3 9KcapmL<pZz]9 p$V KQ Q R Rֻ @{K ֻ  {   AKE 1999 UGA ֻ?*CFBC7ֻ1?.`wCC;3 ?2CCC>(6 {K8I8` p$V TZZ[[p.sx͐.p.s  ,Caustic and diacaustic of Equiangular Spiral|0|&̰p.0sx}kx}Hx͐.sx~ #}U<9}b}8b 0@ |%y@t p$V OQx۠t3]p _pt8 T HCC  "2 p$V AQx۠t3s9p _pt8 T HAC  % 8 p$V MQx۠t3_p _pt8 T HD&C {A  p$V OQx۠t3p _pt8 T HCC{9 p$V KQ Q R R#' 0sx #' s  x#AzThe blue curve is an equiangular spiral. The envelope of blue lines forms the catacaustic of the spiral with respect to point O. The envelope of pink lines is its diacaustic. Caustic is a method of deriving a new curve based on a given curve and a point. Given a curve C and a fixed point O (the light source), catacaustic/diacaustic is the envelope of light rays coming from O and reflected/refracted from the curve C. Drag point O to change its position. Drag point D to change the constant angle of the spiral. Drag point K to adjust the position of point P on the curve. Drag point M to increase/decrease range of curves ploted.xHq`|}xxxK`H?Px!|N |!`|? x|~x|#x|+x| %A p$V AYYZZ8psx 8PsCC " p$V BCQx۠t3<p _pt8 T HAC#  B p$V BQx۠t3qp _pt8 T HCC {%T  p$V jH,@hx8K0 ;|d.x8K0} ; |d.x |CCCC? "Cx  p$V kxxK`8<|8|Cx8(Ag8bT:|c.|iN x8Kҕ`Tq>x~$x8KACB?55CCAC#A@l?  %?Q p$V KQx۠t3^5p _pt8 T HD;C DI? p$V! DQx۠t3Sp _pt8 T HBbEC+ 0#<y p$V m2&8sxx$8T 6!@|.`8|` OK = c.V`R|K`8` K5`|dx(ADD8BbEC+AC?3g p$V Animate 06sx 06soA++W` p$V# p1!`xW<8X|"K`x<T|<8K`; <|ACB?U (uCCBbDC+A@h}l?=#I|U p$V m3`8.8 .xxx8K=xxx8K@xxx8   ln{(:AD}{!:*}tan{(:m{!:A}DBE} =