Connecting Geometric and Algebraic Concepts

The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S=\{(x,y)\in \Bbb R^2: d(\,(x,y),(0,0)\,)=1\}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.

The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $\forall (x,y)\in \Bbb R^2\,(\,(x,y)\in S\iff x^2+y^2=1).$


Geometrically a circle is the locus of all points
on a plane a given distance from a point.

$\{ (x,y) : (x-a)^2 + (y-b)^2 = r^2\}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.

Is there any difference?

Tags:

Geometry

Calculus

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