Questions tagged [3d]
For things related to 3 dimensions. For geometry of 3-dimensional solids, please use instead (solid-geometry). For non-planar geometry, but otherwise agnostic of dimensions, perhaps (euclidean-geometry) or (analytic-geometry) should also be considered.
3,710
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Difficult Vectors Problem (Calculus & Vectors 12)
Find parametric equations of a line that intersects line 1 and line 2 at right angles.
Line 1: $[x,y,z] = [4,8,-1] + t[2,3,-4]$ and
Line 2: $[x,y,z] = [7,2,-1] + k[-6,1,2]$.
I've tried solving this ...
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2
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31
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Unique rotation matrix using two axes when rotating a vector
I have a vector $v1$, and a rotated vector $v2$. I want to find two rotation matrices $Rx$ and $Ry$, which are rotation matrices around x-axis and y-axis, respectively, so that $Rx \times Ry \times v1 ...
2
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22
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3d straightedge and compass
Given a tool that can draw a sphere by given center and a point on it and a surface by given 3 points, is the constructable set of the tool equivalent to the streightedge and compass constructable ...
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A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with....
A parallelepiped is formed using three non collinear vectors whose magnitudes are 1, 2, 3. Angle between any of the vector with normal of the plane determined by other two is π/3. Then the ratio of ...
2
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3
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181
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Find the equation of a plane containing two given points and having a given distance to a third point
This problem is part of examination preparation material for second mid-semester test of 12-th grade in my school:
In the 3D space Oxyz, given 3 points $A(1,0,0)$, $B(0,-2,3)$, $C(1,1,1)$. Let $(P)$ ...
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1
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24
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Angles between vector and z axis [closed]
I've just started studying vector algebra so go easy on me. I'm not sure how to approach this problem.
"The vector OP makes an angle of 60 degrees with the positive x-axis and 45 degrees with the ...
2
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1
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35
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Construct perspective projection of rotating tesseract by perpendicular lines intersecting ellipse
The contruction was used in two different sources on the web: a Geogebra resource and a video using inRm3D so I think it must be documented and proved somewhere, but I didn't find any.
Here is the ...
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Four points in $\Bbb R^3$ form equal angles then |AB|=|CD| and |BC|=|DA|.
I'm interested in the following problem:
If $A,B,C,D$ are four distinct points in $\Bbb R^3$ satisfying $\angle ABC=\angle BCD=\angle CDA=\angle DAB$,
then is it must be true that $|AB|=|CD|,|BC|=|DA|...
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35
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Extending coplanarity of 4 points to 5 or more points.
Context: We were taught collinearity of 3 points (in vectors) and a method for checking 4 points for coplanarity simultaneously. Searching around I found a better method for checking for coplanarity ...
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31
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Axes of an ellipsoid
A 3-Dimensional ellipsoid is given by the equation:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$
Let $a=20, b=15$ and $c=10$. Then as per my understanding, $a$ is the major axes and $c$ ...
2
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1
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55
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The minimum value of the sum $O_1O_2+…+O_{n-1}O_n$
A cube C of side $n > 2, n ∈ N,$ is divided in $n^3$ cubes of side 1, with disjoint interiors two by two.
We say that two of the cubes of side 1 are Olympic, if any plane parallel to any of the ...
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19
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Looking for an algorithm to calculate the boundary of the parallel projection of a spherically truncated convex polyhedron [closed]
Basically, I want to draw a spherically truncated convex polyhedron, i.e. the intersection of a sphere and a convex polyhedron. The planar parts of the intersection are simple to figure out, so the ...
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2
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123
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Spherical Tractrix
A pantoon P initially at North pole NP heading to Greenwich meridian intersection with equator, $( 0^{\circ} E, 0^{\circ} N $ ) towards point G moving down and east. It is dragged by a ship S moving ...
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Given the direction cosines of two mutually perpendicular lines, show the direction cosines of the lines perpendicular to the above two lines are:
I just want to ask that if it is written two equations of relations of l, l1, m, etc. How did he write it in that ratio or fractional form like l upon something and m upon something etc and made it ...
2
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1
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95
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How to convert from 2D points to 3D points on a plane [closed]
I have some 3D coplanar points. The plane is defined by normal vector and constant. I need to work with the points in 2D and then convert them back to 3D. In order to convert the points to 2D I made a ...
2
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1
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How to prove that the shortest path between two points in sphere is a part of the great circle?
It is a known fact that the shortest path between tow points in sphere is a part of the great circle. but I don't know the proof of this claim so I tried to rigorously prove it myself.
My attempt: ...
1
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1
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38
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How to get a 3D vector in the same direction as another vector limited by an angle? [closed]
If I have two 3D vectors as shown by the blue and red vector in the image below, where the red vector is at an angle of more than 20 degrees from the blue vector, how could I calculate a new vector (...
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1
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65
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Regarding solving for the coordinates of reflection points in 3D space
In space, there exists a smooth plane, and additionally, there are two points, denoted as $A (x_a, y_a, z_a)$ and $B (x_b, y_b, z_b)$. A beam of light is emitted from point A, reflects off the mirror ...
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dihedral angles of non-regulare icosahedron given its coordinates
I have the coordinates (3D) of the vertices of an icosahedron (not a regular one) and I want to find all dihedral angles. I am fine with just equations to which they need to fulfill as I would think ...
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76
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Determine the equation of the right circular cone whose intersection with the $xy$ plane is a known ellipse
There is a known ellipse
$ (r - r_0)^T Q_e (r - r_0) = 1 $
that is the intersection of an unknown right circular cone with the $xy$ plane.
Determine the equation of the cone, i.e. determine the ...
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Determine position of sound source from arrival times of a blip ($3d$ version)
As an extension to this problem, I am now considering the problem in $3d$.
The setup is a follows: You have four sound receivers (microphones) at known locations $P_1(x_1,y_1, z_1), P_2(x_2,y_2,z_2), ...
3
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4
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104
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How to calculate the area of a projected path onto a plane? [closed]
How do you "orthogonally project" the shape? After I drew multiple graphs, I still had no idea.
The correct answer is $12$. I guess it is $3 \sqrt 2 \cdot 2 \sqrt 2$ according to the gist of ...
2
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0
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43
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Volume of water in a tilted hemispherical bowl
Suppose you have a hemispherical bowl that is open at the top. You tilt the bowl by an angle $\theta$ in any direction. Then you fill it with water till the water surface reaches the lowest point on ...
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1
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47
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Hyperbolic equations with three terms [closed]
The equation of a hyperbola is
$ x^2 - y^2 = r $
Assuming $ a = b = 0 $
Suppose now the equation
$ x^2 - y^2 + z^2 = r $
What kind of curve would this equation parametrize ? What is its geometry?
...
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2
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70
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Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$
Problem: Find the minimum value of the expression $2a^2 + b^2 + c^2$ if the point $(a,b,c)$ lies on the plane $4a + 3b + c = 7$.
I couldn't get very far, but I do know how would I solve a simpler ...
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0
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49
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Volume of a sphere enclosing six tetrahedron edges [closed]
Six edge lengths of a tetrahedron $ (a,b,c,p,q,r) $ are given.
What relation should be there in order that a tetrahedron can be enclosed among the edge side lengths?
What is the volume $V$ of the ...
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212
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Inverse square law-type integral over two line segments
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in ...
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Unique affine transformation, sending $P_1$ to $P'_1$, $P_2$ to $P'_2$, $P_3$ to $P'_3$ where $\| P_i P_j \| = \| P'_i P'_j \| $
Given three points in $3d$ space $\{ P_i, i = 1, 2, 3 \} $, and corresponding three points $\{ P'_i, i = 1, 2, 3 \} $ that are their images, respectively, such that $\| P_i P_j \| = \| P'_i P'_j \| , ...
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1
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Finding an affine transformation that will map one set of three given points $P_1, P_2, P_3$ to another given set of points $P'_1, P'_2, P'_3$
Suppose you given three points with known coordinates $P_1, P_2, P_3$, and also another set of three points $P'_1, P'_2, P'_3$. Now does there always exist an affine transformation
$ f( r ) = A r + b ...
2
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Equation of the plane that is at given distances from three given points $A,B,C$
Suppose you have the coordinates of three points $A,B,C$. You want to construct a plane $\pi$ such that
$ d(\pi, A) = r_1 $
$ d(\pi, B) = r_2 $
$ d(\pi , C ) = r_3 $
where $d(\pi,v) $ is the ...
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1
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Generate random point on the ellipsoid in 3d.
I want to generate the trajectory of point M in space, such that the sum of its distances to two other points, P and Q in space, remains constant, that is, |PM| + |MQ| = constant. Therefore, I believe ...
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0
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110
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Parameterizing (potentially non-planar) equilateral polygons in 3d
This is similar to Parameterizing equilateral polygons but for 3 dimensions.
I have $n$ points in $\mathbb{R}^3$ ($n \geq 3$), labeled $1$, $2$, ..., $n$. Point $1$ is unit distance from point $2$, ...
2
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2
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75
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3d fractal helix modeling
I'm trying to build a 3d visual to illustrate a concept. Imagine a circular helix. We could define a cylinder that contains that helix. But now imagine this cylinder takes the helicoïd shape too ! We ...
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1
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Shortest distance between vertex of a circular cone and a quarter of its conical helix
I was given with the question below:
...
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38
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Calculate Up Vector of Object on Surface Given Points and Normals
Context
I want to find the up vector an object would have if it were leaning on a surface, given a large amount of points on that surface and their associated surface normals (i.e., an equation that ...
0
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1
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55
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Spatial curve with certain condition
In a certain exercise I have been asked to find all spatial curves such that, given the curve $\gamma$, it is satisfied that $\gamma'' = \gamma' \times a$, $a$ is a certain constant vector.
I am aware ...
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Are two parallel projections sufficient to prove that four points are the vertices of a parallelogram?
This is an analytical proof of Varignon's theorem (of course it is a well-known proof)
I came up with an idea, which is to take advantage of the two-dimensional situation to prove the three-...
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higher dimensional analogue of volume difference of 2 three-dimensional cubes
In the illustration a way of expressing the difference in volume of 2 different sized 3-dimensional cubes is given:
The volume of a black cube of size b is diminished by the volume of a red cube with ...
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0
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Minimum and Maximum distance to the intersection of an ellipsoid with a plane
Given the ellipsoid
$ (r - C)^T Q (r - C) = 1$
and the plane
$ a^T r = b $
where $ r = [ x,y,z]^T $, $C \in \mathbb{R}^3$ is the center of the ellipsoid, and $Q \in \mathbb{R}^{3 \times 3} $ is a ...
3
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1
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Interpretation of change in direction cosines of a variable line: Pythagorean theorem for small angles?
Consider a variable rotating line passing through a fixed point. The angle between two successive/adjacent positions of the line is a small angle $\delta \theta$. If the change in the direction ...
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1
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23
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Closest and Farthest point on curves of intersection between two ellipsoids to the origin
A curve is the intersection of two ellipsoids that are given by
$ (r - C_1)^T Q_1 (r - C_1) = 1 $
$ (r - C_2)^T Q_2 (r - C_2) = 1 $
I'd like to find the points on this curve of intersection that are ...
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0
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19
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Equations with nodes 72 degrees apart
I am interested the nodal ($z=0$) behavior of the following functions for a small research project:
$$z=x(x^2-3y^2)$$
$$z=y(3x^2-y^2)$$
When graphed, there are nodes 60° ($n=6$) apart from each other. ...
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2
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Rotation of $3$-dimensional space vectors
Assuming there are two vectors $P(x_p,y_p,z_p)$ and $Q(x_q,y_q,z_q)$ originating from the origin. Now, if I rotate vector $P$ to another position in space and obtain the coordinates $P'(x_p',y_p',z_p')...
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1
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46
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Minimum and maximum distance of a $3D$ circle from the origin
A $3d$ circle is the intersection of two spheres given by
$ (r - C_1)^T (r - C_1) = R_1^2 $
$ (r - C_2)^T (r - C_2) = R_2^2 $
I'd like to find the points on this circle of intersection that are at a ...
1
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2
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In three-dimensional space, when the sum of the distances from an unknown point to two known points is constant, what is the trajectory of the point?
The unknown point P (x, y, z) in three-dimensional space is at a constant sum of distances dPA and dPB from two known fixed points A and B, i.e., dPA + dPB = constant. How can we express the ...
0
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1
answer
56
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Finding the minimum and maximum distance between the origin and the intersection curve between a cone and a sphere
Suppose you have the cone with its vertex at the origin given by
$ r^T Q r = 0 \tag{1}$
where $r=[x,y,z]^T $, and $Q$ is a $3 \times 3$ symmetric indefinite matrix.
And you have the sphere centered at ...
0
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0
answers
60
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Finding subset in $\mathbb{R}^3$ with under some condition.
Let $S$ be the subset in $\mathbb{R}^3$ of all vectors forming an angle $ \leq 45^{\circ}$ with the vector $(1 , 0 , 1)^T.$
I have no idea how to approach this problem?
Is this the right approach, $S=...
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1
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61
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Volume of tetrahedron detected by four planes in $\mathbb R^3$ [closed]
Given the equation of $4$ planes in $\mathbb R^3$, find the volume of the tetrahedron. The planes are:
$$\begin{align}&P1: x+3y+z=2\\&P2: x-2y+z=2\\&P3: -x+z=4\\&P4: x=3\end{align}$$
I ...
1
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1
answer
32
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cant find the orthogonal proyection of the line on a plane. plane: 10x-6y-12z=7, line: (8-15t,9t,5+18t). [closed]
i have the following:
when replacing x,y,z values of the line on the plane equation: 10(8-15t)-6(9t)-12(5+18t)=7, t=13/420.Then if we replace "t" in the equation of the line, we obtain the ...
1
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0
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17
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Computing the Upward Direction(Z-Axis) Vector from a 3D Rotation
I was tasked on creating a C Program that needs to process this case:
Convert the Initial Rotation Matrix(InitialRotationMatrix) into 3D Rotation(Rotation3DResult1)
Convert the 3D Rotation(...