What constitutes a 'geometrical condition'? The equation of a line is 
$$ax + by +c = 0 $$
which can be completely determined if we are given two 'independent geometrical conditions'. Similarly, for a circle,
$$x^2+y^2+2gx+2fy+c=0$$
This is completely determined if we are given any three 'independent geometrical conditions'. My question, however, is, how do we define a geometrical condition? 
For instance, in the case of the circle, the conditions could be three given points, or three lines which the circle touches. 
What qualifies as a geometrical condition? For example, a circle is also determined if we just provide the radius and center, which would, superficially seem like giving only two geometrical conditions.
Secondly, what is the test that two geometrical conditions are 'independent'? How do we know if two given conditions are not somehow linked?
 A: Replying as per how it looks to me.
The geometric condition is also a geometric parameter like independent distance, slope, curvature etc.
We need 2 constants to fix a straight line, 3 to fix a circle, 4 to determine a parabola, 5 to determine a conic section and perhaps more for elliptic curves. These are respectively:
$$ { y = m x + c ,(x-h)^2 + (y-k)^2 = r^2, y = ( a x + b) + \sqrt {c x + d },  a x^2 + 2 h x y + b y^2 + 2 f x  +  2 g y + 1 = 0.. }$$
The geometric conditions are  euclidean motion, curvature, curvature variations.. 
The number of geometric conditions equals number of arbitrary geometric constants or differentiations to arrive at an equality to zero  in a differential relation of same (order) number.
It is also the number of equations necessary to arrive at a curve definition which bear a characteristic relation among themselves. This statement I shall however attempt to edit after some time in perhaps a more appropriate manner.
I like your question. Text books should address to such students' questions   connecting various subject areas:


*

*Analytical geometry

*Isometry

*Differential equations 


PS EDIT1:
For a circle center we have two co-ordinates, along with a radius make up three constants.
