How to calculate the volume traced by a circle of radius r that rotates x rad while simultaneously translating a distance d?
Pictured are five snapshots of a rotation by pi rad along a distance d, with the path of a single boundary point marked by a dot.
Here's a relevant thread about a 2D equivalent: https://warosu.org/sci/thread/S7337389#p7338480
Each line segment can be thought of as half a circle in side profile.
>>7929567
>Pictured are five snapshots of a rotation by pi rad along a distance d, with the path of a single boundary point marked by a dot.
Mistake: it's pictured here.
Let v=v(t) be the velocity vector and x=x(t) be the unit normal vector from the plane of the circle with radius r.
If v and x have an angle O between them the "circle" (I think you mean disk) will drag out a volume at a rate V'= ||v||*πr*|cos(O)|=π|x•v|
Where • is the dot-product.
Thus V=π*int[ |x(t)•v(t)| ,t,0,T]
If you are restricting to movement along one direction, and rotation about a diameter, then my formula reduces from the dot-product to ordinary multiplication.
>>7929793
Whoops,I lost an "r" along the way. Just multiply the formula by r and it will be correct.
>>7929793
What should I put in? I don't know how to apply this.
>>7929831
Do you know calculus?
>>7929840
Only the basics of differentiation and integration.
>>7929849
Okay then forget what I said about dot products.
x and v are functions of time, but it seems like you only need something rotating with constant angular velocity and moving with constant speed. Specifically with v(t)=d/T and x(t)=cos(π/T*t)
So my formula reduces to
V=πr*int(|cos(π/T*t)|*d/T,t,0,T)
=2πrd
>>7929883
>V=2πrd
With this, how to define the revolution? What if d=0 and the disk just spins pi rad, tracing a sphere?
>>7929567
Parametrize edges and make a function for the least and greatest y values desu. Then find single area
