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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15 views

Newtonian potential of homogeneous ball

Let $x \in B_R(0) \subset \mathbb{R^3}$. To compute $$u(x)=\int_{B_R(0)} \frac{1}{|y-x|} dy$$ The integrand has singularity at $x$, so consider $$u_\epsilon(x)=\int_{\substack{|y|< R \\ |y- x|> ...
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1answer
918 views

Fitting a line curve to 3D data

I have a set of points in 3D which I need to fit a curve (not a plane) to. Essentially these points describe a string with a set order (i.e. point one connects to point two, etc.) and I need to fit a ...
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3answers
14k views

How to find shortest distance between two skew lines in 3D?

If given 2 lines $\alpha$ and $\beta$, that are created by 2 points: A and B 2 plane intersection I want to find shortest distance between them. $$\left\{\begin{array}{c} P_1=x_1X+y_1Y+z_1Z+C=0 ...
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0answers
34 views

Proof that there exists a 3d representation of all graphs

Below is a question and proof that I've done. I was wondering if there is a more formal way of concluding a point must exist that is not in a set composed of a finite number planes. Currently I am ...
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4answers
50 views

The formula for 3D rotation of the perspective of an image in 2D space

Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of ...
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1answer
441 views

Counting the number of cubes in an isometric view

I had seen questions in a sample aptitude test, where an isometric view of an object made up of cubes was given, with some of the cubes removed. We were supposed to count the number of cubes present ...
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1answer
594 views

Best fitting circle to points in 3D

I have a set of n ≥ 3 points in 3D that are measurements of a possible circle. The measured points are "noisy" so best-fitting algorithms are involved. I'm programming in C# and have put together some ...
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2answers
20 views

Formula For 3D Dilation?

If I have a sphere on a 3D grid with it's center being at the origin, and I want to double the size of the sphere, where would it's poles be?
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1answer
417 views

3D Vector defined by 3 angles trigonometry components

What I'm looking for is the trigonomery equations to calculate the x, y and z components of a 3D vector. What I mean: The counterpart formulas for a 2D vector defined by 1 angle: $x = \cos(\alpha)...
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1answer
562 views

Intersection of a line segment and a paraboloid in 3D

Suppose I have a line segment $L$ in 3D: $$x=a_1(1-t)+b_1t$$ $$y=a_2(1-t)+b_2t$$ $$z=(a_1^2+a_2^2-k_1^2)(1-t)+(b_1^2+b_2^2-k_2^2)t$$ Because $L$ is line segment then $0\leq t\leq 1$. And defining ...
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2answers
783 views

What is an isosurface?

I am trying to understand the marching cubes algorithm. I would like very much an easier definition of an isosurface than what is available online. Could anyone please explain it? Thanks.
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3answers
883 views

Identify and sketch the quadric surface?

I'm stuck trying to figure out which type of quadric surface this equation is: $$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$ I have narrowed it down to a hyperboloid, but cannot ...
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1answer
26 views

Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
2
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1answer
19 views

Non-trivial 3D curve that projects as a line or a segment onto the faces of the quadrant

I want to illustrate how high dimensional objects may have misleading projections. Examples are for instance given with HiSee software, with nD bouquets of circles. Are there non-trivial (not a 3D ...
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0answers
29 views

Area of cross section in the intersection of two cylinders

A Steinmetz solid "B" is formed on the intersection of two cylinders, given as: x^2 + z^2 = r^2 and y^2 + z^2 = r^2 Although, referring to Gardner's work, I have successfully calculated the total ...
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0answers
13 views

Best 3D reconstruction method.

I'm looking for a method to generate a 3D point cloud from one or more cameras (I know the position of each camera). I need the point cloud to be generated each frame (of video from the camera) and ...
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2answers
9k views

How to find line parallel to direction vector and passing through a specific point?

I am give the point $(1,0,-3)$ and the vector $2i-4j+5k$ Find the equation of the line parallel to vector and passing through point $(1,0,-3)$ Could one use the fact that the dot product between the ...
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0answers
25 views

Translation by tensors

According to this question, quaternions would not be the right choice to handle both rotation and translation. In the case of tensors, one might assert that the rotation would be possible by tensors, ...
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1answer
15 views

3-Space Vertices of a Parallelogram

The points (1, -2, 4), (3, 5, 7) and (4, 6, 8) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three ...
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1answer
638 views

Calculating normals for a polygon mesh (3D computer graphics)

I want to write a program to generate arches, a common architectural form, and export them to a wavefront object format for sharing with various three dimensional graphics editors. To do this, I need ...
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1answer
27 views

Determining angle of view from an image with a square or checkerboard in the background.

I take a picture of a square of known dimension (let's say 1x1 units) with the camera at an unknown angle relative to the plane of the rectangle. (The distance to the rectangle is also unknown, but ...
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1answer
5k views

How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i +...
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0answers
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3d-Geometry (paraboloid). Finding the tangent plane

Tangent Planes at two points P and Q of a parabolid mmet in a line RS. show that tthe plane through RS and middle point of PQ is parallel to axis of parabola.
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1answer
1k views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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2answers
119 views

**Location** of shortest distance between two skew lines in 3D?

I can find the shortest distance $d$ between two skew lines $\vec{V_1}$ and $\vec{V_2}$ in 3D space with $d=\left|\frac{(\vec{V_1}\times\vec{V_2})\cdot\vec{P_1P_2}}{|\vec{V_1}\times\vec{V_2}|}\right|$...
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0answers
29 views

Counting balls in face centred cubic close packing

Possibly too easy for stack exchange, but... Consider a cubic close packing, or face centred cubic, arrangement of balls or radius $1$ in dimension $3$. Suppose that the origin is the centre of one ...
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1answer
71 views

Locus of the center of the circle of radius $a$,which always intersects coordinate axes

If the axes are rectangular, show that the locus of the center of the circle of radius $a$,which always intersects coordinate axes is $x\sqrt{a^2-y^2-z^2}+y\sqrt{a^2-z^2-x^2}+z\sqrt{a^2-x^2-y^2}=a^2$ ...
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1answer
49 views

Can there be a limit cycle without a fixed point in 3D space?

I am working with a population dynamics model. Basically, I have a nonlinear ODE in $R^3$ space, (X,Y,Z), and I know that if I start in the an open region ($0<X<1,0<Y<1,0<Z<1$, ...
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1answer
75 views

Trigonometric Word Problem in 3D

The question I am having trouble on is as follows: "As an Expert Mathematics Witness, you have been presented with a Ballistics Report, and a Police Report as your evidence. Use the information ...
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0answers
28 views

Euler Angle + Distance to XYZ coordinate

Background I'm familiar with Euler Angles and 3d space systems but I'm having trouble with Rotation Matrices. Scenario I've converted my Euler Angle to degrees for ...
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2answers
402 views

Calculate the viewing-angle on a square (3d-calc)

I'm in big trouble: My program (Java) successfully recognised a square drawn on a paper (by its 4 edges). Now I need to calculate, under which angle the webcam is facing this square. So I get the 4 ...
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0answers
38 views

Given set of points in 3D, find group of points closest to each other

Given a set of any 8 points in 3D space. I want to find a subset of points that are closest to each other. Application: Assume in a 3D space, I have any 8 colors(represented in RGB). I know how to ...
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0answers
47 views

Draw a line between an observer and the current direction of the sun

My goal is to draw a line between an observer and the current direction of the sun. Given the observers coordinates (Lat, Lon) of (51.50442, -0.08630) a North of (90, 0), an Azimuth of 270 degrees ...
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0answers
25 views

How to determine the equation of shortest path on any 3d surface between two given points?

I am working on draping of woven composite and I have to determine the equation of shortest path on 3D surface (i.e. $z=x^2+y^2$) between two given points in order to get the yarn path between two ...
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1answer
33 views

Prove that as $PP'$ varies,the circle generates the surface $(x^2+y^2+z^2)(\frac{x^2}{a^2}+\frac{y^2}{b^2})=x^2+y^2.$

$POP'$ is a variable diameter of the ellipse $z=0,\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ and a circle is described in the plane $PP'ZZ'$ on $PP'$ as diameter.Prove that as $PP'$ varies,the circle ...
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1answer
94 views

What should happen to an impossible cube at a vertex?

I have automated the process of impossible-cube renders in Blender3D as an exercise. However, while the automator works fine as long as the intersection of the 'impossible' edge and the nearer edge is ...
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2answers
42 views

Quaternion interpolation in 3D

I'm a chemist lost in the captivating world of mathematics thus if you could keep your answers simple it would be awesome! Here is my problem: I have two mobiles (A,B) in 3D. Ideally, I would like to ...
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1answer
40 views

Can two lines lying in different plane be parallel?

In two lines lie in different plane, can they be parallel to each other? I am thinking if two lines are parallel to each other, then their direction cosines must be same so two lines lying in ...
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0answers
20 views

Find radius of curvature of a $3D$ surface at a point if the tangent vector at that point is given.

I have a $3D$ surface with equation $z=x^2 +y^2$ and need to find the radius of curvature at a point $P(x_1,y_1,z_1)$ using the tangent vector at that point. Since, there are many possible radii of ...
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2answers
33 views

Plotting a 3D graph from explicit equation

I´m a 2nd year engineering student and today we learned how to plot 3d graphs from a $XYZ$ equation on paper. For example, I know ($\frac{X^2}{9}+ \frac{Y^2}{16} + \frac{Z^2}{9} =1$) will produce an ...
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1answer
73 views

A 3d integral over a ball

Given $a \in \mathbb{R}^3$ and $r>0$, is it possible to compute $$\int_{B_r(a)} \frac{a\cdot x}{|x|^3} dx$$ where $B_r(a)$ is the ball in $\mathbb{R}^3$ with radius $r$ and centered at $a$.
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1answer
29 views

How to compute the pivot point of a rectangular cuboid to achieve a certain rotation?

Summary: For a video game project, I have an object (craft) that hovers the ground using a soft constraint. Imagine that on the picture below there is an invisible point above the craft whose ...
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1answer
486 views

How to find the points of intersection of the perpendicular vector two skew lines

Correct me if im wrong, but this is what i know so far The cross product of two skew lines is the perpendicular vector between both those lines. The perpendicular vector intersecting two skew lines ...
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1answer
137 views

Photo image to find the screen orientation

I am trying to find the angle of tilts of a screen using projection of a circle from a source $S$. The light beam falls on the photo screen to expose it and what we get is an ellipse with major axis $...
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3answers
55 views

Find the equation of the sphere which touches the sphere $x^2+y^2+z^2+2x-6y+1=0$ at $(1,2,-2)$ and pass through $(1,-1,0)$

Find the equation of the sphere which touches the sphere $x^2+y^2+z^2+2x-6y+1=0$ at $(1,2,-2)$ and pass through $(1,-1,0)$ My Attempt: Let the equation of the sphere be $x^2+y^2+z^2+2ux+2vy+2wz+d=0$...
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3answers
2k views

Minimum distance between point and face

Given a point in 3D space of the form (x, y, z) and a triangle consisting of 3 vectors (also in the (x, y, z) format), how would I calculate the minimum distance between the point and the face of the ...
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1answer
20 views

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2+2x-4y+6z-7=0,$ which intersect in the line $6x-3y-23=0=3z+2.$

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2+2x-4y+6z-7=0,$ which intersect in the line $6x-3y-23=0=3z+2.$ Let the tangent planes be $A_1x+B_1y+C_1z+D_1=0$ and $A_2x+B_2y+...
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4answers
65 views

Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$

Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$ Let the circle passes through the sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ and the plane $Ax+By+Cz+D=0$ So the equation of ...
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1answer
65 views

3D Perspective projection

I have this following question to answer, however I am not sure how I should combine my calculation into one final answer. Suppose the Centre of Projection in a viewing space is at an offset $(0,...
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1answer
11 views

Find the equation of the sphere $OABC.$

$OA,OB,OC$ are mutually perpendicular lines through the origin and their direction cosines are $l_1,m_1,n_1;l_2,m_2,n_2;l_3,m_3,n_3.$If $OA=a,OB=b,OC=c,$prove that the equation of the sphere $OABC$ is ...