In an earlier post I had the problem of logarithmic spiral …

… that I needed a tangent for

As a shortcut, I used a vector orthogonal to the point vector from the origin.

But that's not right. The green line represents the true tangent.

Let's use Mathematica to calculate the real tangent.

### Calculating the true tangent

That was the definition of the spiral for some values of `a`

and `b`

.

That was our original definition of the tangent at point `curl[t]`

.

The real tangent can be calculated with the `Limit[]`

function.

Just like we learned in school, the tangent is the limit of a secant with the two points approaching each other.

Thanks to Mathematica, that is calculated with ease.