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3D Abe: Silhouette = Surface of REV Cross Section

3D Abe: Silhouette = Surface of REV Cross Section
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2019-03-28 23:36
FunctionCoordinate GeometrySolid GeometryGraphing
In collections GeoGebra 3D & AR: Geometry Resources & Challenges
This example demonstrates the generation of a 3D surface from Abraham Lincoln's silhouette using a Revolution around an axis, highlighting the relationship between the surface and its cross-section. Key mathematical concepts include solid geometry, functions, rotation, and cross-sectional analysis. The construction utilizes GeoGebra commands such as Corner, Midpoint, Rotate, Function, Surface, Point, OrthogonalLine, Line, Distance, and Translate to model the shape accurately. Suitable for high school and advanced students studying 3D geometry and calculus applications.

GeoGebra Commands & Parameters

  1. Corner(pic1, 4) → C
  2. Corner(pic1, 3) → D
  3. Midpoint(A, D) → E
  4. Rotate(pic1, π / 2, E) → pic1'
  5. Function({1.5248963794530292, 5.051539539125208, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.992057427589251, 1.9872981250795718, 1.982538822569893, 1.977779520060214, 1.968260915040856, 1.958742310021498, 1.94922370500214, 1.9397050999827818, 1.930186494963424, 1.920667889944066, 1.8778341673569547, 1.8635562598279176, 1.8492783522988807, 1.8350004447698436, 1.8135835834762881, 1.7921667221827324, 1.7826481171633743, 1.7731295121440163, 1.7636109071246584, 1.7540923021053005, 1.7445736970859425, 1.7350550920665844, 1.7302957895569053, 1.7255364870472263, 1.7207771845375472, 1.7160178820278682, 1.7112585795181892, 1.7064992770085101, 1.701739974498831, 1.6969806719891523, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.6922213694794732, 1.701739974498831, 1.7112585795181892, 1.7207771845375472, 1.730295789556905, 1.7398143945762632, 1.7493329995956213, 1.7588516046149794, 1.7683702096343374, 1.7778888146536953, 1.7874074196730534, 1.7969260246924115, 1.8064446297117696, 1.8112039322214486, 1.8159632347311276, 1.8207225372408065, 1.8254818397504853, 1.8302411422601645, 1.8350004447698436, 1.8397597472795226, 1.8445190497892014, 1.8492783522988807, 1.8587969573182388, 1.8683155623375967, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8778341673569547, 1.8825934698666338, 1.8873527723763128, 1.8921120748859919, 1.896871377395671, 1.9016306799053497, 1.9063899824150288, 1.9111492849247078, 1.9159085874343866, 1.920667889944066, 1.930186494963424, 1.9397050999827818, 1.94922370500214, 1.958742310021498, 1.968260915040856, 1.977779520060214, 1.9991963813537696, 2.0206132426473253, 2.042030103940881, 2.0634469652344363, 2.084863826527992, 2.1062806878215476, 2.1157992928409057, 2.1253178978602634, 2.1348365028796215, 2.1491144104086586, 2.1633923179376957, 2.177670225466733, 2.182429527976412, 2.187188830486091, 2.19194813299577, 2.201466738015128, 2.2109853430344857, 2.2205039480538438, 2.225263250563523, 2.230022553073202, 2.234781855582881, 2.23954115809256, 2.244300460602239, 2.249059763111918, 2.249059763111918, 2.249059763111918, 2.249059763111918, 2.249059763111918, 2.23954115809256, 2.230022553073202, 2.2205039480538438, 2.177670225466733, 2.172910922957054, 2.1681516204473748, 2.1633923179376957, 2.14197545664414, 2.1205585953505843, 2.099141734057029, 2.0777248727634734, 2.034891150176362, 1.992057427589251, 1.94922370500214, 1.9063899824150288, 1.8635562598279176, 1.8540376548085595, 1.8445190497892014, 1.8350004447698436, 1.8135835834762881, 1.7921667221827324, 1.7874074196730534, 1.7826481171633743, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7778888146536953, 1.7874074196730534, 1.7969260246924115, 1.8064446297117696, 1.8112039322214486, 1.8159632347311276, 1.8207225372408065, 1.8254818397504853, 1.8302411422601645, 1.8350004447698436, 1.8397597472795226, 1.8445190497892014, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8492783522988807, 1.8445190497892014, 1.8397597472795226, 1.8350004447698436, 1.8302411422601645, 1.8254818397504853, 1.8207225372408065, 1.8159632347311276, 1.8112039322214486, 1.8064446297117696, 1.7921667221827324, 1.7778888146536955, 1.7636109071246584, 1.742194045831103, 1.7207771845375472, 1.6993603232439916, 1.677943461950436, 1.668424856931078, 1.65890625191172, 1.649387646892362, 1.606553924305251, 1.5637202017181397, 1.5208864791310286, 1.4780527565439174, 1.4352190339568063, 1.3923853113696951, 1.349551588782584, 1.3067178661954728, 1.2638841436083617, 1.2067725134922136, 1.1853556521986581, 1.1639387909051024, 1.1544201858857444, 1.1449015808663863, 1.1353829758470284, 1.1258643708276703, 1.1163457658083122, 1.1068271607889542, 1.1020678582792751, 1.0973085557695963, 1.0925492532599173, 1.0925492532599173, 1.0925492532599173}) → f
  6. Surface(f, n, Line[E, H]) → b
  7. Point(f) → J
  8. Rotate(b, π / 2, J, Plane[J, E, I]) → b'
  9. Rotate(pic1', ((-π)) / 2, J) → pic1''
  10. Rotate(f, ((-π)) / 2, J) → f_2
  11. OrthogonalLine(J, z = 0) → h
  12. Rotate(f, n, Line[E, H]) → f_1
  13. Rotate(f_1, π / 2, J, Plane[J, E, I]) → a
  14. Line(J, J + (0, 0, 1)) → j
  15. Line(N, O) → k
  16. Line(E, H) → l
  17. Rotate(Line[E, H], π / 2, J, Plane[J, E, I]) → m
  18. Distance(J, m) → c
  19. Translate(J, Vector[((-c), 0, 0)]) → J'