# knp.de

– Tangents on a Logarithmic Spiral

## Tangents on a Logarithmic Spiral

03/29/2015

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`.

```curl[t_] := {
Exp[a t] Cos[b t],
-Exp[a t] Sin[b t]
}```

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

```tangent[t_, o_] := Module[{
p = curl[t],
q},
q = {p[[2]], -p[[1]]};
p + o q
]```

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.

```tangent[t_, o_] := Module[{
p = curl[t],
q},
q = Limit[
(p - curl[t + h])/h,
h -> 0];
p - o Normalize[q]
]```