Logarithmic Spiral - Is Point on Spiral (cartesian coordinates

Question

Lets Say I have a 3d Cartesian grid. Lets also assume that there are one or more log spirals emanating from the origin on the horizontal plane.

If I then have a point in

Answer 1

These are two functions defining an anti-clockwise spiral:

PolarPlot[{

  Exp[(t + 10)/100],
  Exp[t/100]},

 {t, 0, 100 Pi}]

Output:

These are two functions defining a clockwise spiral:

PolarPlot[{

 - Exp[(t + 10)/100],
 - Exp[t/100]},

 {t, 0, 100 Pi}]

Output:

Cartesian coordinates

The conversion Cartesian <-> Polar is

  (1)  Ro = Sqrt[x^2+y^2] 
        t = ArcTan[y/x]

  (2)  x  = Ro Cos[t]
       y  = Ro Sin[t]  

So, If you have a point in Cartesian Coords (x,y) you transform it to your equivalent polar coordinates using (1). Then you use the forula for the spiral function (any of the four mentinoned above the plots, or similar ones) putting in there the value for t, and obtaining Ro. The last step is to compare this Ro with the one we got from the coordinates converion. If they are equal, the point is on the spiral.

Edit Answering your comment

For a Log spiral is almost the same, but with multiple spirals you need to take care of the logs not going to negative values. That's why I used exponentials ...

Example:

PolarPlot[{

  Log[t],
  If[t > 3, Log[ t - 2], 0],
  If[t > 5, Log[ t - 4], 0]

}, {t, 1, 10}]

Output: