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5 votes

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

I think this problem is intended to be a purely logical problem with very little vector geometry. For starters, the problem states (without proof) that there IS a plane that passes through the ...
Bobby Ocean's user avatar
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4 votes
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Shortest distance between vertex of a circular cone and a quarter of its conical helix

The trick is to unwrap the side of the cone. Since the side is $AC=12$, you get a circular sector with the same radius. To find the angle, the length is $2\pi \frac 62=6\pi$. Then the angle of the ...
Andrei's user avatar
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3 votes

Volume of the two parts of a pyramid cut by a plane

Using Geogebra 3D (that I advise you : some hours to spend at the beginning, with great profit afterwards), I have generated this figure : I have found the four points of intersection : $$\begin{...
Jean Marie's user avatar
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3 votes

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Adopt a Cartesian coordinate system such that the bee begins at the origin and east/north/up is in the direction of increasing $x/y/z$, respectively. Then the trajectory of the bee is a series of ...
DarkLordOfPhysics's user avatar
3 votes

How to prove that the shortest path between two points in sphere is a part of the great circle?

First, I'll define the following unit vectors: $u_0 = \dfrac{b - a}{\| b - a\|}$ $u_1 = \dfrac{a + b}{\| a + b \| }$ $u_2 = \dfrac{a \times b }{ \| a \times b \| }$ then the axis of rotation that ...
of course's user avatar
  • 21.1k
2 votes
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How to convert from 2D points to 3D points on a plane

Attach a reference frame to the plane. You know the plane's normal $n$ and you know the constant $d$. The plane equation is $n \cdot r = d $ , where $r =(x,y,z)$. Such a frame is not unique. The ...
of course's user avatar
  • 21.1k
2 votes

3d fractal helix modeling

In 2020 I constructed a similar object using signed distance fields based on a helix as a sheared stack of toruses: https://github.com/claudeha/fragm-examples/blob/...
Claude's user avatar
  • 5,617
2 votes
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3d fractal helix modeling

I think that most direct analog from the library of classic fractals is likely to be Weierstrass's nowhere differentiable function, which is defined by $$\sum_{n=1}^{\infty} a^n \cos(b^n x),$$ for ...
Mark McClure's user avatar
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2 votes
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How to get a 3D vector in the same direction as another vector limited by an angle?

This is really simple. You need to construct a vector that is perpendicular to the blue vector and at the same time in the same plane as both the blue and red vectors, such that the angle between ...
of course's user avatar
  • 21.1k
1 vote

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Here I'll develop an analytical approach that hopefully will address the theoretical aspects of the original question. As the bee moves in 3D space, and easst/west, norht/south, and up/down are ...
cjferes's user avatar
  • 2,191
1 vote

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Following is the code I developed to solve this problem. The square of the area output by the program is $A^2 = 12$ ...
of course's user avatar
  • 21.1k
1 vote

Hyperbolic equations with three terms

Take the surface $$x^2-y^2+z^2=r,$$ for some $r>0$. If we project onto a plane $x=c$, we get $$-y^2+z^2=r-c^2,$$ a hyperbola (depending on the sign of $r-c^2$ we may get a rotated hyperbola). Doing ...
Julio Puerta's user avatar
1 vote

Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$

By application of Cauchy-Schwarz Inequality $$ \left( \sum_{i=1}^{n} a_i ^2 \right)\left( \sum_{i=1}^{n} b_i ^2 \right) \geq \left( \sum_{i=1} a_i b_i \right)^2$$ We have $$ (2a^2 + b^2 + c^2)((4/\...
Shivansh Jaiswal's user avatar
1 vote

Minimizing $2a^2 + b^2 + c^2$ given $4a + 3b + c = 7$

Hint, Let $f(a,b,c)=2a^2+b^2+c^2$ and $g(a,b,c)=4a+3b+c-7$ , Now consider $\nabla f=\lambda \nabla g$ with $g(a,b,c)=0$.
DARK's user avatar
  • 737
1 vote

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 $P'_1, P'_2, P'_3$

Extend $P_2-P_1,P_3-P_1$ to a basis $P_2-P_1,P_3-P_1,v$ then your choice of where the first two vectors is sent is forced, but you can send $v$ anywhere. Thus, in this basis, the third column of the ...
Joshua Tilley's user avatar
1 vote
Accepted

Spatial curve with certain condition

The mapping $\ x\mapsto x\times a\ $ is linear with eigenvalues $\ 0\ ,i\|a\|\ $ and $\ {-}i\|a\|\ ,$ corresponding to eigenvectors $\ a, u-iv\ $ and $\ u+iv\ ,$ respectively, where $\ u\ $ and $\ v\...
lonza leggiera's user avatar
1 vote
Accepted

Interpretation of change in direction cosines of a variable line: Pythagorean theorem for small angles?

Without loss of generality, we take the fixed point to be the origin. Let $L$ and $L'$ be two close lines in ${\mathbb R}^3$ passing through the origin with angle $\delta\theta$ between them. Let $P=L\...
Three aggies's user avatar
  • 4,318
1 vote

Rotation of $3$-dimensional space vectors

As peterwhy noted in his comments, an axis of rotation for the pair $P$ and $P'$ is any line that lies in the perpendicular bisecting plane of the line segment $PP'$. This plane has the equation $ n \...
of course's user avatar
  • 21.1k
1 vote
Accepted

Rotation of $3$-dimensional space vectors

Assuming $P$ is not parallel to $P'$, the unit-length axis of rotation is $k=\frac{P\times P'}{\lvert P\times P' \rvert}$, where $\times$ denotes the vector cross product. To find the angle of ...
FabrizzioMuzz's user avatar
1 vote
Accepted

Minimum and maximum distance of a $3D$ circle from the origin

The furthest I get is as follows: The equation $\nabla_rg = 0$ gives you an expression for $r$ in terms of the Lagrange multipliers $\lambda_1,\lambda_2$: $$ r = \frac{\lambda_1C_1+\lambda_2C_2}{1+\...
Levie B's user avatar
  • 146
1 vote

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?

Yes, you are correct in noting that the surface is that of an ellipse for each cross-section. The shape is essentially that which is formed by considering the 2D ellipse formed with A and B as focii, ...
SarthakC's user avatar
  • 349
1 vote

Volume of tetrahedron detected by four planes in $\mathbb R^3$

The four points are $(-1,0,3)(3,0,-1)(3,-8/3,7)(3,4,7)$. We have 3 position vectors $\bar{a}=(4,0,-4),\bar{b}=(4,-8/3,4),\bar{c}=(4,4,4)$ wrt $(-1,0,3)$. The volume of the tetrahedron with relative ...
RandomGuy's user avatar
  • 1,167
1 vote
Accepted

cant find the orthogonal proyection of the line on a plane. plane: 10x-6y-12z=7, line: (8-15t,9t,5+18t).

The normal vector to the plane is $n = [10 , -6, -12]$ and the direction vector of the line is $d = [-15,9, 18]$. The direction vector of the projected line (let's call it $v$ ) satisfies $ v \cdot n ...
of course's user avatar
  • 21.1k
1 vote

get dihedral angles of octahedron given all triangles

This boils down to finding dihedral angles in a pyramid with quadrilateral base, given all its edges. Let's take then a pyramid with base $ABCD$ and vertex $V$ and give its edges the names in figure ...
Intelligenti pauca's user avatar
1 vote

Solving a Lagrange multiplier optimization problem

Working on the idea by 'whpowell96' in his comment above, the second (linear) constraint: $ a^T r = b \tag{1}$ Leads to the solution $ r = r_0 + V u \tag{2}$ where we can choose $ r_0 = \dfrac{b}{a^T ...
of course's user avatar
  • 21.1k
1 vote

Equation of pair of lines created by the intersection of 2 double cones?

The two given cones share the same vertex which is the origin. The axis of the first is the unit vector $u_1 = (a,b,c)$ while the second's axis is the unit vector $u_2 = (a',b',c')$. The semi-apical ...
of course's user avatar
  • 21.1k

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