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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
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
Description
What 3D shape is this??
Ruled surface around an epicycloid
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.
What 3D shape is this?
Ruled surface constructed around a cardioid
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
What 3D shape is this?
Ruled surface constructed around a ‘bowtie curve’
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])
What 3D shape is this?
Solid Pacman
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.
What 3D shape is this?
Ruled surface constructed around an 8-petal flower
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]
What 3D shape is this?
Ruled surface constructed around a ‘spring curve’
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])
What 3D shape is this?
Ruled surface constructed around an ‘8-curve’
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
What 3D shape is this?
Figure of the lemniscate of Bernoulli
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)
What 3D shape is this?
Figure of a “folium de Descartes”
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
What 3D shape is this?
Figure of a “folium de Kepler”
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
What 3D shape is this?
Figure of a “butterfly” curve
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))
What 3D shape is this?
Figure of an 8-tooth cog
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
What 3D shape is this?
Figure of a “pyriform” plane curve
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
What 3D shape is this?
Figure of a “lituus” plane
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.
What 3D shape is this?
Figure of Nielsen’s spiral
Description
A ruled surface formed around the Nielsen spiral, a plane curve parametrically represented by:
f[a](u) = (aCosIntegral[u],aSinIntegral[u])
What 3D shape is this?
Figure of a “scarab” curve
Description
Ruled surface formed around a “scarab” curve, a plane curve parametrically represented by:
f[a,b](u) = ((aCos[2u] - bCos[u])Cos[u], (aCos[2u] - bCos[u])Sin[u])
What 3D shape is this?
Figure of a diamond curve
Description A ruled surface formed around a diamond curve, a plane curve parametrically represented by:
f[n,a,b](u) = (a (Cos[u]2)(n-1)/2 Cos[u], b (Sin[u]2)(n-1)/2 Sin[u])
In architecture there are many mathematical-geometrical elements, such
as friezes, mosaics, cones, symmetries, curved surfaces, arches , etc.
A very clear sample of that can be found in Granada, in the Alhambra and Generalife.
Let’s watch now a 3D sample of these world marvels.
One of the tools that can be used to make 3D figures is computer science programing.
To show what I mean, I’d like to refer to the use of Cabri. Now, we’ll watch some figures made with that program:
Carlos V Palace
CARLOS V PALACE MADE WITH THE CABRI 3D PROGRAM
OTHER FIGURES …
Merry go round
Snow man
Trampoline
The fountain
A trip on a boat
Bubblegum
Moebius strip
Submerged icosahedron
The shadow
Multiple pendulum
The Schools participating in the Ne.M.O. project
are the following:
PROYECT Ne.M.o.
I.E.S “Arabuleila”Cúllar Vega Granada (Spain)
Istituto Tecnico Commerciale e Per il Turismo “Feliciano Scarpellini”
Foligno (Italy)
Istituto Comprensivo Statale “Monte Grappa” Bussero (Italy)
Lycée Couffignal Strasburgo (France)
Kiuruveden Lukio Kiuruvesi (Finland)
It happened first in Foligno
Then in Kiuruvesi
Thirdly inStrasburgo
And it is happening now
inCúllar Vega
CO-PRODUCER: MANUEL QUESADA
EXECUTIVE PRODUCER: RAFAEL BLASCO
AFTER OF PROYECT NE.M.O.
PRODUCED BY: PACO NAVARRO
TO BE CONTINUED
• GRACIAS A TODOS• GRACIE A TUTTI• THANK YOU VERY MUCH• MERCI A TOUS
• KIITOKSET KAIKILLE• THE END
