How can I roll the plane with shader or mesh warp on the unit?

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plane geometry: how to simplify this dotted expression.

I feel like I have a good understanding of the vector equation of a plane, however I am stumbling over manipulating the following equation at: Publishing the equation I need help to understand how the author simplified the equation of the plane to the solution t, specifically how to move the operation of points from one side to another or how to factor t to the other side. Thank you.

$ Ray P = Po + tv $

$ Plane Eq = P cdot N + d = 0 $

Substituting Flat Equalizer with P-Ray:

$$ PlaneEq = (Po + tv) N + d = 0 $$
This is where I lose the author:
$$ t = – (Po cdot N + d) / V point N $$
How do I step from PlaneEq to t, step by step?
In particular, how do I move the point expression inside the equation?

dnd 5e: What happens when the Ensnaring Strike target enters the ethereal plane?

Ensnaring Strike states the following (abbreviated for brevity):

The next time you hit a creature with a weapon attack before this spell ends, a writhing mass of thorny vines will appear at the point of impact, and the target must either succeed on a Force save cast or be restrained by magic vines until the spell ends. If the target succeeds in salvation, the vines wilt.

Tangle Strike has a Focus duration of up to 1 minute.

What happens if the Ensnaring Strike target enters the etheric plane? They could accomplish this through methods like the Etherealness spell.


  • Do the spell's effects remain while on the ethereal plane?
  • Can they traverse the vines that support them while on the etheric plane?
  • Are the spell's effects restored when they return from the ethereal plane?

matlab: get an adaptive solution from the neuro-diffuse inference system (ANFIS) for the inverse kinematics of a 3R plane manipulator

I made that part of ANFIS using direct kinematics, but now how can I validate it on a range of x and y components using IK? as x = 0: 0.1: 2; ey = 8: 0.1: 10;
This is the link that helped me, but here it is done in 2 RR glider but I want 3 RRR.

clear all
link1 = 10;
link2 = 7;
link3 = 5;
theta1 = 0:0.1:pi/2;
theta2 = 0:0.1:pi/2;
theta3 = 0:0.1:pi/2;

(theta1,theta2,theta3) = meshgrid(theta1,theta2,theta3); 

X = link1 * cos(theta1) + link2 * cos(theta1 + theta2) + link3 * cos(theta1 + theta2 + theta3);
Y = link1 * sin(theta1) + link2 * sin(theta1 + theta2) + link3 * cos(theta1 + theta2 + theta3); 

data1 = (X(:) Y(:) theta1(:));
data2 = (X(:) Y(:) theta2(:));
data3 = (X(:) Y(:) theta3(:));
axis equal;
title('X-Y coordinates generated for all theta1 and theta2 combinations using forward kinematics formula','fontsize',10)
opt = anfisOptions;
opt.InitialFIS = 7;
opt.EpochNumber = 15;
opt.DisplayANFISInformation = 0;
opt.DisplayErrorValues = 0;
opt.DisplayStepSize = 0;
opt.DisplayFinalResults = 0;
disp('--> Training first ANFIS network.')
anfis1 = anfis(data1,opt);
disp('--> Training second ANFIS network.')
opt.InitialFIS = 7;
anfis2 = anfis(data2,opt);
disp('--> Training third ANFIS network.')
opt.InitialFIS = 7;
anfis3 = anfis(data3,opt);```

air travel: refund of a plane ticket, purchased through a third party

I am planning to fly from Dubai in November. I still haven't bought a ticket. I received a $ 100 gift voucher from a friend to purchase the ticket from a specific external agency / aggregator, which I would like to use soon.

Therefore, I am choosing an airline that allows me to refund this ticket directly through them, even if the ticket itself was purchased from an outside agent.

plane geometry: can it be improved for another way to test Kepler's second law?

  1. Is it possible to find linear equations for all $ n $ Lines extended from the focus of the ellipse to the perimeter of the ellipse in Kepler's second law? If so, how? (Kepler has stated that when a planet orbits its sun, it sweeps out equal areas at each given time interval. $ n $ is the time interval here; for example, $ n $ days, $ n $ months etc.) (suppose we have $ 12 $ lines here to $ 12 $ months)
  2. Can Kepler's second law be generalized to all ellipses (horizontal or vertical) with any given semi-major and semi-minor axis or is it only applicable to the orbits of the planets?

I tried to get a parametric answer to the first question by letting one of the lines cross the ellipse focus (point A): $ A = ( sqrt {a ^ 2-b ^ 2}, 0) $ and a point on the perimeter of the ellipse (point M): $ M = (m, frac {b} {a} sqrt {a ^ 2-m ^ 2}) $. my goal was to calculate $ m $. so I found the area between $ m $ Y $ to $ (semi-major axis) plus a triangular area with the base of: $ m- sqrt {a ^ 2-b ^ 2} $ and height of: $ frac {b} {a} sqrt {a ^ 2-m ^ 2} $ with the help of comprehensive; so I equated the result to $ frac { pi {ab}} {12} $. since i had $ arcsin ( frac {m} {a}) $ in the result, I substituted $ m $ with $ a sin theta $ and the end result was $ frac { pi} {3} = theta + cos theta sqrt {1 – ( frac {b} {a}) ^ 2} $. so he $ theta $ It was $ frac { pi} {3} – sqrt {1 – ( frac {b} {a}) ^ 2} < theta < frac { pi} {3} + sqrt {1 – ( frac {b} {a}) ^ 2} $ . now remember that $ m = a sin theta $. so, we have $ a sin ( frac { pi} {3} – sqrt {1 – ( frac {b} {a}) ^ 2}) <m <a sin ( frac { pi} {3} – sqrt {1 – ( frac {b} {a}) ^ 2}) $.

Is this a valuable form?

co.combinatorics – Refined Inverse Plane Partition Generation Function

I have a simple question about the generation function for inverse plane partitions:

$$ sum _ { pi in RPP ( lambda)} z ^ {| pi |} = prod_ {s in lambda} frac {1} {1-z ^ {h _ { lambda} (s)}} $$

There is a natural refinement of the right side:

prod_ {s in lambda} frac {1} {1-t z_1 ^ {a _ { lambda} (s)} z_2 ^ {l _ { lambda} (s)}}

Or maybe just with $ t = z_1, z_2 $.

I suspect that there should be a right side equivalent to this identity, that is, to count some "refined weight" of the inverse plane partition. Maybe along diagonals? I was wondering if there is any known generating function. I think there is a natural geometric interpretation as a Poincar polynomial in $ z_2 $.

If it helps, the right side is something like $ c _ lambda (q, t) $ of Macdonald's polynomial theory.

Thank you

air travel – Where is the safest area / seat to sit on a plane as a passenger to avoid the coronavirus?

Seen this in some places where they claim that a window seat is & # 39; safer & # 39; since it has fewer neighbors, etc.

But wondering given the air movement, the fact that airlines balance passengers on the ship, business versus economy, etc., if this could affect it or not.

air travel: what is the safest seat to sit on a plane to avoid the coronavirus?

Seen this in some places where they claim that a window seat is & # 39; safer & # 39; since it has fewer neighbors, etc.

But wondering given the air movement, the fact that airlines balance passengers on the ship, business versus economy, etc., if this could affect it or not.

3d – How to calculate the vertices of the flat corners from the point of origin of the plane and the normal plane?

Here is one way I can think about doing it.

Find a point on the plane a apart from the origin, o. The vector a - o It lies on the plane, so it is guaranteed perpendicular to the normal plane. Normalize the vector a - o and call it x.

Take the cross product of x and the normal plane and they call it y. y it is guaranteed that it is perpendicular to both x and the normal plane.

The four points of a square are now o + x, o + y, o - x, o - y.