Is there an infinite range of perfect circles.

[Jehanne]

And we similarly define a sphere in three dimensions the same way. The same definition even works in any metric space, although there are some counter-intuitive situations in that generality.

There is a another curved shape that follows as a set of all points that are equidistant from a given point. Unfortunately I can't remember what the geometry was called and can't find it anywhere now. I'll keep looking though.

It's hard to take what you are posting seriously.  From the perspective of modern mathematicians, this topic is very well understood:

Wikipedia -- Curvature

The fact that you are searching on this topic strains credulity.

[GrandizerII]

And we similarly define a sphere in three dimensions the same way. The same definition even works in any metric space, although there are some counter-intuitive situations in that generality.

There is a another curved shape that follows as a set of all points that are equidistant from a given point. Unfortunately I can't remember what the geometry was called and can't find it anywhere now. I'll keep looking though.

Do you mean another shape in 2d realm besides the circle? I fail to see what that could be.

[polymath257]

There is a another curved shape that follows as a set of all points that are equidistant from a given point. Unfortunately I can't remember what the geometry was called and can't find it anywhere now. I'll keep looking though.

Do you mean another shape in 2d realm besides the circle? I fail to see what that could be.

He may be confused about equi-diameter figures. These do not have to be circles.

[Fireball]

Eh, he strikes me as uneducated in geometry and algebra. Maybe he's channeling parabolas, and left out the directrix?

[highdimensionman]

Do you mean another shape in 2d realm besides the circle? I fail to see what that could be.

He may be confused about equi-diameter figures. These do not have to be circles.

Sort of It's a video that I saw on youtube math video about a month and a half ago can't find it now that furthered my thinking, there was something odd about the centre measurement and the shape had 3 curved corners and 3 curved sides i think and it was closeish to circular. 
It's a shame I can't find the video now.

[highdimensionman]

I was thinking about other shapes and solids of constant width but the video I had watched had confused me a little.

[Thumpalumpacus]

He may be confused about equi-diameter figures. These do not have to be circles.

Sort of It's a video that I saw on youtube math video about a month and a half ago can't find it now that furthered my thinking, there was something odd about the centre measurement and the shape had 3 curved corners and 3 curved sides i think and it was closeish to circular. 
It's a shame I can't find the video now.

"Corners" implies a different length, because corners are declivities in a given space. If you've got corners, your radii aren't equal all the way 'round, which is the definition of a circle.

[polymath257]

Sort of It's a video that I saw on youtube math video about a month and a half ago can't find it now that furthered my thinking, there was something odd about the centre measurement and the shape had 3 curved corners and 3 curved sides i think and it was closeish to circular. 
It's a shame I can't find the video now.

"Corners" implies a different length, because corners are declivities in a given space. If you've got corners, your radii aren't equal all the way 'round, which is the definition of a circle.

Take a standard equilateral triangle. From each vertex, draw a circle with radius to the other vertices. You will obtain a 'fat' triangle with circular sides.

That figure is an equidiameter figure: it is the same width in every direction. But it is not a circle. It has 'corners'.

This can be generalized for any regular polygon with an odd number of sides.

[UniversesBoss]

Yes and they are getting tighter.