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Set of All Points That ...

In Mathematics we often say "the set of all points that ... ".
What does it mean?

A set is just a collection of things with some common property.
If you collect ALL points that share a property you can end up with a line, surface or other interesting thing.

Points can make a line

Example: A Circle is:

"the set of all points on a plane that are a fixed distance from a central point".

As you can see, just a few points start to look like a circle, but if you collect ALL the points, you will actually have a circle.


Now imagine all the points that are a fixed distance from a center (like the circle) but no longer just on a plane, let them be in 3D space ... you would have a sphere!


The idea of "the set of all points that ..." is used so much it even has a name: Locus.

A Locus is a set of points that share a property.

So, a circle is "the locus of points on a plane that are a fixed distance from the center".

Note: "Locus" usually means that the points make a continuous curve or surface.

Example: An ellipse is the locus of points whose distance
from two fixed points add up to a constant.

So, no matter where you are on the ellipse, you can add up the distance to point "A" and to point "B" and it will always be the same result.

(The points "A" and "B" are called the foci of the ellipse)

The idea of "Locus" can be used to create some weird and wonderful shapes!