Three-dimensional shapes
description
Transcript of Three-dimensional shapes
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Three-dimensional shapes
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Hyperboloid of one sheet
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In the real world...
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Paraboloid
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In the real world...
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What 3D shape is this?
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Ruled surface around a prolate cycloid
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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
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What 3D shape is this??
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Ruled surface around an epicycloid
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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.
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What 3D shape is this?
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Ruled surface constructed around a cardioid
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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
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What 3D shape is this?
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Ruled surface constructed around a ‘bowtie curve’
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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])
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What 3D shape is this?
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Solid Pacman
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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.
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What 3D shape is this?
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Ruled surface constructed around an 8-petal flower
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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]
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What 3D shape is this?
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Ruled surface constructed around a ‘spring curve’
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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])
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What 3D shape is this?
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Ruled surface constructed around an ‘8-curve’
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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
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What 3D shape is this?
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Figure of the lemniscate of Bernoulli
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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)
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What 3D shape is this?
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Figure of a “folium de Descartes”
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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
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What 3D shape is this?
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Figure of a “folium de Kepler”
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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
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What 3D shape is this?
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Figure of a “butterfly” curve
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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))
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What 3D shape is this?
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Figure of an 8-tooth cog
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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
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What 3D shape is this?
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Figure of a “pyriform” plane curve
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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
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What 3D shape is this?
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Figure of a “lituus” plane
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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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What 3D shape is this?
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Figure of Nielsen’s spiral
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Description
A ruled surface formed around the Nielsen spiral, a plane curve parametrically represented by:
f[a](u) = (aCosIntegral[u],aSinIntegral[u])
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What 3D shape is this?
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Figure of a “scarab” curve
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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])
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What 3D shape is this?
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Figure of a diamond curve
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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])
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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.
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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:
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Carlos V Palace
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CARLOS V PALACE MADE WITH THE CABRI 3D PROGRAM
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OTHER FIGURES …
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Merry go round
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Snow man
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Trampoline
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The fountain
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A trip on a boat
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Bubblegum
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Moebius strip
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Submerged icosahedron
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The shadow
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Multiple pendulum
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The Schools participating in the Ne.M.O. project
are the following:
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PROYECT Ne.M.o.
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I.E.S “Arabuleila”Cúllar Vega Granada (Spain)
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Istituto Tecnico Commerciale e Per il Turismo “Feliciano Scarpellini”
Foligno (Italy)
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Istituto Comprensivo Statale “Monte Grappa” Bussero (Italy)
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Lycée Couffignal Strasburgo (France)
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Kiuruveden Lukio Kiuruvesi (Finland)
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It happened first in Foligno
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Then in Kiuruvesi
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Thirdly inStrasburgo
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And it is happening now
inCúllar Vega
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CO-PRODUCER: MANUEL QUESADA
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EXECUTIVE PRODUCER: RAFAEL BLASCO
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AFTER OF PROYECT NE.M.O.
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PRODUCED BY: PACO NAVARRO
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TO BE CONTINUED
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• GRACIAS A TODOS• GRACIE A TUTTI• THANK YOU VERY MUCH• MERCI A TOUS
• KIITOKSET KAIKILLE• THE END