Hi, I’m a bit confused by the drawing in “Multiple Planes”.
From the rest it seems that the planes intersect at the origin. So the 3 lines here should intersect at the origin here, right?
So something like this:
There is no relation to origin, think of the lines as planes 2D not just lines, and of course the planes will interesect and each individual space created is a hash region.
Thanks @gent.spah . I understand that they are 2D planes and that they intersect. However, the planes are at the end of the plane defining vector, right? As you see in this picture from “Hash tables” lab
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As all the random plane vectors start at the origin, so all the planes will intersect at the origin, making the drawing I posted before confusing.
If you wanted to have these little regions in the middle, wouldn’t we have 1/ put the plane at the arrowhead of the plane vector and 2/subtract the plane vector before doing the dot product?
Im not understanding your point, if all planes start at origin they will not intersect anywhere else so there will not be enclosed regions separate.
They define separate regions, but there is no “enclosed region” correct. This is why the drawing in the lecture is confusing compared to the calculations with the dot product and the random plane normal vectors.
The drawing should be more something like this in my opinion:
Hi Christian,
it was long ago actually I did the NLP specialization, but it doesnt look right for all planes to intersect at the same place, one reason i can think of is the distribution of hash regions which in your figure is uniform (or linear if you wish to call it) and the hash function is very naive. As far as i remember the hash buckets are more complex than just dividing regions and grouping one after the other in sequence. Also yes the there is no requirement for the region to be enclosed.
I have the same question as you. One of the ways to define a plane is using a [point-normal vector form] (Plane (geometry) - Wikipedia) and there was no point defined here.
To be concrete - in 2D if the normal vector is (1, 1) that defines the plane (line as it’s 2D) x+y=0 but that’s the plane(line) passing through the point (0, 0). If you take the normal vector (1, 1) and point (1, 0), the plane would be x+y=1. It’s still parallel to the earlier plane but they are not the same planes. There are infinity many planes with a given normal vector (all parallel to each other) but you can get a specific one by also giving a point it passes through. This also extends to higher dimensions - 3D examples for the same on khan academy.
Another way we can confirm this is knowing that a 2D line is defined by ax+by+c=0 so we need 3 values i.e. a, b,c to get a unique line, and in the course, we define just 2 numbers for a plane in 2D (line).
Thank you for raising this, Christian!