What is the edge of the circle?

The term "circumferential direction" is commonly used in engineering, but in physics the term "azimuthal direction" is used. I use "circumferential direction" when talking/writing to engineers and/or technical managers about something that is circular or cylindrical. I would use "azimuthal direction" with mathematicians and physicists.

A common example from engineering is the circumferential stress, which acts in the circumferential direction. One may also speak of the circumferential component of some object. I have seen the German phrases "Spannung in Umfangsrichtung", used to indicate the stress in the circumferential direction, and "Umfangskomponent" for the circumferential component, but I suspect these expressions are rare.

The term "azimutale Richtung" is common in German engineering and physics textbooks.

To keep it clear and simple, I would write "The points are displaced along the circumference as the radius of the circle increases." My German is poor, but perhaps something like "Wenn der Radius des Kreises sich vergrößert, die Punkte werden entlang dem Umfang verschoben" would get the idea across.

I was talking to a friend about stretching something over a circular drum's "edges". We gave google a shot and it seems like people are unsure if there are one, zero, or infinite edges. On a semantic level, is it ok to say I stretched it around the "edges" or should it be "edge"?

No, it cannot. A chord of a circle is a line segmentwhich has its endpoints on the curve (or circumference) of the circle. A line segment is part of a line, and a line is straight. It's that simple.

Definitions and formulas for the radius of a circle, the diameter of a circle, the circumference (perimeter) of a circle, the area of a circle, the chord of a circle, arc and the arc length of a circle, sector and the area of the sector of a circle 
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The radius of a circle:
	The radius of a circle is the distance from the center of the circle to the outside edge.

The diameter of a circle:
	The diameter of a circle is longest distance across a circle.  (The diameter cuts through the center of the circle.  This is what makes it the longest distance.)

The circumference of a circle (the perimeter of a circle):
	The circumference of a circle is the perimeter -- the distance around the outer edge. Circumference =  where r = the radius of the circle and pi = 3.141592...

A chord of a circle:
	A chord of a circle is a line segment that connects one point on the edge of the circle with another point on the circle. (The diameter is a chord -- it's just the longest chord!)