There are two different ways you can approach drawing the spiral.
1) Using GeneralPath to define the path from a fixed coordinate system in which you define the coordinates of the segments to be drawn on a fixed x-y plane.
2) Using coordinate transformations (translate, rotate) to alter the coodinate system as you draw each segment. With translation and rotation, drawing the path becomes trivial. You simply draw a line in a fixed direction (which ever way you want to go from your origin, like 0,0 to 0,100, translate to the end of that line, rotate 90 deg left or right, draw a slightly shorter line in the same direction as the first (perhaps 0,0 to 0,90), and continue until you reach the endpoint you want. By moving and rotating your origin as you draw, you only have to draw increasingly shorter line segments in the direction of path traversal.

There are some examples of using coordinate transforms here:
http://www.java2s.com/Code/Java/2D-G...ndRotation.htm

Hope that helps get you started.