Escape Circle

Escape Circle is the circumference of Escape Envelope assuming the motion of the target is linear and uniform.

Calculation

Assuming uniform linear motion, consider a situation where a bullet fired from the firer A hits the target C at some future position B. A is the angle opposite to a, B is opposite to b, and C is opposite to c.

Let the target C be the origin, the distance and orientation from the target to the firer be the unit length and positive x-axis respectively.

Let θ be the angle between b and a, we have c / a = Vb / Vr, b = 1, where Vb and Vr is the velocity of the bullet and the target respectively. Now consider cosine formula:

a² + 1² - 2 a cosθ = (Vb / Vr a)²

Let x = a cosθ, y = a sinθ, k = 1 / ((Vb / Vr)² - 1), we have:

(x + k)² + y² = k² + k

The points defined by (x, y) satisfying above formula is a circle e with center P(-k, 0), and radius sqrt(k² + k).

See also

• Escape Envelope
• Maximum Escape Angle