Q: A set of numbers placed at fixed distances?

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Is that the 'scale'?

It could be a ruler. Just a number line.

an ellipse.

The set of numbers which 3 does not belong is the set of even numbers.

the set of points equidistant from a fixed point

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Is that the 'scale'?

It could be a ruler. Just a number line.

A number line or, in 2 or more dimensions a grid.

In the branch of algebra called group theory, they are called equivalent classes or residual classes. They are generated with the fixed distance as the basis of the modulo relationship.

an ellipse.

Ocular scale

The set of all points in a plane for which the sum of the distances to two fixed points equals a certain constant. - APEX

Sounds like an elliptical to me.

No. Triangle numbers are a fixed set of numbers that can't be altered to make one certain number turn into a triangle number.

You're trying to describe an "ellipse".

It is a set of metal wrights calibrated to measure fixed numbers of units of mass. They will be used to weight things in a mass balance.

The foci (plural of focus, pronounced foh-sigh) are the two points that define a hyperbola: the figure is defined as the set of all points that is a fixed difference of distances from the two points, or foci.