GSP4QThat document was created with a newer version of Sketchpad and cannot be opened.|This document was created with a newer version of Sketchpad. Some information may be lost. The document will open as a copy.$b ` a b c d e f g h i j k l m n o p q _ ,#GSP4_W_0498/Jan 29 20020#` Arial '''$'f$'$'f$'f$'$'$'$'l$'$' $' $' %'&'8'8'8'9':' & K-Disk Center@y@Pq@f@& K-Disk Radiusm@v@f@  @.#@42nQ\aAt[1]k"fn(K-Disk Radius' n@v@> @    %,nQ\aAShow Disk Controlse)fnPw@o@4(f@ 4\.B@: &@   nQ\aAx6(fn 6\.B@: &n   nQ\aAy='n   -# {A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LKlein line Arialol &nQ\aA K-Disk CenterK`!KK&nQ\aA K-Disk Radius"1$nQ\aA #2 on lineu$nQ\aA #1 on lineu?   $gQ@ Klein Disku -g nQ\aAA -n nQ\aABn /#Klein Line Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A and B in Klein Disk Returns: The Klein line (i.e. chord of the circle) passing through A and B Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein segment Arialsi &nQ\aA K-Disk Center&nQ\aA K-Disk RadiusK$nQ\aA Endpoint #2$nQ\aA Endpoint #1  $gQ@ Klein Disk/#Klein Segment Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, B in Klein Disk Returns: Klein segment AB Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003/{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LKlein length Arial &nQ\aA K-Disk Center&nQ\aA K-Disk Radius $nQ\aA Endpoint #2$nQ\aA Endpoint #1?   $gQ@ Klein Disk -g nQ\aAA -n nQ\aAB%0n   N<0>%0n   9<0>%0n   $<0>%0n   <0>0 !n  c<0>Q b0  ``  ``  "distancek/#Klein Length Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, B in Klein Disk Returns: The Klein length of the segment AB. Note that since the Klein model is not isometric, the Klein length of the segment is generally different from its Euclidean length. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$$LKlein angle measure Arial@ s&nQ\aA K-Disk Center&nQ\aA K-Disk Radius*nQ\aAon angle side #2n nQ\aAVertexe*nQ\aAon angle side #1n?  ?  ?   $gQ@ Klein Diskd -g  nQ\aAA -n  nQ\aAB -n  nQ\aAC -n  nQ\aAD -n  nQ\aAE -n  nQ\aAF%0n   <0>%0n   <0>%0n   <0>%0n   <0>%0n   <0>%0n   <0>%0n   <0>%0n   <0>%0n   N<0>%0n   9<0>%0n   $<0>%0n   <0>0 !n <0> apy b0  ``  ``  m170 ! <0> apy b0  ``  ``  m120 ! c<0> apy b0  ``  ``  m50 !t <0> T0  `` m200 !t <0> T0  `` m190 !t <0> T0  `` m180 !t <0> T0  `` m70 !t {<0> T0  `` m60 !!# "| <0>{ \0 ```  `` Theta^/#Klein Angle Measure Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, C, B in Klein Disk Returns: The measure of Klein angle ACB. Note that the Klein Disk is not conformal, so angle measures cannot be visually estimated. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003v{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein angle bisector Arial0 e&nQ\aA K-Disk Center&nQ\aA K-Disk Radius*nQ\aAon angle side #1s  nQ\aAvertexe*,pAA*nQ\aAon angle side #2s?  ?   $gQ@ Klein Diskd -g nQ\aAA -n nQ\aABn  n  /n  nQ\aAj/n  nQ\aAk -n  nQ\aAC? n  -n nQ\aAD -n nQ\aAEn /#Klein Angle Bisector Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, C, B in Klein Disk Returns: The Klein line which is the angle bisector of angle ACB in the sense of Klein. Note that since the Klein model is not conformal, this line is generally not the Euclidean angle bisector of angle ACB. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein raise perpendicular Arial p&nQ\aA K-Disk Center&nQ\aA K-Disk RadiusPK.nQ\aAfoot of perpendicularb$nQ\aA #2 on linep `?   $gQ@ Klein Diskp -g nQ\aAA -n nQ\aABn  -n  nQ\aAC -n  nQ\aADn  n  /n  nQ\aAj/n  nQ\aAk -n nQ\aAE? n  -n nQ\aAF -n nQ\aAGn /#Klein Raise Perpendicular Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, B in the Klein Disk Returns: The Klein line through A which is perpendicular (in the Klein sense) to the Klein line containing A and B. Both lines are drawn in this construction. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 20032{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein drop perpendicular Arial e&nQ\aA K-Disk Center??&nQ\aA K-Disk RadiusL&nQ\aA #1 on Line l '>&nQ\aA #2 on line l&nQ\aA not on line l?   $gQ@ Klein Disk  -g nQ\aAA -n nQ\aABn  n  n /n  nQ\aAj/n  nQ\aAk -n  nQ\aAC? n  -n nQ\aAD -n nQ\aAEn /#Klein Drop Perpendicular Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Non-collinear points A, B, C in the Klein Disk Returns: The Klein line through C which is perpendicular (in the Klein sense) to the Klein line l containing A and B. Both lines are drawn in this construction. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003G{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$)LKlein perpendicular bisector Arial u&nQ\aA K-Disk Center&nQ\aA K-Disk Radius1P$nQ\aA Endpoint #2nQ\aAA$nQ\aA Endpoint #1 ?   $gQ@ Klein Disk -g nQ\aAA -n nQ\aABn   -n  nQ\aAC -n  nQ\aADn  n  /n  nQ\aAj/n  nQ\aAk -n nQ\aAE? n ? n  -n nQ\aAF -n nQ\aAG -n nQ\aAH -n nQ\aAIn n  -n nQ\aAJ? n  -n nQ\aAK -n nQ\aALn  -n nQ\aAM -n nQ\aANn n  /n !nQ\aAl/n  "nQ\aAm -n #$nQ\aAO? n % -n &nQ\aAP -n &nQ\aAQn ('H/#Klein Perpendicular Bisector Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, B in Klein Disk Returns: The Klein perpendicular bisector of segment AB. Segment AB is also drawn in this construction. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003 {A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$g"LKlein circle by center and point Arial  &nQ\aA K-Disk Center&nQ\aA K-Disk Radius&nQ\aA circle center"nQ\aA on circle'nQ\aAS[1]r@    $gQ@ Klein Diske -g nQ\aAAn  n  B3n    <0>A3n    <0>A3n    <0>B3n    <0>B3n    <0>0 !n^ <0>um >0`A3n    <0>0 !n^ <0>um >0`B3n    <0>0 !n^ <0>um >0`A3n    <0>0 !n^ <0>um >0`0 !n ^ <0>um >0`0 !n ^ <0>um >0` n & K-Disk radius0 ! ^ *<0>um >0`0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 ! ^ <0>um >0`0 ! ^ k<0>um >0`0 !^ R<0>um >0`0 !^ 9<0>um >0`%0   <0>r0 !~ <0>O ^0 ``  ``  m200 !!" ~ <0>O ^0 ``  ``  m70 !$ <0> j0` ` ``  ``  m240 !%  <0> j0` ` ``  ``  m110 !$& <0>O d0` ``  `` `m260 !$& <0>O d0` ``  `` `m250 !%'  <0>O d0` ``  `` `m130 !%' <0>O d0` ``  `` `m12E8)(nQ\aABE8n+*nQ\aAC? n - -n . nQ\aAE -n . nQ\aAF? n 0/n /0$An濘53^?) F@adiu? ^ 2/? ^ 2-? ^ 02 ^5nQ\aAG? n 6/ -n 74nQ\aAH? n 08 -n 93nQ\aAI? n 6: -n ;1nQ\aAJ? n ,<n ,<n </n <>nQ\aAjn <>nQ\aAc[1] n?nQ\aAK -n @AnQ\aALn B< -n  DnQ\aAMn <E -n @FnQ\aAN -n =FnQ\aAO? n HC.n GInQ\aAk -n =JnQ\aAPn K -n .LnQ\aAQ? n M<n M< -n NFnQ\aAR/n <OnQ\aAln <OnQ\aAc[2] -n QFnQ\aAS -n QRnQ\aAT? n PT.n SUnQ\aAm -n NVnQ\aAUn ,W nXnQ\aAV/n YXnQ\aAn -n .ZnQ\aAWn [, n\nQ\aAXn ]B3n ]   <0>0 !n_^ <0> >0`A3n ]   <0>0 !na^ <0> >0`$0n^  <0>d0 !c`# <0> l0` ``  `` `m500 !cb# <0> l0` ``  `` `m49E8ednQ\aAD#*n4 f\]^_`abcdef  (-DT!@(nQ\aAL[1]/#Klein Circle by Center and Point Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points O, P in Klein Disk Returns: The Klein circle centered at O with radius OP. The circle is generated as a locus of points. The number of points on the circle can be changed in the "Preference" menu. Author: Steve Szydlik/December 2001 Revised for GSP 4.0 in February 2003 {A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$&LKlein circle by center and two points Arial  &nQ\aA K-Disk Center&nQ\aA K-Disk Radius&nQ\aA circle center8nQ\aAEndpoint #2 determining radiusk8nQ\aAEndpoint #1 determining radiusk'nQ\aAS[1]i@    $gQ@ Klein Disk  -g nQ\aAAn  n  B3n    <0>A3n    <0>A3n    <0>B3n    <0>B3n    <0>0 !n^ <0>um >0`A3n    <0>0 !n^ <0>um >0`B3n    <0>0 !n^ <0>um >0`A3n    <0>0 !n^ <0>um >0`B3n    <0>0 !n^ <0>um >0`A3n    <0>0 !n^ <0>um >0`0 !n ^ <0>um >0`0 !n^ <0>um >0` n & K-Disk radius0 !^ <0>um >0`0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`%0   <0>r0 !" #!~ <0>O ^0 ``  ``  m400 !&$'%~ <0>O ^0 ``  ``  m270 !*(+)~ <0>O ^0 ``  ``  m140 !- ! <0> j0` ` ``  ``  m440 !.$% <0> j0` ` ``  ``  m310 !/() <0> j0` ` ``  ``  m180 !-0! <0>O d0` ``  `` `m460 !-0  <0>O d0` ``  `` `m450 !.1% <0>O d0` ``  `` `m330 !.1$ <0>O d0` ``  `` `m320 !/2) <0>O d0` ``  `` `m200 !/2( <0>O d0` ``  `` `m19E843nQ\aABE8n65nQ\aACE8n87nQ\aAD? n ;:n ;:? n ;@ n ; n=nQ\aAF -n > nQ\aAG -n > nQ\aAH/n ;?nQ\aAj/n @=nQ\aAk? n BAn AB$An濘53^?) F@adiu -^ C nQ\aAI? n FA? n F;? n BFn G nJnQ\aAJ/n GKnQ\aAl? n LA -n ?M"Invertedd - INnQ\aAKn O;? n BP nQnQ\aAL -n HRnQ\aAM/n SQnQ\aAm -n DUnQ\aAN? n LTn VO -n EWnQ\aAO-n X nQ\aAP-n  XnQ\aAQn Y? n Z[ n\nQ\aAR -n ]<nQ\aASn ^Y? n 9_n 9_n _ -n  `nQ\aAT/n _bnQ\aAnn _bnQ\aAc[1]t ncnQ\aAUn Yd -n efnQ\aAVn g_ -n  jnQ\aAWn _k -n elnQ\aAX -n alnQ\aAY? n ni.n monQ\aAo -n apnQ\aAZ? n qYn qY -n rhnQ\aAA[1]t/n YsnQ\aApn YsnQ\aAc[2]t -n uhnQ\aAB[1]t -n uvnQ\aAC[1]t? n tx.n wynQ\aAq -n zrnQ\aAD[1]tn { -n >|nQ\aAE[1]t? n }Yn }Y -n ~hnQ\aAF[1]t/n YnQ\aArn YnQ\aAc[3]t -n hnQ\aAG[1]t -n nQ\aAH[1]t? n .n nQ\aAs -n ~nQ\aAI[1]tn q nnQ\aAJ[1]t/n nQ\aAt -n >nQ\aAK[1]tn q nnQ\aAL[1]tn B3n    <0>0 !n^ <0> >0`A3n    <0>0 !n^ <0> >0`$0n  <0>d0 !, <0> l0` ``  `` `m630 !, <0> l0` ``  `` `m62E8nQ\aAE#*n4  (-DT!@(nQ\aAL[1]@/#Klein Circle by Center and Two Points Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points O, A, B in the Klein Disk Returns: The Klein Circle centered at point O with radius OP congruent to AB. Author: Steve Szydlik/December 2001 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LIntersect Klein circles Arial  &nQ\aA K-Disk Center@:@:&nQ\aA K-Disk Radius 2nQ\aAcenter of Klein circle #1K-Disk Centerenn.nQ\aAp1 on Klein circle #12nQ\aAcenter of Klein circle #2`:`: `.nQ\aAp2 on Klein circle #2'nQ\aAS[1]  @  nQ\aAjn $gQ@ Klein Disk  -g  & P-Disk radius    B3    <0>A3    <0>A3    <0>B3    <0>B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`  & K-Disk radius%0 6  <0>r%0 6  <0>r0 !(&)'~ <0>O ^0 ``  ``  m244k0 !,*-+~ <0>O ^0 ``  ``  m237k0 !0.1/~ <0>O ^0 ``  ``  m230k0 !4253~ <0>O ^0 ``  ``  m223k0 !9&' <0> j0` ` ``  ``  m274k0 !:*+ <0> j0` ` ``  ``  m268k0 !;./ <0> j0` ` ``  ``  m262k0 !<!23 <0> j0` ` ``  ``  m256k0 !9=' <0>O d0` ``  `` `m276k0 !9=& <0>O d0` ``  `` `m275k0 !:>+ <0>O d0` ``  `` `m270k0 !:>* <0>O d0` ``  `` `m269k0 !;?/ <0>O d0` ``  `` `m264k0 !;?. <0>O d0` ``  `` `m263k0 !<@3 <0>O d0` ``  `` `m258k0 !<@!2 <0>O d0` ``  `` `m257kE8BA p2'E8DC o2'E8FE p1'E8HG o1'?  J?  L - M nQ\aAE -n M nQ\aAF -n N nQ\aAG -n N nQ\aAH? n POn OP$An濘53^?) F@adiu? ^ RQ^ QR$An濘53^?) F@adiu? ^ TO? ^ TJ? ^ PT? ^ VQ? ^ VL? ^ RV ^YnQ\aAI n\nQ\aAJ? n ]O? n ^Q -n X_nQ\aAK -n [`nQ\aAL? n Pa? n Rb -n WcnQ\aAM -n ZdnQ\aAN? n ]e? n ^f -n SgnQ\aAO -n UhnQ\aAP? n Iin Iin i? n Kjn Kjn j/n ilnQ\aAkn ilnQ\aAc[1] nmnQ\aAQ/n jonQ\aAln jonQ\aAc[2] npnQ\aAR -n qrnQ\aASn si -n tunQ\aATn vj -n  xnQ\aAU -n  znQ\aAVn i{n j| -n q}nQ\aAW -n k}nQ\aAX -n t~nQ\aAY -n n~nQ\aAZ? n w? n y.n nQ\aAm.n nQ\aAn -n knQ\aAA[1] -n nnQ\aAB[1]n n  -n MnQ\aAC[1] -n NnQ\aAD[1]? n in i? n jn j -n }nQ\aAE[1]/n inQ\aAon inQ\aAc[3] -n ~nQ\aAF[1]/n jnQ\aApn jnQ\aAc[4] -n }nQ\aAG[1] -n nQ\aAH[1] -n ~nQ\aAI[1] -n nQ\aAJ[1]? n ? n .n nQ\aAq.n nQ\aAr -n nQ\aAK[1] -n nQ\aAL[1]n In K nnQ\aAM[1] nnQ\aAN[1]/n nQ\aAs/n nQ\aAt -n MnQ\aAO[1] -n NnQ\aAP[1]n In K-n nQ\aAA -n nQ\aABn n B3n    <0>0 !n^ <0> >0`A3n    <0>0 !n^ <0> >0`$0n  <0>dB3    <0>0 !^ <0> >0`A3    <0>0 !^ <0> >0`$0  <0>d0 !"7 <0> l0` ``  `` `m2880 !#7 <0> l0` ``  `` `m2870 !$8 <0> l0` ``  `` `m2820 !%8 <0> l0` ``  `` `m281E8nQ\aACE8nnQ\aAD /#Intersect Klein Circles Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Two Klein circles defined by their respective centers and points on the circles. Returns: The point(s) lying on the intersection of the circles. Since Klein circles are ellipses drawn as loci (which cannot be intersected in current Sketchpad versions), this tool maps the Klein disk isomorphically onto the Poincare disk, where "circles are circles," finds the intersection points there, then maps the results back to the Klein disk. (See _Euclidean and Non-Euclidean Geometries_, 3rd Ed. by Greenberg, p. 236.) Author: Steve Szydlik/January 2001 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$&LIntersect Klein circle and Klein line Arial  &nQ\aA K-Disk Center.&nQ\aA K-Disk Radius>$nQ\aA #1 on lineu$nQ\aA #2 on lineu::0nQ\aAcenter of Klein circle (nQ\aAon Klein circle'nQ\aAS[1]e @  nQ\aAjn $gQ@ Klein Diskr -g  &  P-Disk radius    B3    <0>A3    <0>A3    <0>B3    <0>B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`  & K-Disk radius0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ *<0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ k<0>um >0`0 !^ R<0>um >0`0 !^ 9<0>um >0`%0 &  <0>r%0 &  <0>r0 !)'*(~ <0>O ^0 ``  ``  m630 !-+.,~ <0>O ^0 ``  ``  m500 !1/20~ C<0>O ^0 ``  ``  m200 !5364~ <0>O ^0 ``  ``  m70 !9'( <0> j0` ` ``  ``  m670 !:+, <0> j0` ` ``  ``  m540 !;/0 <0> j0` ` ``  ``  m240 !<!34 <0> j0` ` ``  ``  m110 !9=( <0>O d0` ``  `` `m690 !9=' <0>O d0` ``  `` `m680 !:>, <0>O d0` ``  `` `m560 !:>+ <0>O d0` ``  `` `m550 !;?0 <0>O d0` ``  `` `m260 !;?/ <0>O d0` ``  `` `m250 !<@4 <0>O d0` ``  `` `m130 !<@!3 <0>O d0` ``  `` `m12E8BA P'E8DC O'E8FE D'E8HG C'?  J LK@  L - M nQ\aAE -n M nQ\aAF nNnQ\aAG/n LOnQ\aAk? n QPn PQ$An濘53^?) F@adiu/^ RNnQ\aAl -n S nQ\aAH? n UP? n UJ? n QUn W nZnQ\aAI/n W[nQ\aAm? n \P -n O]nQ\aAJ -n ^YnQ\aAKn _L? n Q` nanQ\aAL -n bXnQ\aAM/n canQ\aAn? n \d -n eVnQ\aAN -n fTnQ\aAOn g_? n Ihn Ihn h -n i nQ\aAP -n  inQ\aAQ/n hknQ\aAon hknQ\aAc[1] nlnQ\aARO Cninm - opnQ\aASn qh -n  tnQ\aATn hu -n ovnQ\aAU -n jvnQ\aAV? n xs.n wynQ\aAp -n jznQ\aAWn { -n M|nQ\aAX? n }hn }h -n ~vnQ\aAY/n hnQ\aAqn hnQ\aAc[2] -n vnQ\aAZ -n nQ\aAA[1]? n .n nQ\aAr -n ~nQ\aAB[1]n I nnQ\aAC[1]/n nQ\aAs -n MnQ\aAD[1]n I-n rnQ\aAA -n rnQ\aABn n B3n    <0>0 !n^ <0> >0`A3n    <0>0 !n^ <0> >0`$0n  <0>dB3    <0>0 !^ <0> >0`A3    <0>0 !^ <0> >0`$0  <0>d0 !"7 <0> l0` ``  `` `m960 !#7 <0> l0` ``  `` `m950 !$8 <0> l0` ``  `` `m860 !%8 <0> l0` ``  `` `m85E8nQ\aACE8nnQ\aAD /#Intersect Klein Circle and Klein Line Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Klein line defined by two points, Klein circle defined by its center and a point on the circle Returns: The point(s) lying on both the line and the circle. Since Klein circles are ellipses drawn as loci (which cannot be intersected in current versions of Sketchpad), this script maps the Klein disk isomorphically onto the Poincare disk, where "circles are circles," finds the intersection points there, then maps the results back to the Klein disk. (See _Euclidean and Non-Euclidean Geometries_, 3rd Ed. by Greenberg, p. 236.) Author: Steve Szydlik/January 2001{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$*LIntersect Klein circle and Klein segmentS Arial  &nQ\aA K-Disk Center &nQ\aA K-Disk Radius6nQ\aAendpoint #1 of Klein segmente6nQ\aAendpoint #2 of Klein segmente0nQ\aAcenter of Klein circleg N:(nQ\aAon Klein circle?'?nQ\aAS[1]e訚?? @ ? nQ\aAjn $gQ@ Klein Diskr -g  &  P-Disk radius    B3    <0>A3    <0>A3    <0>B3    <0>B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 !^ <0>um >0`B3    <0>0 !^ <0>um >0`A3    <0>0 ! ^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`0 !^ <0>um >0`  & K-Disk radius%0 6  <0>r%0 6  <0>r0 !(&)'~ <0>O ^0 ``  ``  m530 !,*-+~ <0>O ^0 ``  ``  m400 !0.1/~ <0>O ^0 ``  ``  m200 !4253~ <0>O ^0 ``  ``  m70 !9&' <0> j0` ` ``  ``  m570 !:*+ <0> j0` ` ``  ``  m440 !;./ <0> j0` ` ``  ``  m240 !<!23 <0> j0` ` ``  ``  m110 !9=' <0>O d0` ``  `` `m590 !9=& <0>O d0` ``  `` `m580 !:>+ <0>O d0` ``  `` `m460 !:>* <0>O d0` ``  `` `m450 !;?/ <0>O d0` ``  `` `m260 !;?. <0>O d0` ``  `` `m250 !<@3 <0>O d0` ``  `` `m130 !<@!2 <0>O d0` ``  `` `m12E8BA P'E8DC O'E8FE D'E8HG C'?  J LK@  L - M nQ\aAE -n M nQ\aAF nNnQ\aAG/n LOnQ\aAk? n QPn PQ$An濘53^?) F@adiu/^ RNnQ\aAl -n S nQ\aAH? n UP? n UJ? n QUn W nZnQ\aAI/n W[nQ\aAm? n \P -n O]nQ\aAJ -n ^YnQ\aAKn _L? n Q` nanQ\aAL -n bXnQ\aAM/n canQ\aAn? n \d -n eVnQ\aAN -n fTnQ\aAO,nLgKnQ\aAon g_? n Ihn Ihn h -n ijnQ\aAP/n hlnQ\aApn hlnQ\aAc[1] nmnQ\aAQQ CnKnL - opnQ\aARn qh -n  tnQ\aASn hu -n ovnQ\aAT -n kvnQ\aAU? n xs.n wynQ\aAq -n kznQ\aAVn { -n M|nQ\aAW? n }hn }h -n ~vnQ\aAX/n hnQ\aArn hnQ\aAc[2] -n vnQ\aAY -n nQ\aAZ? n .n nQ\aAs -n ~nQ\aAA[1]n I nnQ\aAB[1]/n nQ\aAt -n MnQ\aAC[1]n I-n rnQ\aAA -n rnQ\aABn n B3n    <0>0 !n^ <0> >0`A3n    <0>0 !n^ <0> >0`$0n  <0>dB3    <0>0 !^ <0> >0`A3    <0>0 !^ <0> >0`$0  <0>d0 !"7 <0> l0` ``  `` `m860 !#7 <0> l0` ``  `` `m850 !$8 <0> l0` ``  `` `m760 !%8 <0> l0` ``  `` `m75E8nQ\aACE8nnQ\aAD /#Intersect Klein Circle and Klein Segment Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Klein segment defined by two endpoints, Klein circle defined by its center and a point on the circle Returns: The point(s) lying on both the segment and the circle. Since Klein circles are ellipses drawn as loci (which cannot be intersected in current versions of Sketchpad), this script maps the Klein disk isomorphically onto the Poincare disk, where "circles are circles," finds the intersection points there, then maps the results back to the Klein disk. (See _Euclidean and Non-Euclidean Geometries_, 3rd Ed. by Greenberg, p. 236.) Author: Steve Szydlik/January 2001 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$ LKlein pole Arial &nQ\aA K-Disk Center&nQ\aA K-Disk Radius0nQ\aAPoint #2 on Klein linet*nQ\aA#1 on Klein line ?   $gQ@ Klein Disk  -g nQ\aAA -n nQ\aABn n n /n  nQ\aAj/n  nQ\aAk -n  nQ\aAC/#Klein Pole Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Two points defining a Klein line m, i.e. a chord m of the defining circle of the Klein disk. Returns: The pole of m, i.e. the point of intersection of the tangents to the defining circle of the Klein disk at the endpoints. Author: Steve Szydlik/December 2001 Revised for GSP 4.0 in February 2003p{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein midpoint Arial &nQ\aA K-Disk Center &nQ\aA K-Disk Radius$nQ\aA Endpoint #2$nQ\aA Endpoint #1  ?   $gQ@ Klein Disk -g nQ\aAA -n nQ\aABn   -n  nQ\aAC -n  nQ\aADn  n  /n  nQ\aAj/n  nQ\aAk -n nQ\aAE? n ? n  -n nQ\aAF -n nQ\aAG -n nQ\aAH -n nQ\aAIn n  -n nQ\aAJj/#Klein Midpoint Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points A, B in Klein Disk Returns: The midpoint M of segment AB, so that M lies on AB and AM is congruent (in the sense of Klein) to BM. Segment AB is drawn in this construction. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$LKlein point reflection Arial l&nQ\aA K-Disk Center&nQ\aA K-Disk Radius ӖSd@͛N=d@@.nQ\aAM on reflecting lined=$v@؞I@@*nQ\aAA to be reflected ӖSd@͛N=d@? @ @ $gQ@ Klein Diskl -g nQ\aAA -n nQ\aABn ?t -  nQ\aAC -n  nQ\aADn  n  /n  nQ\aAj/n  nQ\aAk -n  Q?  ?n?   -  Sig'  - nQ\aAE -n nQ\aAF? n n ?m -  Omega?   -  A'g/#Klein Point Reflection Setting: Klein Disk defined by points "K-Disk Center" and "K-Disk Radius" Given: Points M, A in Klein Disk Returns: The image A' of point A upon reflection in the line m which is perpendicular (in the Klein sense) to segment AM. In other words, AA' is perpendicular to M and AM is congruent (in the Klein sense) to A'M. Both lines through M are drawn in this construction. Author: Steve Szydlik/July 2000 Revised for GSP 4.0 in February 2003{A tool in this document was created with a newer version of Sketchpad and cannot be read. The document will open as a copy.aA tool in this document was created with a newer version of Sketchpad and may not work correctly.$(LMap Klein point to Poincare Disk pointi Arial  &nQ\aA K-Disk Center::&nQ\aA K-Disk Radius6nQ\aAto be mapped to Poincare Diskn drop perpendic'cnQ\aAS[1]  $gQ@ Klein DiskdA3g    <0>B3g    <0>A3g    <0>B3g    <0>B3g    <0>0 !g ^ <0>um >0`A3g    <0>0 !g ^ <0>um >0`0 !g^ <0>um >0`0 !g^ k<0>um >0`0 !g^ R<0>um >0`0 !g ^ 9<0>um >0`0 !g~ <0>O ^0 ``  ``  m470 !   <0> j0` ` ``  ``  m510 !  ,<0>O d0` ``  `` `m530 !  <0>O d0` ``  `` `m52E8nQ\aAA