$${(x-h)^2 + (y-b)^2 = r^2}$$ What does a point lying on a circle mean? Does it mean that I can show this equation to anyone and they would be able to understand that it really is a circle? Edit: The equation of a circle is made out of coordinate axes 'X' and 'Y'. If I input numbers x,y that satisfy the equation then a Circle will emerge after joining those points? But it is also possible that the user might join points in random order and a circle might never emerge. I have a diagram below to show you what I understand in mathematics.
construct circle by joining points?
In previous post, I asked what is the meaning of equation of the circle and equation of a cube. The answer was, if I input x,y that satisfies the equation then the points lie on the circle. This post is not duplicate of my previous post. In this post I want to know what the meaning of points lying on the circle is. If points are lying somewhere then do I need to join them to make a circle? But there is a possibility that I can draw a line segment connecting those points in random order. I have illustrated my understanding in the diagram above.
Let's look at $$x^2+y^2 = 1.$$ You can imagine this as all the points in $\mathbb{R}^2$ i.e. tuples $(x,y) \in \mathbb{R}^2$, which have distance 1 from the origin $(0,0) \in \mathbb{R}^2$. Another way of rewriting this would be the set $$\{ (x,y) \in \mathbb{R}^2 | x^2 + y^2 = 1\}$$ which is a circle in $\mathbb{R}^2$. It may be best if you try to plot this set to see that it is indeed a circle.
So those points satisfying $x^2+y^2=1$ lie indeed on a surface namely on a circle.
The other case is similar.