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Three-dimensional shapes

Three-dimensional shapes. Hyperboloid of one sheet. In the real world. Paraboloid. In the real world. What 3D shape is this?. Ruled surface around a prolate cycloid. Description. Ruled surface constructed around a prolate cycloid , plane curve parameterized by:

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Three-dimensional shapes

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  1. Three-dimensional shapes

  2. Hyperboloid of one sheet

  3. In the real world...

  4. Paraboloid

  5. In the real world...

  6. What 3D shape is this?

  7. Ruledsurfacearound a prolatecycloid

  8. Description Ruled surface constructed around a prolate cycloid, plane curve parameterized by: f[a,b](u) = (a u - b Sin[u],a - b Cos[u])   This curve is the geometric plot of the points on the plane which describe a circumference of radius b when a circumference cocentric of radius a turns without slipping along a fixed straight line, where a<b

  9. What 3D shape is this??

  10. Ruled surface around an epicycloid

  11. Description Ruled surface constructed around an epicicloide, plane curve parametrized by: f(u) = ((a+b)Cos[u] - bCos[((a+b)/b)u], (a+b)Sin[u] - bSin[((a+b)/b)u])   Parameterized curve which describes a point P with a circumference of radius b which revolves around another circumference with radius a.

  12. What 3D shape is this?

  13. Ruled surface constructed around a cardioid

  14. Description Ruled surface constructed around a cardioid, plane curve parameterized by: f[a](u) = (2 a Cos[u](1+Cos[u]), 2 a Sin[u](1+Cos[u])) The implicit equation of the cardioid is: and its polar equation

  15. What 3D shape is this?

  16. Ruledsurfaceconstructedaround a ‘bowtie curve’

  17. Description Ruled surface constructed around a “bowtie", a plane curve parameterized by: f[a,b](u) = (a(1+Cos[u]2)Sin[u], (b+Sin[u]2)Cos[u])

  18. What 3D shape is this?

  19. Solid Pacman

  20. Description ‘Fun’ constructed around a pacman curve, a plane curve whose form is reminiscent of the popular video game ‘pacman’. This ‘solid’ form has been created by means of the following parameterization :      f[n](q,a) = (Cos[q](Cos[q]n + a),Sin[q](Cos[q]n + a),pm(1 - a)/2) where pm takes the values 1 y -1, y a varies between 0 and 1.

  21. What 3D shape is this?

  22. Ruledsurfaceconstructedaroundan 8-petal flower

  23. Description Ruled surface constructed around a flower of 8 petals, plane curve parameterized by: f[n,a](u) = (a Cos[n u]Cos[u],a Cos[n u]Sin[u]) We create a flower of n petals if n is odd, and of 2n petals if n is even. The polar equation of the flower is: r = a Cos[n q]

  24. What 3D shape is this?

  25. Ruledsurfaceconstructedaround a ‘spring curve’

  26. Description Ruled surface constructed around a ‘spring’ curve, a plane curve parameterized by: f[a,b,c](u) = (aCos[u], aCos[c]*Sin[u] + buSin[c])

  27. What 3D shape is this?

  28. Ruled surface constructed around an ‘8-curve’

  29. Description Ruled surface constructed around an ‘8-curve’, a plan curve parameterized by:f(u) = (Sin[u],Sin[u]Cos[u]) Ruled surface constructed around an ‘8-curve’, a plane curve given in implicit form by the equation: : y2 - c2 a2 x4 + c2 x6 =0

  30. What 3D shape is this?

  31. Figure of thelemniscate of Bernoulli

  32. Description A ruled surface formed around a lemniscate of Bernoulli, a plane curve with the parametric representation of: f[a](u) = (a Cos[u]/(1+Sin[u]2),a Sin[u]Cos[u]/(1+Sin[u]2)) The implicit equation of the Bernoulli lemniscate is: (x2+y2)2 = a2(x2-y2)

  33. What 3D shape is this?

  34. Figure of a “folium de Descartes”

  35. Description A ruled surface formed around a “Folium of Descartes", a plane curve parametrically represented by: f(u) = (3u/(1 + u3), 3u2/(1 + u3)) The implicit equation of the Folium of Descartes is: x3 + y3 - 3 x y = 0

  36. What 3D shape is this?

  37. Figure of a “folium de Kepler”

  38. Description A ruled surface formed around a “Folium de Kepler", a plane curve with an implicit equation of:     ((x - b) 2 + y2)(x(x-b) + y2) - 4a(x - b)y2 = 0

  39. What 3D shape is this?

  40. Figure of a “butterfly” curve

  41. Description A ruled surface formed around a “butterfly” curve, one of the various curves found in the catastrophe theory, with a parametric equation of: f[a,c](u) = (c(8 a t3 + 24 t5),c(-6 a t2 - 15 t4))

  42. What 3D shape is this?

  43. Figure of an 8-tooth cog

  44. Description A ruled surface formed around an “8-tooth cog”, a plane curve that is well known in the catastrophe theory, expressed with the implicit form of: x4 - 6 x2y2 + y4 = a

  45. What 3D shape is this?

  46. Figure of a “pyriform” plane curve

  47. Description A ruled surface formed around a pyriform curve, a plane curve parametrically represented by: f[a,b](u) = (a(1+Sin[u]), bCos[u](1+Sin[u]) ) The implicit equation of the pyriform curve is: a4y2 - b2x3 (2 a - x) = 0

  48. What 3D shape is this?

  49. Figure of a “lituus” plane

  50. Description A ruled surface formed around a “lituus”, a plane curve parametrically represented by: f[a](u) = (a u/(u^2)(3/4)Cos[Sqrt[u2]], a u/(u^2)(3/4)Sin[Sqrt[u2]]) Polar equation: r = a q (1/2) This curve is the geometric plot of points P where the square of the distance between P and the origin is inversely proportional to the angle that P forms with the horizontal axis.

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