You can make an oblique cone that is circular with respect to its axis. This would end up being an oblique elliptical cone (but still an oblique cone) since its base would not be circular.
You are describing an oblique circular cone (also still an oblique cone), since it has a circular base. This would necessitate an elliptical cross section perpendicular to its axis.
I think we are getting there, but I would venture to suggest that you have gotten the definitions reversed, and that what I described in the OP is in fact an elliptical oblique cone, or we could even dispense with the "oblique " and simply call it an "elliptical "cone, since any section of the cone perpendicular to the axis will yield an ellipse, whereas any section of a right cone perpendicular to the axis will yield a circular section, whether it is oblique or not.
Problem solved, I thank you all , I love this forum.
On a lathe, the item will rotate about the axis. How are you making an elliptical cross-section perpendicular to the axis? I would expect you would always get a cross-section that is circular perpendicular to the axis. ETA: if you then cut the base perpendicular to the axis, you get a right circular cone. If you cut it at an angle, you can call it an oblique elliptical cone.
I don’t believe this is true. Ignoring the base, the same cone can be described as elliptical or circular. A right circular cone can be called an oblique elliptical cone. For any elliptical cone, you can find an angle to cut it so that the cross-section is circular, so any cone can be called a (possibly oblique) circular cone.
Just to be clear, looking at this picture engineer_comp_geek linked to, the dashed line is not the axis. The axis would be a line connecting the vertex and the center point of the base.
See, I’d say it was a “cone with one side straight up and down, but not flat.” Not only because I don’t think a lot of people know the word apex, but also because I don’t know too many people who conceptualize cones as starting from the apex. The start from the circle/ellipse, and meet at a point.
Whether it is circular or elliptical depends on the shape of the base, not the shape of a slice perpendicular across its axis. Since your cone has a circular base, it’s a circular cone.
You could call it a cone, you could call it a circular cone (though some might assume it’s a right cone if you called it that without specifying), you could call it an oblique cone, or you could call it an oblique circular cone.
Reminds me of the old “you can call me Ray” comedy routine from the 70s…
Who said anything about whether it “starts” vs. “ends” at the apex? But, fair enough on the first point; replace the word “apex” with “tip” or “point” or such things as you like.
Okay, here’s the picture I’m getting from this description, and why I think at least some other people are getting confused.
From your description, if your original (right circular) cone is a solid block of rigid kryptonite, and you “tilt” it as you describe, it is still a right circular cone! You can move (translate) or tilt (rotate) it any why you want, and it is still the same shape, and that is still a right circular cone.
Now, if you slice it as you describe, you are just passing a plane through the cone, parallel to the plane of the original base, but not parallel to the base of the cone in its new position. So yes, this will create a new “base” (let’s call it a pseudo-base) perpendicular to whatever you just said it’s perpendicular to. But that is by no means a true base of anything. It’s just a plane passed through the (now tilted) cone, creating an elliptical section. You might be able to set the cone on this face and not have it tip over. So the new “shape” you have is nothing more than the original right circular cone you originally had, tilted, with a piece of it sliced away. I doubt there’s a more specific name for it than that.
NOW: As to what almost everyone else seems to be thinking (maybe): Picture a deck of playing cards. Skew it so that each card sits just a little to the left of the card below it. I think in analytic geometry this is called “shear” or maybe “skew”. Here’s the best picture I can quickly find. Now do this process with your cone. This creates the “oblique cone” that most everyone else seems to be thinking of, and I think you could make one line of the surface of the cone be perpendicular to the base. This base would still be the same circular base as the original. But it sounds like this isn’t what you are describing.
NOW: Upon re-reading your OP, it becomes less clear which of these figures you are trying to describe. Your OP seems to be describing the sheared cone that I just described. But the post that I quoted above (post #9 I think) is clearly just the tilted sliced-off block of kryptonite. These are two distinctly different shapes. Now I see why we are all confuzled.
