How to catch up a boat.Apollonius circleList of animations posted on this page.(Click the text to watch animation.)Case (a) Facing each other Case (b) Chasing behind on a line Case (c) General case Suppose a ship leaves point B and steams in a fixed direction at a constant speed. A second ship,leaving point A at the same time, can go K times as fast as the first ship. Assuming a plane ocean,what course should the first ship steer in order to intercept the slow ship as quickly as possible ? ******* Apollo_circle_00.dwg ******** Let us begin with two very simple hypothetical cases when both ships are on the same line. Case (a) Facing each other
It is easy to guess that two ships will meet at Point C,
******* Apollo_circle_01.dwg ******** ******* Apollo_circle_01A.dwg *******
You can see the process in animation.
Case (b) Chasing on the same straight line
In this case the Chaser will catch up the slower ship at Point D,
You can see the process in animation.
Case (c) More general case
What direction the boat "A" should take when the slower boat is moving in the direction
******* Apollo_circle_00.dwg ******** Suppose the point of interception is P. Then this point P has to satisfy the following condition: AP:BP = K : 1For example let the value of K be set equal to 2. Draw a circle with its center at B and radius R1. Then draw a circle with its center at A and radius K times R1. Locate the intersection points. Do this process a few times. The result is as follows:
You will notice the following:
To create this drawing and animation:

If angles APC, CPB, BPD, and DPE are measured, we will find out that points C & D bisect both internal and external angles at apex P of triangle APB.
Rigorous proof is given in this page later.
******************* pursuit_Apollonius_01.dwg ********************
More on the Apollonius Circle:
OD = r* (K+1)/(K1)
The discussion above are based on the following Theorems proved by Apollonius.
CB/BD = AC/AD
AC/CB = AD/BD = AP/BP = K
In the following drawing, CP and DP are the angle bisectors, and
CP is parallel to BE, and BG is parallel to DP.
In Triangle ADP,since BG is parallel to DP, AD/BD = AP/GP = AP/BP
To create this drawing and animation:
Load pursuit_Apollonius.lsp (load "pursuit_Apollonius")
To see case (a),(b),& (c)
from command line, type pursuit_test
Case (a), (b) & (c) runs continuously.
Note* : This program requires red_boat.dwg & blue_boat.dwg .
****************** Apollo_circle_03.dwg *********************
All comments/suggestions should be sent to Takaya Iwamoto
Last Updated Aug10th, 2006
Copyright 2006 Takaya Iwamoto All rights reserved.