# Math Help - Turning radius algorithm

Hello - total senior citizen newbie here - decent math skills but over my head with this issue. Hopefully this is the correct forum for my question.

Problem relates to aviation.

Aircraft travelling from point A along line AB to point B and turning to course along line BC to point C. If the aircraft waits until it reaches pointBbefore starting the turn it will swing way to wide.

The speed is constant, the angle formed by ABC is constant and the turn rate is 3 deg/sec.

I need a formula that - given any speed and any angle ABC - will determine the turn start point from AB which will bring it closest to B and come out on line BC.

I visualize a circle with two tangent lines forming an intersection (B). Where the first line contacts the circle is turn start point A and where the second line contacts the circle is the turn end point C. Just don't have the math skills to calculate what I need.

Hope this is clear.

Thanx,

Vic Baron

2. Originally Posted by vgbaron
Hello - total senior citizen newbie here - decent math skills but over my head with this issue. Hopefully this is the correct forum for my question.

Problem relates to aviation.

Aircraft travelling from point A along line AB to point B and turning to course along line BC to point C. If the aircraft waits until it reaches pointBbefore starting the turn it will swing way to wide.

The speed is constant, the angle formed by ABC is constant and the turn rate is 3 deg/sec.

I need a formula that - given any speed and any angle ABC - will determine the turn start point from AB which will bring it closest to B and come out on line BC.

I visualize a circle with two tangent lines forming an intersection (B). Where the first line contacts the circle is turn start point A and where the second line contacts the circle is the turn end point C. Just don't have the math skills to calculate what I need.

Hope this is clear.

Thanx,

Vic Baron
let $\theta$ = magnitude of the aircraft's actual turn angle ... note that $\theta = 180 - m\angle{ABC}$

$v$ = aircraft speed in miles/min

based on a $\frac{3^{\circ}}{sec}$ turn rate ...

turn radius, $r = \frac{v}{2}$ miles

let $d$ = distance to the turn point (in miles) to initiate the turn.

I calculate $d = r\tan\left(\frac{\theta}{2}\right) = \frac{v}{2} \cdot \tan\left(\frac{\theta}{2}\right)$ miles

to check, say you had to make a 90 degree turn. let the aircraft speed be 240 kts, or 4 miles/min.

$d = \frac{4}{2} \cdot \tan\left(\frac{90^{\circ}}{2}\right) = 2 \cdot 1 = 2$ miles ... which equals the turn radius, as it should.

for a 120 degree turn ...

$d = \frac{4}{2} \cdot \tan\left(\frac{120^{\circ}}{2}\right) = 2 \cdot \sqrt{3} \approx 3.5$ miles

for a 30 degree turn ...

$d = \frac{4}{2} \cdot \tan\left(\frac{30^{\circ}}{2}\right) = 2 \cdot \tan(15^{\circ} \approx 0.5$ mile

hope this helps.

3. Yup - that looks like it. REALLY appreciate this.

Thanks!

Vic