What I am trying to represent is a 3D solid which is similar to a cone, but subtly different.
If you imagine a cone as being made up of an infinite number of infinitely thin circles of decreasing radius, , with the **center **of each circle directly above the adjacent underlying circle, the figure which I have tried to draw above is what I can only call an “offset cone” ie a cone in which the circles have been pushed to one side so that there is one point on the circumference of each circle which is directly above the adjacent circle, ie there is a perpendicular which can be drawn at one point (and one point only) along the height of the " offset cone" .
This is not an imaginary construct btw. This “offset cone” is what is produced when a tapered workpiece is turned on a lathe with the tailstock center offset from the headstock center. Many engineering workers labor under the misapprehension that this is one way to turn a true cone, but any graduate engineeer will realize otherwise. The only way to turn a true cone on a lathe is with the headstock and tailstock centers in perfect alignment relative to the lathe bed, and with a compound top slide set at whatever angle you want the cone to be (or if you want to nitpick , to half the angle you want the cone to be).
But I digress.
What I really want to know is the correct mathematical/geometrical name for the solid which I have described.
There*** has ***to be a better name than "offset cone " …
Are you talking about an oblique (as opposed to right) circular cone, in which one of the edges is perpendicular to the base instead of the axis being perpendicular to the base?
It most likely*** is ***an “oblique cone”, but just to be certain (and it is not clear from the Wiki article) is it the case that the base of an oblique cone is circular, and not elliptical?
What I mean is, the construct I have described is most definitely not a normal right circular cone tilted over and sliced through so that one side is perpendicluar to the base.
This would result in the base being elliptical.
What I am describing is a “cone” in which the base is circular, just like a right cone, but “offset”.
So just to reiterate, and to reassure me, is this in fact an oblique cone ?
If the base is circular, it’s technically an “oblique circular cone”. If it had an oval base it would just be an “oblique cone”.
ETA: Just for clarification, oblique cones can be circular or not. An oblique circular cone is just a type of oblique cone. So calling it an oblique cone is correct, while calling it an oblique circular cone is just a bit more descriptive and detailed.
If you take a standard right cone, tilt it over so that one side becomes perpendicular to the base, then slice it through so that it sits on the base with one side perpendicular to the base, does it then become an “oblique cone”?
It’s times like this you really miss not having the [IMG] code …
It’s a particular kind of oblique circular cone (not all oblique circular cones are of the form the OP gives), but nevermind that. It’s a circular cone whose directrix is perpendicular to one generatrix, but nevermind that either.
The easiest way to get anyone to understand what you’re talking about is just to give a description like you did in the OP. Alternatively, you can always say “The convex hull of a circle and a point directly above one on the circle” or such things.
Basically, it’s like a cone only the cross-section perpendicular to the central axis would be an ellipse. (I.e. if the axis, seen by side view, went through the center viewed from all side views like a regular, circular cone).
Cut at the proper angle, the cross-section would be a circle.
On a regular cone, the pependicular-to-central-axis cross section is a circle not an ellipse. (Radial symmettry)
If I am understanding you correctly, as soon as you "tilt"it, it becomes an oblique cone instead of a right cone. It doesn’t matter if it’s tilted so that the side is perpendicular to the base or not. It can be just little off center, or it can be way big crazy off center. Anything that isn’t centered is an oblique cone.
What I now know is that what I have described is*** not ***in fact an oblique cone.
I don’t know what it is called, and maybe there is in fact no name for it.
One thing for sure, the Ancient Greeks didn’t have lathes which had the facility to offset the tailstock center, so there is no reason why they would ever have encountered this construct, so maybe that is why it doesn’t have a name.
What (I understand that) you described is an oblique cone.
Also, I don’t understand your comment about the lathe. Doesn’t the piece of wood revolve about an axis? If you’re just referring to the base not being parallel to the cone axis, what you’d get isn’t what you described above. What you described above would have an elliptical cross section perpendicular to the axis. That would be hard to turn on a lathe unless you had something that moved in-and-out with the lathe turning (do they even make something like that?).
My understanding of what you’ve described is that it is very much an oblique cone. If you want to get very specific about it, it’s an oblique circular cone whose apex happens to be directly above the rim of its base (to borrow from Indistinguishable’s post). There isn’t a specific name to distinguish it from any other type of oblique cone, but it’s an oblique cone.
I followed you up until the “slice it through” part. Do you mean that you’re slicing off the (tilted) base to create a new base, perpendicular to the vertical side of the cone?
If so, then you are creating an oblique cone, but it isn’t a circular one. The new base will be an ellipse.
EDIT: But if I follow the OP, the figure you described IS an oblique circular cone with one vertical side.
I think what you guys are missing is that there is a substantive, qualitative difference between the construct which I am describing (and which, incidentally, can be turned very easily on a lathe) and a normal cone.
Zenbeam spotted it when he said “What you described above would have an elliptical cross section perpendicular to the axis”.
That is precisely the point.
The “cone” I am describing does have a elliptical cross section perpendicular to the axis, but as I understand it so far, that is not the case in a “oblique cone”, which will always have a circular cross section perpendicular to the axis.
So I ask again, is it an oblique cone or is it not ?
