an unknown size. Both objects are two dimensional, and take the form of rectangles. They are not coplanar, but I know ...
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A 3D problem on the range of angles

$CD$, of length $14$ units, is a line fixed on the ground. $AB$ is a thin rod fixed in the air. A ring, which is attached to $AB$, can slide freely on $AB$. An elastic band (remaining taut all the ...
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24 views

Arc Length in 3D space

So given a function $$f(x,y)=x^3+30xy+2y^3,$$ I am able to find the maximum value in a specific region. However, I need to be able to calculate the distance that would be traversed by a particle ...
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42 views

3D coordinates rotation — new direction for Z axis

I need to rotate 3D coordinate system so Z axis points in new direction. So, I have a direction defined by spherical coordinates ($\theta$, $\phi$), where $\theta$ (in $[0, \pi]$ range) is polar and ...
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Finding 2 vectors in a circle in 3D space

I'm trying to find 2 vectors in a given circle (not a sphere). This circle can be at a random position, rotation. Given the position, rotation, and radius of the circle, where the center position of ...
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34 views

Given a line and a point in 3D, how to find the closest point on the line?

I have a point given by $P = (P_x, P_y, P_z)$, and a line give by the two points $Q = (Q_x, Q_y, Q_z)$ and $R = (R_x, R_y, R_z)$. I'd like to know a general formula to figure out the closest point to ...
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3d rotation of a triangle onto surface of sphere?

I'm trying to develop a 3D model of a sphere, and I want to place triangles near the surface, normalized to the surface at that point. I am having trouble with the mathematics of the rotation of the ...
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1answer
36 views

How to find the best-fit transformation between two sets of 3D observations

Let us assume that I have two sets of observations, A and B. Each of them is a list of 3D points. They describe the same set of world 3D points at the same order. However, they are given in different ...
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Equation of circle in 3d Plane?

Suppose I have a sphere centered at origin. $$ x^2+y^2+z^2=5 $$ and a plane $$ \vec{r}.(\hat{i}+\hat{j}+\hat{k})=3\sqrt{3} $$ And this plane cuts the sphere at a circular region. How do I write the ...
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30 views

Plotting a function in 3D (theory)

I made a program which draws/plots a function like it is given in the picture ...
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33 views

Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
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Kelvin problem with restriction to only one type of polyhedron

If we only allow one type of polyhedron, what would be the answer to the Kelvin problem? Would the Kelvin conjecture be true?
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1answer
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Why must the outside limits of an iterated be constant?

My book claims that in an iterated integral $$\int_a^b \int_{g(x)}^{h(x)} f(x, y) \, dy \, dx$$ $h$ and $j$ are allowed to be any functions of $x$ not containing $y$, but $a$ and $b$ must be constant ...
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Limits of integration of a multivariable function?

My book is filled with problems such as $\int_{1}^{x} f(x, y) dy$ . My question is if there is a good reason why the top limit of integration is an $x$. When we integrate the function with respect to ...
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How to obtain 3D coordinates of the point by the length of the vector?

How can I obtain R3 position of a point? For example, I've got two points linked by a vector: p1 = (-4000;250;-5000) p2 = (428;776;-300) |v| = 6926.32 I'd like to find a point which lies on the line ...
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Drunk tankist problem

Imagine that we have a military tank. At time t0, we get an initial rotation matrix M0, that specify the initial position of the tank acording to world's frame F0. At time t0 we also have an initial ...
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What is the purpose of $v$ in the parametric equation for a sphere?

The longitude / latitude parameterization of a sphere is described by: $x = cos(φ) * cos(θ) \quad y = cos(φ) * sin(θ) \quad z = sin(φ)\quad$ where $\quadθ = 2 π u$ and $φ = π v - π / 2$ I ...
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Using Lagrange multipliers to find max and min of a function?

The function is $f(x, y, z)=x^2+y^2-z$ subjected to $z=2x^2y^2+1$ . My first step was to define $g(x, y, z)=z-2x^2y^2=1$ So $\nabla f = 2xi+2yj-k$, and $\lambda \nabla g = -4xy^2\lambda i ...
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Finding the maximum volume of a box bounded by a plane?

The box is in the first octant, and one corner is located at the origin while the opposite corner is located on the plane $x+2y+3z=6$. My approach was to write the volume as $F(x, y, z) = xyz$, ...
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Why is the maximum increase given by $||\nabla f(x, y)||$?

I understand the steps of the proof in the book, but I don't see intuitively the of maximum increase at a point $P$ must be given by the $||\nabla f(x, y)||$. A graph has infinite directional ...
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22 views

In which cases is there a need for U and V in parametric equations

I'm am reviewing parametric equations (to get a better grasp over how they are used to make shapes in computer graphics) and currently I have an understanding of how the parameter $t$ is used to ...
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63 views

How similar can the nets of distinct polyhedra be?

My school, and most math books do not cover 3-d geometry well, especially the topics of polyhedron nets. However, I see quite a few questions here are being answered about them. I was wondering about ...
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“The gyroelongated triangular bipyramid can be made with equilateral triangles”

According to Wikipedia article Gyroelongated bipyramid The gyroelongated triangular bipyramid can be made with equilateral triangles I can only imagine that this would result in a cube, could ...
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38 views

A curve of constant curvature and zero torsion must be a circle

From Elementary Differential Geometry by Pressley I don't understand the last paragraph. Why does it show $\gamma$ lies on the sphere $\mathcal S$ with center $\mathbf a$ and radius $1/\kappa$?
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42 views

Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
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1answer
26 views

How to find the intersection of a line and a plane with interpolation ( given two points in the opposite side of the plane)

I have two points in the opposite side of a plane (P1,P2) in 3D space, and i know their signed distances to the plane(D1,D2). how can i use interpolation to calculate the point that is the ...
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27 views

Modify Equation of sphere intersection

I got to know how to modify formula for radius of circle formed after intersection of two spheres when centres of both spheres where at origin or at x axis to centres at any arbitary positions ...