Catacaustic

Roman Chijner, 04.03.2014

Inhaltsverzeichnis

  1. Catacaustic
    1. Catacaustic of a Hyperbola - Radial Rays
    2. Duté hyperb., elipt., kulové a parab. - zdroj v nekonečnu
    3. Katakaustik einer Ellipse
    4. Un café ?
    5. Reflected beam at ellipse
    6. Duté parabolické - zdroj v konečné vzd. i v nekonečnu
    7. Riflessione al biliardo
    8. 스트링아트2
    9. 스트링아트
    10. Ellipse as Envelope
    11. Hyperbola as Envelope
    12. Capareda curve
    13. Tennis ball
    14. Constructing the Catacaustic of a Circle for Radial Rays
    15. Flight of an Insect - Radial Source of Light
    16. 스트링아트2
    17. Caustic of a cycloid.
    18. Нефроидa 6.
    19. Circle Catacaustic
    20. Caustic
    21. Constructing the Catacaustic of an Ellipse -Parallel Rays
    22. The Catacaustic of a Hyperbola - Parallel Rays
    23. deltoid_caustics
    24. ellipse_Caustic
    25. equiangular_spiral_caustic
    26. catenary and evolute
    27. Evolute of hypocycloid
    28. hypocycloid pedal curve
    29. hypocycloid
    30. Rétrogradation de Mars
    31. Multiplication Wheel:
    32. Circular Reasoning
    33. Multiplication wheel
    34. rebonds entre ellipses
    35. Kardioide als Kaustik von Punktlicht am Kreis
    36. Kaustik der Kardioide ist Nephroide
    37. Kardioide als Hüllkurve von Kreisen
    38. Nephroide als Kaustik parallelen Lichtes am Kreis
    39. Quadrik-Muster
    40. Kreis-Punktzuordnung
    41. Times Tables
    42. 包絡線：橢圓
    43. 包絡線：雙曲線
    44. Copy of Catacaustic of a Hyperbola - Radial Rays
    45. Copy of Catacaustic of a Parabola - Tschirnhausen Cubic
    46. Constructing the Catacaustic of a Circle for Parallel Rays
    47. Curvas cáusticas en la deltoide y enla astroide
    48. Curvas cáusticas en la elipse
    49. Soustraction ?
    50. Modulare Multiplikationstafel
    51. Cardioid and nephroid catacaustics
    52. osculating circles of hypocycloid
    53. Càustica en un cercle - I
    54. Modular Times Table on a Square
    55. Modular Times Table Between Two Sine Curves
    56. Modular Times Table
    57. Circle Multiplication Table
    58. ttt
    59. Závod s plácnutím o eliptickou zeď z F_1 do F_2
    60. Vieleck-Rotation
    61. Katakaustik einer Hyperbel
    62. Katakaustik einer Parabel
    63. Katakaustik im Kreis
    64. Modulare Rechnung
    65. The involute of the evolute of a parabola is a curve parallel to the parabola
    66. Elliptic billiard table
    67. Parabola as Envelope
    68. Caustic 1. Heart.
    69. Fun with lines and modular arithmetic
    70. Pickover's Mygalomorph Patterns
    71. Zykloiden
    72. Complex Powers of the Unit Circle
    73. Parametric Rays
    74. Goormaghtigh's generalization of the Droz-Farny Line
    75. Wallace Simson Line
    76. Càustica en un cercle - II
    77. L'el·lipse plegant paper.
    78. Sequence
    79. epicycloid
    80. Negative Pedal curve of an ellipse
    81. Doppeltes Linienmuster
    82. Envelope
    83. Iterates of a circle map
    84. Movimientos sincrónicos
    85. Giros en el reloj
    86. Conics Papiro
    87. Modular multiplication
    88. Modular arithmetic
    89. Modular times table
    90. Curve Stitching, Point Map
    91. Katakaustik3a_Zip
    92. Curve Stitching in 3D
    93. Curve Stitching in a Pentagon
    94. Wallace-Simson, Deltoid Envelope, Orthopole
    95. curve stitching in 3D
    96. 副産物
    97. フェルマ螺旋
    98. ぐるぐる
    99. Curve Stitching with Rotations and Reflections
    100. Curve Stitching with Rotations
    101. Spiral Patterns.
    102. Spiral Patterns with Shape
    103. Caustica di una conica
    104. Modular Multiplication in a Square with Inversion
    105. A Run for Two - revised
    106. The Doubling Map for Selected Mods
    107. Rolling Circle
    108. Continuous Multiples for Selected Mods
    109. Complex Number Tripling Map
    110. The Tripling Map for Selected Mods
    111. Rolling Circle Tripling Map
    112. Reflecting Light Rays Tripling Map
    113. Reflecting Light Rays
    114. Modular Addition for Selected Mods and Inversion
    115. Reflective Properties of the Conics
    116. Lines Forming Curves
    117. Envelope
    118. beauty of mathematics
    119. Reflexión en la elipse
    120. 스트링 아트(원)
    121. 포물선에서의 빛 반사(Reflection of Light on a Parabola)
    122. 타원에서의 빛 반사(Reflection of Light on a Ellipse)
    123. 포락선 도구활용 쌍곡선
    124. A ray in a corner (between two lines)
    125. Ray in a corner, 2
    126. Reflect a lamp
    127. Epicycloïdes
    128. Présentation
    129. Focus of the parabola
    130. Hyperbel-Spiegel 1
    131. Hyperbel-Spiegel 2
    132. envelope of circles
    133. デルトイドとシムソン線と三角形
    134. Conics as envelopes: Family of ellipses and hyperbolas
    135. Epicicloide. Hilorama
    136. String art, cuadrado
    137. Circles Segments
    138. Dva rotující vektory -- tlumení
    139. Geraden-6-Ecknetz / Satz von GRAF u. SAUER
    140. Modulo-Muster im Kreis
    141. Número de diagonales en un polígono regular.
    142. Mathologer TimesTable
    143. Circular Pattern Stepping V1.1
    144. Doblegant el paper per mostrar l'envolupant
    145. Ellipse - Many Reflections
    146. Animation #6
    147. HILORAMAS
    148. Reflexión en la Parábola
    149. Times Table
    150. Circle to Circle mapping
    151. The dance of the terrestrial planets
    152. Spinning Segments (Epicycloids)
    153. Curvas cáusticas de la esquiral logarítmica o equiangular
    154. Mandala - Cardioid formula on a pentagon
    155. shifted circle - moveable
    156. rotating cardioide mandala
    157. truncated octahedron
    158. MOD-Multiplikation am Kreis
    159. rotating cardioid mandala #3
    160. superimposed triangles
    161. Cardioid formula in a triangle
    162. superimposed square
    163. 6 time cardioid mandala
    164. Kreisschar als siebenzählige Rosette
    165. Hamid Naderi Yeganeh: Fish
    166. 411　楕円の反射
    167. Hüllkurve
    168. Linien-Muster
    169. Billar circular
    170. Nephroide mit Evolute
    171. Vznik nefroidy odvalováním kružnice po kružnici + jako obálka
    172. La flor de Venus
    173. Mars and Jupiter
    174. Duté zrcátko a svazek rovnoběžných paprsků
    175. Kreismuster
    176. VYZRC -- svazek rovnoběžných paprsků, šikmých k optické ose
    177. Kardioide mit Evolute
    178. Kaustika - odkazy
    179. DUZRC: svazek paprsků vycházejících z bodu na ose
    180. DUZRC: KOMPLETNÍ KAUSTIKA!
    181. Rovnice kaustiky a poloha hrotů
    182. DUZRC: svazek paprsků vycházejících z bodu mimo osu
    183. DUZRC: svazek paprsků z bodu mimo osu -- paraxiální aproximace
    184. chi-ball
    185. cardioid - v2
    186. Design mit Std-Min-Zeiger
    187. Seka3
    188. Venus-Earth curve
    189. Two celestial bodies orbiting the sun
  2. 2. Catacaustic- Bilder
    1. Catacaustic- Bilder
    2. Times Tables, Mandelbrot and the Heart of Mathematics

1.

Catacaustic

  • 1. Catacaustic of a Hyperbola - Radial Rays
  • 2. Duté hyperb., elipt., kulové a parab. - zdroj v nekonečnu
  • 3. Katakaustik einer Ellipse
  • 4. Un café ?
  • 5. Reflected beam at ellipse
  • 6. Duté parabolické - zdroj v konečné vzd. i v nekonečnu
  • 7. Riflessione al biliardo
  • 8. 스트링아트2
  • 9. 스트링아트
  • 10. Ellipse as Envelope
  • 11. Hyperbola as Envelope
  • 12. Capareda curve
  • 13. Tennis ball
  • 14. Constructing the Catacaustic of a Circle for Radial Rays
  • 15. Flight of an Insect - Radial Source of Light
  • 16. 스트링아트2
  • 17. Caustic of a cycloid.
  • 18. Нефроидa 6.
  • 19. Circle Catacaustic
  • 20. Caustic
  • 21. Constructing the Catacaustic of an Ellipse -Parallel Rays
  • 22. The Catacaustic of a Hyperbola - Parallel Rays
  • 23. deltoid_caustics
  • 24. ellipse_Caustic
  • 25. equiangular_spiral_caustic
  • 26. catenary and evolute
  • 27. Evolute of hypocycloid
  • 28. hypocycloid pedal curve
  • 29. hypocycloid
  • 30. Rétrogradation de Mars
  • 31. Multiplication Wheel:
  • 32. Circular Reasoning
  • 33. Multiplication wheel
  • 34. rebonds entre ellipses
  • 35. Kardioide als Kaustik von Punktlicht am Kreis
  • 36. Kaustik der Kardioide ist Nephroide
  • 37. Kardioide als Hüllkurve von Kreisen
  • 38. Nephroide als Kaustik parallelen Lichtes am Kreis
  • 39. Quadrik-Muster
  • 40. Kreis-Punktzuordnung
  • 41. Times Tables
  • 42. 包絡線：橢圓
  • 43. 包絡線：雙曲線
  • 44. Copy of Catacaustic of a Hyperbola - Radial Rays
  • 45. Copy of Catacaustic of a Parabola - Tschirnhausen Cubic
  • 46. Constructing the Catacaustic of a Circle for Parallel Rays
  • 47. Curvas cáusticas en la deltoide y enla astroide
  • 48. Curvas cáusticas en la elipse
  • 49. Soustraction ?
  • 50. Modulare Multiplikationstafel
  • 51. Cardioid and nephroid catacaustics
  • 52. osculating circles of hypocycloid
  • 53. Càustica en un cercle - I
  • 54. Modular Times Table on a Square
  • 55. Modular Times Table Between Two Sine Curves
  • 56. Modular Times Table
  • 57. Circle Multiplication Table
  • 58. ttt
  • 59. Závod s plácnutím o eliptickou zeď z F_1 do F_2
  • 60. Vieleck-Rotation
  • 61. Katakaustik einer Hyperbel
  • 62. Katakaustik einer Parabel
  • 63. Katakaustik im Kreis
  • 64. Modulare Rechnung
  • 65. The involute of the evolute of a parabola is a curve parallel to the parabola
  • 66. Elliptic billiard table
  • 67. Parabola as Envelope
  • 68. Caustic 1. Heart.
  • 69. Fun with lines and modular arithmetic
  • 70. Pickover's Mygalomorph Patterns
  • 71. Zykloiden
  • 72. Complex Powers of the Unit Circle
  • 73. Parametric Rays
  • 74. Goormaghtigh's generalization of the Droz-Farny Line
  • 75. Wallace Simson Line
  • 76. Càustica en un cercle - II
  • 77. L'el·lipse plegant paper.
  • 78. Sequence
  • 79. epicycloid
  • 80. Negative Pedal curve of an ellipse
  • 81. Doppeltes Linienmuster
  • 82. Envelope
  • 83. Iterates of a circle map
  • 84. Movimientos sincrónicos
  • 85. Giros en el reloj
  • 86. Conics Papiro
  • 87. Modular multiplication
  • 88. Modular arithmetic
  • 89. Modular times table
  • 90. Curve Stitching, Point Map
  • 91. Katakaustik3a_Zip
  • 92. Curve Stitching in 3D
  • 93. Curve Stitching in a Pentagon
  • 94. Wallace-Simson, Deltoid Envelope, Orthopole
  • 95. curve stitching in 3D
  • 96. 副産物
  • 97. フェルマ螺旋
  • 98. ぐるぐる
  • 99. Curve Stitching with Rotations and Reflections
  • 100. Curve Stitching with Rotations
  • 101. Spiral Patterns.
  • 102. Spiral Patterns with Shape
  • 103. Caustica di una conica
  • 104. Modular Multiplication in a Square with Inversion
  • 105. A Run for Two - revised
  • 106. The Doubling Map for Selected Mods
  • 107. Rolling Circle
  • 108. Continuous Multiples for Selected Mods
  • 109. Complex Number Tripling Map
  • 110. The Tripling Map for Selected Mods
  • 111. Rolling Circle Tripling Map
  • 112. Reflecting Light Rays Tripling Map
  • 113. Reflecting Light Rays
  • 114. Modular Addition for Selected Mods and Inversion
  • 115. Reflective Properties of the Conics
  • 116. Lines Forming Curves
  • 117. Envelope
  • 118. beauty of mathematics
  • 119. Reflexión en la elipse
  • 120. 스트링 아트(원)
  • 121. 포물선에서의 빛 반사(Reflection of Light on a Parabola)
  • 122. 타원에서의 빛 반사(Reflection of Light on a Ellipse)
  • 123. 포락선 도구활용 쌍곡선
  • 124. A ray in a corner (between two lines)
  • 125. Ray in a corner, 2
  • 126. Reflect a lamp
  • 127. Epicycloïdes
  • 128. Présentation
  • 129. Focus of the parabola
  • 130. Hyperbel-Spiegel 1
  • 131. Hyperbel-Spiegel 2
  • 132. envelope of circles
  • 133. デルトイドとシムソン線と三角形
  • 134. Conics as envelopes: Family of ellipses and hyperbolas
  • 135. Epicicloide. Hilorama
  • 136. String art, cuadrado
  • 137. Circles Segments
  • 138. Dva rotující vektory -- tlumení
  • 139. Geraden-6-Ecknetz / Satz von GRAF u. SAUER
  • 140. Modulo-Muster im Kreis
  • 141. Número de diagonales en un polígono regular.
  • 142. Mathologer TimesTable
  • 143. Circular Pattern Stepping V1.1
  • 144. Doblegant el paper per mostrar l'envolupant
  • 145. Ellipse - Many Reflections
  • 146. Animation #6
  • 147. HILORAMAS
  • 148. Reflexión en la Parábola
  • 149. Times Table
  • 150. Circle to Circle mapping
  • 151. The dance of the terrestrial planets
  • 152. Spinning Segments (Epicycloids)
  • 153. Curvas cáusticas de la esquiral logarítmica o equiangular
  • 154. Mandala - Cardioid formula on a pentagon
  • 155. shifted circle - moveable
  • 156. rotating cardioide mandala
  • 157. truncated octahedron
  • 158. MOD-Multiplikation am Kreis
  • 159. rotating cardioid mandala #3
  • 160. superimposed triangles
  • 161. Cardioid formula in a triangle
  • 162. superimposed square
  • 163. 6 time cardioid mandala
  • 164. Kreisschar als siebenzählige Rosette
  • 165. Hamid Naderi Yeganeh: Fish
  • 166. 411　楕円の反射
  • 167. Hüllkurve
  • 168. Linien-Muster
  • 169. Billar circular
  • 170. Nephroide mit Evolute
  • 171. Vznik nefroidy odvalováním kružnice po kružnici + jako obálka
  • 172. La flor de Venus
  • 173. Mars and Jupiter
  • 174. Duté zrcátko a svazek rovnoběžných paprsků
  • 175. Kreismuster
  • 176. VYZRC -- svazek rovnoběžných paprsků, šikmých k optické ose
  • 177. Kardioide mit Evolute
  • 178. Kaustika - odkazy
  • 179. DUZRC: svazek paprsků vycházejících z bodu na ose
  • 180. DUZRC: KOMPLETNÍ KAUSTIKA!
  • 181. Rovnice kaustiky a poloha hrotů
  • 182. DUZRC: svazek paprsků vycházejících z bodu mimo osu
  • 183. DUZRC: svazek paprsků z bodu mimo osu -- paraxiální aproximace
  • 184. chi-ball
  • 185. cardioid - v2
  • 186. Design mit Std-Min-Zeiger
  • 187. Seka3
  • 188. Venus-Earth curve
  • 189. Two celestial bodies orbiting the sun

1.1.

Catacaustic of a Hyperbola - Radial Rays

A light source at the radiant point S produces some radial light rays (in yellow). The rays are reflected by the hyperbola. If the rays have more than one intersection point with the hyperbola, they are reflected at both points. The reflected rays and their continuations are shown in light blue. Definition: The envelope of the reflected rays by the hyperbola is called the [i]catacaustic[/i] of the hyperbola (also called [i]caustic[/i]). [list] [*]Drag the slider - "numberRays" to change the number of light rays. [*]Drag the slider - "d" to change the distance from the center of the hyperbola to the radiant point. [*]Drag the slider -"[math]\alpha[/math]" to rotate the radiant point around the center of the hyperbola. [*]To experiment with the catacaustic, select different values for the distance ([math]c[/math]) from the center of the hyperbola to its foci, and for the distance ([math]a[/math]) from the center of the hyperbola to the vertices of the hyperbola . [/list]
– Irina Boyadzhiev

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2.

2. Catacaustic- Bilder

  • 1. Catacaustic- Bilder
  • 2. Times Tables, Mandelbrot and the Heart of Mathematics

2.1.

Catacaustic- Bilder

– Roman Chijner

2.2.

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Information