Locus and Definition of a Circle and Sphere - Concept
A locus is a set of points that meet a given condition. The definition of a circle locus of points a given distance from a given point in a 2-dimensional plane. The given distance is the radius and the given point is the center of the circle. In 3-dimensions (space), we would define a sphere as the set of points in space a given distance from a given point.
A term in Geometry that's a little bit confusing is Locus and we're not talking about one of the plagues here. we're talking about a set of points that meet a given condition, so to examine what this means let's look at 2 examples, first one says draw all the points 5 centimeters away from point p in the plane. So here's point p the plane would be the plane of the board. Remember planes are 2 dimensional and they extend infinitely in every direction. So if I start to think about well 5 centimeters you know just probably about right there and a point 5 centimeters away from point p could be right there. I can also have a point on the other side of point p and I can continue finding points that are about 5 centimeters away from point p until I'm blue in the face.
And as you're going to see eventually if I did this for long enough I would have a circle. So a locus is all of these points that meet a specific condition in which case this one said 5 centimeters away from point p in that plane. Which leads us to our definition of a circle, which is the locus of points at given distance away from a given point in a plane. So it's kind of a long way of saying if I give you a point a circle by definition is all those points that are certain distance away from that point that I gave you.
Let's go back to the second example, second example says draw all the points 5 centimeters away from point o in space. So we've got point o and obviously I won't be able to draw space which is coming out at you and it's going into the board but I can kind of make a guess. Well what it's going to look like is we're going to have all of these points o that'll be in the plane of the board and then we're going to have points that are going to come out of the board so I guess what I can do is I can draw a kind of a curve here which will show all the points that are coming up and I'm going to draw a dotted line for all the points that go into the board back into the board that are 5 centimeters away. So what we've drawn here is a sphere so that's the definition of a sphere which is related to a circle plus spheres talking about 3 dimensions. So it's the locus of points that are a given distance away from a point in space. So we're talking about 3 dimensions which is the space that we live in. So the key concept there is that a locus is all of the points that meet a given condition.