Finding gradient of equiangular (logarithmic) spiral

Question

I encountered a problem in determining the gradient in cartesian coordinates (x,y) of a logarithmic spiral (or equi-angular spiral) profile. The log-spiral definintion is as shown below (similar to a previous question of mine):

I have also generated several points in accordance to the profile using the equation `r=a*e^(θtan m):

Point x y

0 9.9999997 2.700000258

1 9.805274245 2.030963458

2 9.452271678 1.338905277

3 8.921115486 0.654692774

4 8.196483978 0.016283107

5 7.269524783 -0.531552828

6 6.139893897 -0.937932945

7 4.817842108 -1.147059284

8 3.326250384 -1.099962912

9 1.702494595 -0.736840057

10 0 0

with (xc,yc) = (7.699656589, 4.680792423); a = 2.013727242; and m = 30 degrees

The plot of the points:

Using cartesian equation from the book.

The cartesian equation of a log spiral is (excerpt of the book):

What I then did next was to bring the term y/x to the left hand side of the equation so that the cartesian equation equates to zero.

To find the gradient at any point of the log spiral profile, I used the following code in Mathematica:

D[ , x]

The output is (note that s = tan (m) ):

However, when I evaluate the gradient based on the equation above, it yielded only positive values at those points I generated. This does not make sense as I expect that the gradient at point no. 8 (for example) to be negative.

Anyone knows what I'm doing wrong?

Thank you.

Regards Corse

Answer 1

I may have misunderstood the intent but post this in case it is helpful.

Note:

• I have tried to mimic the spiral, clockwise rotation with displaced centre of spiral
• I have displayed tangent to curve, the slope and angle.

Parametrization seems the most useful approach:

sp[t_, m_, a_, v_] := 
 v + a {Exp[Tan[m] t] Cos[t], -Exp[Tan[m] t] Sin[t]}
der[t_, m_, a_, v_] := D[sp[u, m, a, v], u] /. u -> t
arrow[t_, m_, a_, v_] := 
 With[{b = sp[t, m, a, v], ar = 5 Normalize@der[t, m, a, v]}, {Red, 
   PointSize[0.02], Point[b], Black, Arrow[{b, b - ar}]}]

where sp is parametrization of logarithmic spiral, der is tangent, arrow is just for visualization purposes.

Visualizing:

Manipulate[
 ParametricPlot[
  sp[t, 30 Degree, 2.013727242, {7.699656589, 4.680792423}], {t, 0, 
   Pi}, Epilog -> 
   arrow[angle, 30 Degree, 2.013727242, {7.699656589, 4.680792423}], 
  Frame -> True, PlotRange -> {{-10, 20}, {-10, 10}}, 
  PlotLabel -> 
   Row[{"slope of tangent: ", 
     slope = Divide @@ 
       Reverse[N@
         der[angle, 30 Degree, 
          2.013727242, {7.699656589, 4.680792423}]], " (", 
     180 ArcTan[slope]/Pi, , Degree, ")"}]], {angle, Pi, 0}]

Post script: I have not aimed to correct angle but hope this promotes your own solution.