This is a question, that came up whilst working on a scientific representation of a cell membrane. In the following image you can see the particle system applied to a plane:  What I now want to do is to apply that particle system to a cylinder or circular object, which will show the cross-section of a vesicle. The particles should therefore all face outwards from the origin point of the circle. In the Rotation menu of the particle system there is no option for the particle to be orientated 90° away from the surface. Here is a bad sketch of how it should look.  Q: How can I orient particles on a round object?
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I want to replicate a protein structure around an icosahedron. Basically a vesicle of protein structure is what we want:  When I am instancing the protein structure using each vertex of the icosahedron it looks like:  However, the tails are supposed to be directed towards the center. How can I make that work?
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The cylinder is given by the equation $x^2 + (y-\frac{a}{2})^2 = (\frac{a}{2})^2$.  The region of the cylinder is given by the limits $0 \le \theta \le \pi$, $0 \le r \le a\sin \theta$ in polar coordinates.  We need to only calculate the surface from a hemisphere and multiply it by two. By implicit functions we have:  $$A=2\iint\frac{\sqrt{\left(\frac{\partial F}{\partial x}\right)^2 + \left(\frac{\partial F}{\partial y}\right)^2 + \left(\frac{\partial F}{\partial z}\right)^2}}{\left|\frac{\partial F}{\partial z} \right|} dA$$  where $F$ is the equation of the sphere.  Plugging in the expressions and simplifying ($z \ge 0)$, we get:  $$A=2a\iint\frac{1}{\sqrt{a^2 - x^2 - y^2}} dxdy$$  Converting to polar coordinates, we have:  $$A = 2a \int_{0}^\pi \int_{0}^{a\sin(\theta)} \frac{r}{\sqrt{a^2 - r^2}} drd\theta$$  Calculating this I get $2\pi a^2$. The answer is $(2\pi - 4)a^2$. Where am I  going wrong?
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I know very little about quantum physics, but I just can't wrap my head around the fact that non-spherical orbitals appear to be "oriented", constituting a set of fixed "frame of reference axes".  Does it really work like that? Are there for a single atom given directions in which the orbitals "protrude"? And if so, what are they based on? If I understand it right, the nucleus is seemingly spherical - symmetrical with no unique axis of symmetry. What would then decide the orientation of the orbitals in space (with respect to the surroundings)?  Edit: I understand any orientation works to satisfy the wave equation. I was more curious why the atom would choose one direction, instead of any of the others. Is that fundamentally random? And is this setup retained further, or can it change at any time?
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I have tried using the Subframes option. It hasn't done a thing.  So I have an object which gives off 3 particles into the air.  I have an Icosphere object on a different layer with a particle instance modifier on it.  That allows me to add a particle system to the Icosphere to create some awesome particle trails    But the particle system that is attached to the Icosphere creates clumps of particles on each frame.  I am wanting a nice, smooth stream of particles being emitted from the main particles.  Is there a way to fix this, or does the Subframes option not work with the Particle Instance modifier.  Here is my .blend file:  I am using Blender 2.72
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Position probability densities of eigenstates of hydrogen-like systems have axial symmetry, so that the wavefunction too much resembles the circular orbits in Bohr's model. I'd like to have a demonstration of correspondence principle, where an electron would be localized to look somewhat like a classical particle, and move in an elliptic (non-circular) orbit around a nucleus.  It seems though that if we try to make a wave packet too localized, then it'll disintegrate (get scattered by the nucleus) too fast to see anything resembling an elliptic orbit. OTOH, if we take it too spread out, it'll have to be quite far from the nucleus so as to avoid hitting it, and thus it'll have high total energy, which may appear to be above ionization threshold (at least partially), after which it'd be quite hard to analytically calculate evolution of the wave packet.  Thus my question is: is it possible to form a more or less localized wave packet, which would (on average) move in an obviously-elliptic orbit (major/minor axes ratio of 4:3 or higher), and only require bound states to fully represent it? If yes, then what properties (FWHM, apocenter, angular momentum etc.) should it have for this to be possible?
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I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.   I know that on the 2-sphere we have $ds^2 = d\theta^2+\sin^2(\theta)d\phi^2$ (in spherical coordinates) but I'm not sure how this generalizes to $n$ dimensions.  Added note: If anything can be discovered only about the determinant of the tensor (when presented in matrix form), that would also be quite helpful.
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Suppose there's a force $F$ of magnitude 10 newtons in the direction of positive y-axis acting on a particle A. I know that the particle would not experience any force in the positive x-direction because,  component of force $F$ in the positive x-direction $=|F|cos90°=0$.   But if we first take the component of $F$ at an angle of 45° to the positive y-axis, it is,   $|F|cos45°=\frac{10}{\sqrt2}\text N$.   If we now consider this a vector (can we? that is the main question), we can find its component in the positive x-direction (it makes an angle of 45° with the positive x-axis) ,   $\frac{10}{\sqrt2}\times cos45°=5\text N$. According to this result, the particle should experience a force of 5 newtons in the positive x-direction, which is impossible. I want to know what all is wrong in the above reasoning. Thanks!
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My intuition has always been that when any sphere is projected into 2D space that the result will always mathematically be an ellipse (or a circle in degenerate cases).  In the past when I was actively doing my own graphics programming and brought this up with other people they were adamant that I was wrong. If I recall correctly they believed the result could be something vaguely "egg shaped".  Who was correct?    Since there is already one answer submitted, I don't wish to totally change my question but I realize I left out important details due to losing familiarity with the field over the years.  I intended to ask specifically about perspective projection where the projection is a linear application.  The other projections are of course interesting for many uses so I wouldn't want them removed at this point. But it would be great if answers could have perspective projection as their most prominent section.
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I'm trying to find a way to achieve an extruded version of what the image below shows without having to manually scale every cylinder.  I tried particle systems (hair, both vertex and face modes), but I seem to be unable to reach a satisfying compromise between gaps and intersections. My cylinders, in an ideal world, should not intersect at all, and every cylinder must be touchng at least another one.  Here is my reference image:   Any ideas?  Thanks in advance :)
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I'm using cycles and particles that form text. I've got every letter separated, and while the particles form the shape, it doesn't properly 'fit' the shape or cover all of the surface of the object, so it doesn't work properly for me.  I've already tried making another particle system on the same object, with more particles, but it looks weird...is there any solution to this?  Thanks in regard, I'll be waiting for a response.
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I have a seed object with some particle hair strands. I've specified this object to be instanced in another particle system, then specified that object to be instanced in yet another particle system.  To clarify:      I have a cube with hairs strands I have a Suzanne coated in cubes with hair strands (the hair strands only appear in the render, which is fine) I have another Suzanne coated in Suzannes, but these Suzannes don't have cubes or strands like I was hoping..     Is there a way to get the particle systems to keep recursing, so that the last Suzanne is coated in Suzannes coated in cubes coated in strands?
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Given a general equation for a plane through the origin $$\vec{n}\cdot\vec{r}=0$$ With no assumptions made on $\vec{n}$ except having unit modulus, real $3\times1$ vector. How can you describe a unit circle, centred at the origin, laying in this plane?  I can only seem to find parametric equations that rely on knowing two vectors in the plane, but with no knowledge of the vector $\vec{n}$ you can't generally create two such vectors, as some component(s) of $\vec{n}$ may be zero. All the information you need to define such a circle is contained within the normal to the plane, so I am confused as to why there is not a form defined only with reference to this vector.  EDIT#1: With reference . Can we start with in the $xy$ plane  $$(x,y,z)=(\cos(\theta),\sin(\theta),0)$$ Then rotate this about the axis ($\vec{u}$ in the link)  $$\vec{u}=(-n_2,n_1,0)$$ about an angle $\phi$ that satisfies $$\tan(\phi)=\frac{n_3}{\sqrt{n_1^2+n_2^2}}.$$ I claim that $\vec{u}$ is the axis of rotation as this vector is perpendicular to the normal of the plane $\vec{n}$ and lies in the $xy$ plane. Also that $\phi$ is the angle which the $xy$ plane is rotate about $\vec{u}$ by.  Therefore by substituting into the matrix linked to at the beginning of this edit, transforming $(x,y,z)=(\cos(\theta),\sin(\theta),0)$ by said matrix will give parametric coordinates for the tilted circle in terms of $\vec{n}$?  EDIT #2: I find this for the rotation matrix from the $xy$ plane to the plane with normal $\vec{n}$, from the method described above.  $$Q=\small{\left(\begin{array}{ccc} {\mathrm{n_2}}^2 - {\mathrm{n_2}}^2\, \sqrt{1 - {\mathrm{n_3}}^2} + \sqrt{1 - {\mathrm{n_3}}^2} &amp; \mathrm{n_1}\, \mathrm{n_2}\, \left(\sqrt{1 - {\mathrm{n_3}}^2} - 1\right) &amp; \mathrm{n_1}\, \mathrm{n_3}\\ \mathrm{n_1}\, \mathrm{n_2}\, \left(\sqrt{1 - {\mathrm{n_3}}^2} - 1\right) &amp; {\mathrm{n_2}}^2\, \sqrt{1 - {\mathrm{n_3}}^2} + {\mathrm{n_3}}^2\, \sqrt{1 - {\mathrm{n_3}}^2} - {\mathrm{n_2}}^2 - {\mathrm{n_3}}^2 + 1 &amp; \mathrm{n_2}\, \mathrm{n_3}\\ - \mathrm{n_1}\, \mathrm{n_3} &amp; - \mathrm{n_2}\, \mathrm{n_3} &amp; \sqrt{1 - {\mathrm{n_3}}^2} \end{array}\right)}$$  This is found from this .  EDIT #3: Using $\vec{n}=(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ I find this parametrically plots
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Typically when drawing a 3D figure on paper (or a black/white board), a mathematician draws the z-axis to point due north on the page, the y-axis to point due east, and the x-axis to point due southwest (as if coming out of the page). And then the 3D objects are drawn on that; e.g., when a sphere is drawn on those axes, it looks head-on like a circle, without distortion, so that its cross-section in the yz-plane is a perfect circle.  In the answer  by @Max to my question , he shows how to create a 3d set of axes that look like that, using a Cabinet projection.  However, when the Cabinet projection is applied to a sphere, it distorts the sphere's shape, as he shows.  Is there a way to create with TikZ a 3D drawing that does draw the axes as I've described, but does not distort solid object such as a sphere?
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I'm having a little trouble understanding how to orient a surface in $\mathbb{R}^3$ For example, how would I orient the ellipsoid given by:  $$x^2+y^2+z^2+xy+xz+yz=\frac12$$   for $(x,y,z) \in \mathbb{R}^3$ ?
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Quite impressed by all of Gleb Alexandrov's scenes with some amazing dust floating around.   (To see what I'm talking about > )  And I started working last night on a scene that is going to need the same sort of atmospheric dust in the air. But when I looked around on how to create dust particles in the air there was nothing there. Not even anything I could adapt from a tutorial. Everything was from 2009 before volumetrics was even introduced.  Someone in an older tutorial did use a particle system and used a scaled down icosphere as a mesh for each particle, but experimenting with that made everything look really tacky.  Is there a way to do dust particles that I'm not thinking of?
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I've been experimenting with node-based procedural materials in Cycles lately, and after spending a few hours working out a brick material that works pretty well along walls, I noticed that it has a few issues. Here's my initial setup:    It's not perfect, since it switches the different orientations in an all-or-nothing way, but for my purposes, it looks good enough.  So I went ahead and used this throughout my test scene. But then I turned the camera around and looked at the other side, and... there's something wrong with the side of the cube facing away on the X axis!    This isn't a huge issue, because I don't plan to render my objects from that angle, but I'm concerned about the way it will affect reflections, light bounces, etc. so I started looking around for other solutions. I happened on this thread at Stack Exchange:   Particularly, Rich Sedman's answer seemed promising (since I'm trying to avoid using UV mapping). Following what he posted, I created this:    This new setup works great for the cube - on every side - but you can see pretty plainly that the cylinder, strangely, has problems. By the way, there is a part of the node setup that I omitted from the screenshot that isn't really relevant here - it checks the Z value of the geometry surface and subs in the default vectors for the texture coordinates if Z is over a certain amount, for sides that face upward.  I've been experimenting with this for a few more hours now, but I'm afraid I've reached my limits in terms of vector math.  Is there anything obvious I'm missing here that might help fix this, and make this setup work for both the cube and cylinder?
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Given x points that must be on the surface of a sphere, how should I place them so they are the maximum distance from each other? I'm attempting to create a script that will place any given number of objects on the surface of a sphere. I think the simplest way to represent the position of each point would be two degrees, the first representing horizontal position and the second the vertical. (0, 0) would the highest point, (90, 90) would the leftmost point, (270, 90) the rightmost point, etc. (x, y) would be like:  Is there a way to get the positions for each point on the sphere given the number of total points? For example, if I had 2 points, the furthest they could be is (0, 0) and (180, 180). 6 would have a point on each of the furthest points (top, bottom, front, back, left and right) 2 points example:  Is this possible? Also is there a better way to do the coordinate system?