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How does a plane calculate its position if it has three unknowns?

I have been working on a fairly simple 3d engine in javascript (my own, no libraries) and I’m wondering about how to calculate a plane’s 3D location based on the 3 unknowns (p1, p2, p3) and the planar equation ax+by+cz=d.
For the purposes of this question we only need to consider the xy plane; all of the following will be done with a single plane equation of the form Ax+By+Cz=D.
Assuming all vectors are normalized to unit length, what does a plane need to know in order to calculate its position?
For instance, how would the following equation be calculated?
x3 = mx1 + nx2 # mn3

A:

If you are calculating for a simplex, it is definitely best to use the Z-axis. You don’t need to calculate using the planar equation.
Now, if you are calculating in 3D space, you need to know the location of your 3D origin, so you need to define which axis the plane is going to be normal to. Then you can find all points relative to the origin.
If the plane is defined to be normal to the XZ plane, then the equation will look like:
x3 = mx1 + nx2 + mz

Where m and n are vectors that are parallel and perpendicular to your Y-axis. So m = [0, -1, 0] and n = [0, 0, 1]. Note that these vectors must be normalized to unit length.
Also, if you are normalizing a vector like this, then you can get rid of the constant multiple (that happens when you divide all values by the length of the vector). So instead of this:
v = [x, y, z]
v/length(v) = [x/length(v), y/length(v), z/length(v)]

You can get rid of the first division (slightly more confusing, but less prone to errors), and just leave it like this:
v = [x, y, z]
v/length(v) = [x, y, z]

Note that this is

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College of Natural Resources and Life SciencesSuggestions for a better title

I submitted the following suggestion to Discover Magazine in 2014 (in response to the question ‘If you could change the world, what would your legacy be?’):

‘It’s time for science to move beyond the fixed, hard-wired mind-body divide.’

Today I learned that the title ‘Knowledge is power’ for the new book, ‘The Plagiarism Epidemic’, was proposed by Willimon. I am pretty sure this is how the book was pitched to Discover Magazine, so I am adding a suggestion for another title as well:

‘Your life depends on data. Act.’

If you want to read the book, you can find it here:

(Note that it is not an easy read, but it is well worth reading if you think about what the book is saying).
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Animal Man #61: Spinning out of Last Son’s last issue, Animal Man is once again in the same universe as his original series, at the same time. I kind of hope it’s something like that, because it would be strange if the DC/Vertigo line had been folded into his universe but not his history. Plus, on the subject of the ‘V-Men’ — and I think I’m going to be really critical of this book when we get here — they tried to kill another time travelling villain in the last issue. Instead of just running away with the body, Batman and Robin find a nurse trying to revive the dying (because, you know, no one would have volunteered to save a villain) and this nurse happens to be the daughter of this villain. This villain is also the villain we first saw in World’s Finest #180 (this was before Wally became the Flash), and they knew who they were, and they told their daughter who they were, and she laughed at them. And then they came back in the future after the daughter’s murder and found the nurse’s body with a «WTF?» on her forehead, implying that the villain hadn’t actually killed her. And he’s got another son who’s using all his powers to destroy the city, so he’s still a threat. Speaking of villains, Animal Man is having an absolute bloody battle with a supervillain here. They make some connection in the past, but Animal Man hasn’t been trained or given the tools to defeat him and he’s mostly just hitting the villain over the head and running away. And then Animal Man’s foe starts talking — I think it’s his ‘friend’ — and it’s not a long time before we get to the part where he reveals that he’s not the only one, and Animal Man is left without a chance to ever defeat the ‘V-Men’ because he now knows how dangerous they are to everyone. And hey, here’s the trade in which they give us a new Animal Man:Cem Cesur

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