Calculate the Tangent Line of a Circle

Given a circle with it's center point $M$, the radius $r$ and an angle $\alpha$ of the radius line, how can one calculate the tangent line on the circle in point $T$? The trick is, that the radius line and the tangent line are perpendicular. This means, that their dot-product must be zero.

So, how can this schematic be derived? The intersection point $T$ is the center point $M$ plus the radius in the direction of the angle:

$$T = \left(\begin{array}{c} T_x\\ T_y \end{array}\right) = M + r \cdot \left(\begin{array}{c} \sin\alpha\\ \cos\alpha \end{array}\right)$$

Now we need to rotate the vector by 90° in order to get a vector that is perpendicular to the radius line:

$$\vec{v} = \left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right) \cdot \vec{MT}$$

Which leads to the line equation:

$$L(s) = T + s \cdot \frac{\vec{v}}{|\vec{v}|}$$

The implementation is therefore also quite simple:

function tangentLine(M, r, alpha) {

   T = {
      x: M.x + Math.cos(alpha) * r,
      y: M.y + Math.sin(alpha) * r
   }

   v = {
      x: T.y - M.y,
      y: M.x - T.x
   }

   norm = Math.hypot(v.x, v.y)
   v.x/= norm
   v.y/= norm

   return [T, v]
}