Todays Program

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Todays Program

• Surfaces of Revolution.
• Analytic Description of Surfaces of Revolution.
• Quadrics of Revolution.
• Classification of Quadrics, Parameterization and
  Examples.
• General Quadrics.
• Classification of Quadrics, Parameterization and
  Examples.
• Sample of the Exam Example

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Quadrics
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Surfaces of Revolution

• Axis of revolution (o z).
• Curve k (planar or spatial).
• A point X on the curve k.
• Distance of the point X from the axis o radius.
• Circle (in the plane perpendicular to the axis o)
  with the centre on the axis o and the radius r.
• Circles through all points of the curve k.
• Meridian section of the surface of revolution
  with a plane containing the axis o.
• Surface of revolution is union of all circles.
• Contour envelope of the generating circles.

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Analytic Description of Surfaces of Revolution

• Parametric form (axis of revolution z)

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Analytic Description of Surfaces of Revolution

• Non-Parametric form (implicit) (axis of
  revolution z)

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Analytic Description - Summary
Basic position o z, conic section
S/V0,0,0, F(x,z)0 or X(t)x(t),0,z(t)

• Parametric equation
• X(t,u)x(t)cosu x(t)sinu z(t),u??0,2p?,t?J

• Implicit equation

Shifted axis of revolution oz,
S/V0,0,0?m,n,p

• Parametric equation
• X(t,u)mx(t)cosu nx(t)sinu
  pz(t),u??0,2p?,t?J

• Implicit equation

Changed axis of revolution oy, conic section
S/V0,0,0, F(y,z)0, X(t)0,y(t),z(t)

• Parametric equation
• X(t,u)z(t)cosu y(t) z(t)sinu,u??0,2p?,t?J

• Implicit equation

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Quadrics of Revolution.
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Sphere

• Revolving of a circle about its any diameter.
• Sections circle, point.

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Pavilion USA, Worlds Fair, Montreal, 1967
Motion-picture Theatre "Deoda" (Screen 1000 m2,
70 m height
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Opera-house, Sydney, Australia
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Ellipsoid

• Revolving of an ellipse about its axis.
• Prolate axis o major axis
• Oblate axis minor axis
• Sections circle, ellipse, point

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National Statuary Hall, United States Capitol, USA
John Quincy Adams 6th president of USA (1825
1829)
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Paraboloid

• Revolving of a parabola about its axis.
• Sections circle, ellipse, parabola, point

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Parabolic Antennas, Hand lamps, Roofing
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Hyperboloid of One Sheet
Revolving of a hyperbola about its minor
axis. Sections circle, ellipse, parabola,
hyperbola, intersecting straight lines, parallel
straight lines, point
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Hypoid Gearing
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London Bridge over Corporation Street
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Dukovany Cooling Tower
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Hyperboloid of Two Sheets
Revolving of a hyperbola about its major
axis. Sections circle, ellipse, parabola,
hyperbola, point Application altimetry (shape of
the geoid in sea areas)
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Right Cylinder
Revolving of a straight line that is parallel to
the axis of the revolution. Sections circle,
ellipse, straight line, parallel straight lines,
point Application columns, pipelines ...
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Right Cone
Revolving of a straight line that intersects the
axis of the revolution. Sections circle,
ellipse, parabola, hyperbola, straight lines,
intersecting straight lines, point Application
air conditioning, transition surface ...
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General Quadrics
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General Quadrics

• Elliptic quadrics derived from the quadrics of
  revolution by proper scale transformation.
• Ellipsoid, elliptic paraboloid, elliptic
  hyperboloids, elliptic cone, elliptic cylinder.

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General Quadrics

• Cylinders Surface formed by set of parallel
  lines that go through points of a conic section.
  The conic section is named curved director of the
  cylinder.

Parabolic Cylinder lateral straight lines z
Parabolic Cylinder lateral straight lines x
Hyperbolic Cylinder lateral straight lines z
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General Quadrics

• Singular Quadrics two intersecting planes, two
  parallel planes, one plane, point.

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General Quadrics

• HP - surface
• Translation surface (two parabolas).
• Warped surface (warped quadrilateral ABCD)

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F. Calatrava, 1982, Oceanographic Museum, Spain
Church, Prague
Felix CandelaLos Manantiales
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Sample
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• Calculation, Sketching, Determination from
  Implicit Equation
• Problem Find the type, important points and
  basic parameters of the given general quadric.
  Sketch it in an oblique projection.
• Common procedure
• Completing squares and other modifications to
  transform the equation into the form resembling
  the standard form (see the list).
• Two questions
• Are there all variables in the equation?
• Are all variables squared?
• If there are three different denominators then
  the surface is not of revolution.
• If there are two identical denominators of the
  fractions with the same signs then the surface is
  of revolution.
• Sketching
• Draw important points, lines, curves which are
  cross-sections with some coordinate planes.

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• Problem Find the type, important points and
  basic parameters of the given general quadric.
  Sketch it in an oblique projection.