Planes in 3D Space

https://alexharri.com/blog/planes

by alexharri • 2 years ago

160 44 2 years ago

alexharri.com

randv• 2 years ago

super cool...thanks

shiandow2• 2 years ago

Shame this doesn't include my favorite description, a 4d vector in projective affine coordinates.

Instead of a normal and point or constant you get (x,y,z,1) . P = 0. The translation between the two is trivial. If you want a plane spanned by 3 points you just can use the generalized cross product to find P.

One advantage is that you can avoid all the special cases with 3 intersecting planes. There exists exactly 1 point that is on all 3 planes, but as this is in projective coordinates it might lie at infinity.

boppo1• 2 years ago

Do you have any good resources for developing an intuition about the projective plane?

alexharri4• 2 years ago

I haven't heard of this before, I'd be interested in learning more about how this is useful. Do you know of a good resource to read more about this?

shiandow2• 2 years ago

I honestly don't know, I encountered the concept during my study, but it took lots of practice to get comfortable with it. Most articles I encounter are too practical to really foster understanding, or too technical to work as an introduction.

You can start with a description of how projective matrices work (and how translation and rotation are related to it). After that, best tips I can give are start with 2D until you can't bear to see another cross product. Then get familiar with Cramer's rule and higher dimensions. You'll need sone fluency in linear algebra.

My first practical use of the concept was to rectify photographs where e.g. a building was not quite upright. That might be a good starting point.

memco• 2 years ago

Are bivectors a 4th way to express planes in 3D space or is that what the 3rd method the article describes is. I didn’t see them mentioned anywhere in the article. I only have a cursory knowledge of their existence and use so not totally sure how they fit, but I feel like they should.

tsuru• 2 years ago

If you use 3D Plane-based Geometric Algebra (https://bivector.net) planes are vector elements and bivectors could be the line intersection of two planes. I'm grossly simplifying here, see link for more info on this interpretation. There are more algebras that have other interpretations of bivectors.

mwexler• 2 years ago

Just to get it out of the way, I assumed just from the title that this was about airplanes in 3d space and was excited to see what I was missing, as I thought they all flew in a 3 dimensional context.

I know I'm not the only one...

ReleaseCandidat2• 2 years ago

There are woodworking planes too, with which the title makes more sense.

makerdiety5• 2 years ago

Learning about planes felt abstract and non-intuitive to me. “Sure, that's a plane equation, but what do I do with it? What does a plane look like?” It took some time for me to build an intuition for how to reason about and work with them.

Imagine how far human society can go if people dumped the desire for intuition that appeals to anthropocentric sensibilities.

But I guess that's forbidden dark arts and wasting time with trivial pursuits and stuff is the culture to adopt while you're in the modern day Rome equivalent.

y = mx + b is enough for me.

harshaxnim• 2 years ago

Anthropomorphic intuition has its advantages - it cuts down the ideas search space with its learnt heuristics. We are intuitive beings, and system 2 thinking is very deliberate and not so natural to us. Secondly, it might often be intractable to process things with sys2 thinking. That said, I don’t think anthropomorphic intuition is necessarily static; I can learn new intuitive skills with enough practice - for example, a person grown in a tribe with very rudimentary speech cannot intuit how to communicate even mildly non-simple ideas, but I can do half a decent job very intuitively.

And finally I think even if you don’t care for intuition, it cares for you. I mean, if you play with y=mx+c long enough, you’ll gain an intuition, intuitively. So, with these devices (like visualisers etc.), we’re essentially trying to gain an intuitive understanding deliberately, which I see nothing wrong with. It’s just meta intuition.

makerdiety5• 2 years ago

A preference for level two is what got us a general theory of relativity and, subsequently, working global positioning satellite technology. Having a cognitive bias toward visual presentations of planes in 3D and higher space leads to techno-scientific stagnation.

ReleaseCandidat2• 2 years ago

There is an error in the line's description, any normal of a line is perpendicular to the line itself (all through a point of the line are contained in a plane perpendicular to the line, with two special ones, the principal normal and the binormal, as with every curve) and does not point "in the direction of the line".

A remark: a plane has two (different) unit length normals, which point in exactly opposite directions (one can be obtained by multiplying the other by -1). This determines the positive and negative half-spaces in which each plane splits the 3 dimensional space - the positive and negative direction, which appears for example in the distance calculation on the site.

gary_0• 2 years ago

I was recently algebra-ing down a complex 3D algorithm to speed it up, and it seemed peculiar how so many things can be boiled down into to a crap-ton of dot products. One of the recurring mathematical themes of the universe, I guess.

SardineArmor• 2 years ago

This is because you weren't algebraing, you were pre-Hilberting :)

Let's forget we're in finite dimension for a minute, in which case it is not trivial to expect a scalar product on our vector space.

So now we have two things, which are separate: a vector space, and a scalar product.

The vector space says: you can sum my elements, multiply them by a scalar, and I'm stable by linear combinations. That's all it says, really.

Everything else, the geometry, is a consequence of the scalar product. It even defines the norm in the first place, thus the topology. For instance, there's no talking about convergence in a vector space if you don't have a norm. You can't even tell if a sequence converges.

You can define a norm without a scalar product, of course. But then you lose certain notions, like orthogonality.

Anyways, in the case of finite dimension vector spaces, there's always a scalar product at hand, and this is where the geometry comes from, really. The algebra portion of it - being a group with a scalar multiplication - is certainly necessary, but not the one to define geometry.

11001100• 2 years ago

there is a great project using multiple planes in 3D space: https://cybernetic.dev/cube

jsenn2• 2 years ago

Another useful representation is as a point in spherical coordinates. The polar+azimuth angles encode the normal, and the radius encodes the distance from the origin.

This is handy because it puts similar planes nearby in space. For example, it allows you to efficiently cluster objects by coplanarity using a spatial index.

zodiac• 2 years ago

Interesting, in this representation a plane is represented by the point on it closest to the origin, right?

jsenn2• 2 years ago

Yes, that's a good way of thinking about it.

phkahler• 2 years ago

Fun fact. Take 4 points representing the vertices of a tetrahedron. Put them in a 4x4 matrix, using 1 as the 4th element then invert the matrix. The 4 row vectors of the result will be the coefficients for the plane equations of the faces of the tetrahedron.

They'll be scaled such that the point not on a plane will evaluate to 1 when plugged into the equation. You can see this easily because multiplying the planes matrix by the points matrix is just plugging each of the 4 points into each of the 4 plane equations, and you get the identity matrix by definition.

thechao• 2 years ago

They'll be scaled such that the point not on a plane will evaluate to 1 when plugged into the equation.

Evaluating a plane equation with a point will give a signed distance from the point to the plane. However, unless the equation is normalized, the distance could be scaled in unusual ways that are less than useful. I think a better way to say this is that a point who's shortest straight line distance to the line is 1 will evaluate to 1, after taking the absolute value.

The LRBni ISA had a whole set of instructions designed to take advantage of signed distance fields defined in both plane equation & barycentric forms. (That's what the bit mask for the lanes was for, in part.)

a1369209993• 2 years ago

However, unless the equation is normalized, the distance could be scaled in unusual ways that are less than useful.
> []the[] point not on a plane will evaluate to 1

They mean the vertex (of the tetrahedron). The signed distance function is scaled such that the (single remaining) vertex that isn't at distance 0 (by construction) will instead be at distance 1. So the distance is always scaled in a unique well-defined way (assuming not all four points are coplanar). Whether that way of scaling is useful depends on your use case, of course.

greenbit• 2 years ago

"A plane in 3D space can be thought of as a flat surface that stretches infinitely far, splitting space into two halves."

Fun fact - if you have the coefficients of the equation ax+by+cz+d=0 that represents your plane, you can plug any point (x,y,z) into the expression ax+by+cz+d, and the result will be positive in one half-space, and negative in the other. I think if you divide the values by sqrt(a^2+b^2+c^2), you end up with the distance from the point to the plane. Easy enough to see which half-space is which by plugging in the origin. I.e., if 'd' is positive, the origin is in the positive half.

BenoitP• 2 years ago

What a wonderfully crafted piece. But a part of me can't refrain from saying it would have looked way more elegant and succinct in 3D Projective Geometric Algebra.

Most of the last sections (all intersections) feel like corner cases, when in PGA they are one and the same.

ngruhn• 2 years ago

I keep hearing this and I'm very interested. But most recommended resources I've seen so far are more targeted at Physicists. Any good text book for CS people?

Edit: nevermind, read in other comments that https://bivector.net/ has a ton of resources.

sebastos• 2 years ago

For those interested, this appears to be a really high quality library that provides a 3D PGA C++ API:

https://github.com/jeremyong/klein

I've always wanted to find an excuse to rebuild some projects at work around this.

at_compile_time• 2 years ago

Yup. Geometric algebra has one equation for each of: projection, rejection, join (e.g., two points into a line), and meet (e.g., two planes interacting at a line). The algebraic expression is the same regardless of the elements you're working with or the space you're working in.

You get transformations too, as easy as M=b/a, where M can be applied to any element in the algebra by taking the square root and applying double-sided multiplication such that b = √M a ~√M, where tilde represents the reverse. These transformations are isomorphic to complex numbers, quaternions, and hypercomplex numbers, and understanding them makes other explanations of these concepts feel inadequate and woefully un-geometric.

Add in logarithms and the exponential map for these transformations and we can perform linear interpolation between states and parametrize transformations.

I'm just a motivated amateur and I can do all of these things. The vector algebra I learned in engineering is useful, and it's often all I need for simple 3-dimensional problems, but it's just shy of something far more powerful and far more general.

dargscisyhp• 2 years ago

Great article, would love to know what you're using to create these diagrams!

sleepingreset• 2 years ago

same!

alexharri4• 2 years ago

They're created with ThreeJS and react-three-fiber. My website is open source, you can find all of the 3D scenes here: https://github.com/alexharri/website/tree/master/src/threejs...

pixelpoet• 2 years ago

Nice work, one thing (which you'll see everywhere once you've fixed it yourself) though: in TeX you want to escape functions like cos, i.e. use "\cos" not "cos", which will get rendered as a product of 3 variables c, o and s.

danielvaughn• 2 years ago

Those graphics are beautiful.

absurdcornbread• 2 years ago

Where can I get a job developing software using these kinds of techniques? I had to perform a ray tracing calculation one time (before I knew what ray tracing was and was essentially deriving the same things the author is talking about) and absolutely loved working with 3D geometry and implementing it into software.

What kinds of industries should I look for to essentially work with programming math problems like this?

keithalewis• 2 years ago

https://en.m.wikipedia.org/wiki/Hermann_Grassmann would like to have a word with you. https://en.m.wikipedia.org/wiki/Giuseppe_Peano was so impressed he wrote a book to explain it. It is much easier to just read and try to understand what smarter people than us spent their precious time on earth figuring out. Newton was on to something about standing on shoulders of giants.An option available to anyone.

Planes in 3D Space

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