Hi there

I've posted this in the calculus section as I think that's where most of the work to solve this is gonna be!

Basically, my problem is as follows. I have a path along which a cross sectional area is swept along, hence giving me a volume. I need some help in establishing the limits of this volume, so I can check whether a point lies within them. the path is defined by beziers cubic interpolation between about 10 points. once i have the volume definition, i think i can using triple integrals to check if the point is in the volume, which will work, however i need to be able to define the volume limits before i can do this.

the link shows a screenshot of what i'm doing - i want to find if the centres of the little cubes lie within the blue volume.

http://www.facebook.com/album.php?ai...f5&id=61408485

i was thinking, take given point x, differentiate the equation of the blue bit (that's just an example that used the bezier interpolation, i'm just gonna stick to y=x^2 for now :P), so find tangent, then can find normal to that, which will give me the plane that the cross-section will lie on. knowing, for that value x, what area y and z may be in, then use a double integral (using limits of the cross-section, i'm using z^2+y^2=1, arbitrary circle), see if the point is inside.

doing that, i'm only taking into account the line path varying in x and y, with constant z, but that's suitable for this task. however, i realised that even when the normal to the tangent is found, that only gives me the line on which the cross-section lies, and all 3 co-ordinates change still across this area

as for after the volume is defined, the point checking part might go like this ...

S = integration sign
(x1)S(x2) = integrations with limits
>/ = greater than or equal to
V = volume
d= differential

SSSxzdV where V is bound by x,y,z

For example let x = 0, y = 0, y = 6, z = x^2. z = 4. for
x >/ 0, y >/ 0, z >/ 0

Change the limits
z = x^2....4. Therefore x = 0...2, y = 0...6

therefore SSSx.z.dV = (0)S(2).(0)S(6).(x^2)S(4) x.z.dz.dy.dx

Integrate with respect to dz
= (0)S(2).(0)S(6).[(x.z/2)](x^2)...(4).dy.dx.
Integrate with respect to dy
=(1/2).{(0)S(2).(0)S(6).[(16x -x^5).y](0)...(6). dx
Integrate with respect to dx
= 3[8x^2 - (x^6/6)](0)...(2)
= 3.2^5 (1 - (1/3))
=2^6
= 64

do people have any suggestions on both parts? (the volume definition, and the co-ordinate check)

I'm really struggling with this, and it's beginning to drive me nuts!

Thanks,
Emily