What should be in the blank?

The t of the GeoGebra can move.

Here is an example of such a curve in GeoGebra. Move the t slider to see the path (or locus) of the point P trace out the spiral. Let P be the point described by Archimedes, 0 = [0, O] and 6 be the angle between OP and the positive :2: -axis. According to Archimedes the line OP revolves at a uniform rate. Suppose for example that this rate is 5 : d625 dt Initially, P is on the cc -axis. 80 we can integrate this expression to find 6 = a .- a . We are also told by Archimedes that the distance r = |0P| moves at a uniform rate. Suppose for example that this rate is 15 : Let P be the point described by Archimedes, O = [0, 0] and 6 be the angle between OP and the positive as -axis. According to Archimedes the line OP revolves at a uniform rate. Suppose for example that this rate is 5 : d6 _ = 5 dt Initially, P is on the 4% -axis. 80 we can integrate this expression to find 9: :] We are also told by Archimedes that the distance 7' = |OP| moves at a uniform rate. Suppose for example that this rate is 15 : _ cit15 Initially, 7' = 2 . So we can integrate this expression to find: _ w w {IaS..- Hence we have a parametric description of the point P in terms of time t. We can eliminate t to nd an expression in terms of 7' and 6 alone. This expression is called the polar form: HEB- Note: the Maple syntax for 6 is theta