Just joined, hobby level

[black5f]

Hi

A bowl. Ok, whats your maths like? Sorry as this might seem a little blunt! But, it's the way to do it. I did a quick picture like Jonathan to explain the problem with set steps in z:



Suffice to say, from above they won't be evenly spaced circles either. You need to use vector geometry. A bowl is quite a simple surface, but this applies to all surfaces (ignoring the tool hitting other things!). You know the point on the surface you need to goto, x,y,z. You need to calculate the tangent to this surface, in 2D its a line, in 3D its a planar surface. Differentiate the equation (easier than it sounds) of the surface to get the equation of the tangent, because it's a bowl there are simpler equations that give the tangent at a point, again, a line in 2D, a plane in 3D. You need to calculate the normal vector to the tangent at the point the tangent and surface intersect, ie x,y,z. This is a line perpendicular to the tangent, normally, you would normalise it, ie, make the length = 1. To define it, in plain terms, you need to goto the point you want to cut to, them move away in the direction of the normal. The normal should point to the outside! This is represented by an equation. The actual tool offset is along this vector for a distance of the raduis of the tool (this is the centre of the tool, other trig will give the tool tip). You should end up with a vector equation, (x",y",z") = (x,y,z) + a(x',y',y') where, the modulus (length (pythagoras)) of (ax',by',cy') (multipied out now) is the tool radius and (x',y',y') is normal the vector from (x,y,z). 2D is a little simpler but you still need to work out the tangent to the semi circle at the point you want to cut.
Yes I know how to do this, and thats sad!
The good news is that it's a bowl, so the normal vector is actually the line from the point you want to cut to the bowl 3D centre. So....
If the centre of the bowl is (i,j,k), then the normal vector is always (x,y,z) - (i,j,k), lets call that (q,r,p) = (x-i,y-j,z-k). normalise .. t=sqrt(q2+r2+p2) so
normal vector is now (q,r,p) = (q/t,r/t,p/t). This is the normal from (x,y,z) though! Say the raduis of the tool is 3mm. If I want the tool edge to be at (x,y,z), the tool centre (of radius) must be at (x",y",z") = (x,y,z)+ 3(q,r,p). ie ((x+3q),(y ... etc This assumes the vector points inwards!
So for a hemispherical bowl only ... work out the series of points that form each circle at even z depths (spreadsheet will do this), probaly easier to work out the radius of the circle first. Know where your bowl is in 3D space! Use the vector equation to calculate the tool position (add it to the point on teh circle, if programming the tip just add (0,0,-3) to the answer.
Generally speaking you will get a good fininsh if z steps (and x,y moves on the circle are about 10% of tool diameter). Not trying to be clever or show off or anything, this is the way to do it, 25 years ago I wrote some surface modelling software for making lasts (shoemaking), csplines, bicubic patches etc etc.
Tom