All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Locus of Points (Posted on 2007-12-12)
Let X1, X2, ... , Xn be n≥2 distinct points on a circle C with center O and radius r. What is the locus of points P inside C such that

∑ni=1 |XiP|/|PYi| = n

,where the line XiP also intersects C at point Yi.

 See The Solution Submitted by Bractals Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: exploration with computer -- 2 non-opposing points | Comment 2 of 16 |
(In reply to exploration with computer by Charlie)

When the two points in the two point case are separated by less than 180 degrees, as in the below diagram with a separation of 90 degrees, the lemniscate transforms into a kidney shaped blob (remember, it's the border that forms the locus, not the interior). Due to the vertical stretching/horizontal compression, the below does not look symmetrical, but is indeed so about the line y=x. Again, the circle of .'s is superimposed to mark where the center of the circle is; actual *'s and blanks underly them.

As usual, the locus goes through the center, O, and is tangent to the circle C at the designated points Xn, except that at points Xn, that one point (the point of tangency) is missing from the actually defined locus as the ratio of lengths is not defined there.

`                                           ------- -------                                                                                 ---    ***********    ---                                                                         -       ***************       -                                                                   -        *******************        -                                                              -         *********************         -                                                          -         *************************         -                                                      -           **************************          -                                                  -            ****************************           -                                                            *****************************                                                         -             *******************************             -                                                        ********************************                                                                    *********************************                                                   -                **********************************               -                                 -                 **********************************                -                               -                 ************************************                -                             -                  ************************************                 -                           -                   *************************************                 -                         -                    *************************************                  -                                             **************************************                                                               **************************************                                                               ***************************************                                      -                       ***************************************                   -                                          ****************************************                                                             *****************************************                                  -                         ******************************************                  -                                        *******************************************                               -                          ********************************************                 -                                        ********************************************                            -                            *********************************************                -                                       ***********************************************                        -                              ************************************************             -                                       **************************************************                   -                                ***************************************************          -                                       *****************************************************                                                 ******************************************************            -                                  *******************************************************      -                                        ********************************************************                                              ********************************************************       -                                     *********************************************************   -  -                                      ********************************************************   -  -                                       ********************************************************  -                                           ********************************************************                                              *******************************************************   -                                           ******************************************************* --                                            ****************************************************** --                                             ******************************************************--                                              **...************************************************--                                              ..***..**********************************************--                                              . ****.**********************************************--                                             .  *****.*********************************************--                                             .   ****.*********************************************--                                             .    ***.*********************************************--                                              .     .**********************************************--                                              ..   ..**********************************************--                                                ...   *********************************************--                                                       ********************************************--                                                        ****************************************** --                                                         ***************************************** -                                                           ***************************************                                                               **************************************    -                                                           ************************************  -  -                                                            **********************************   -  -                                                              ********************************   -                                                                  *****************************                                                                          **************************          -                                                                  ***********************      -                                                                         *******************                                                                                     *************                 -                                                                                             -                                                                                                            -                                                                                           -                                                                                                              -                                                                                         -                                                                                                                -                                                                                       -                                                                                                                  -                                                                                     -                                                                                                                                                                                                                          -                                                                                 -                                                                                                                                                                                                                                                                                                                                    -                                                                           -                         -                                                                         -                           -                                                                       -                             -                                                                     -                               -                                                                   -                                 -                                                                 -                                                                                                                                                                                                                                                -                                                         -                                                                                                                                                  -                                                   -                                                  -                                               -                                                      -                                           -                                                          -                                       -                                                              -                                   -                                                                   -                             -                                                                         ---                   ---                                                                                 ------- -------                                                                                                                                                `

 Posted by Charlie on 2007-12-12 19:52:15

 Search: Search body:
Forums (0)