Constructive Geometry

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Constructive Geometry

• Jana Vecková
• Department of Mathematics
• Room B-304
• Office hours Wednesday 12-1 p.m.
• veckova_at_mat.fsv.cvut.cz
• http//mat.fsv.cvut.cz/veckova

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Todays programme

• Programme of the course
• lecture, tutorial.
• Introduction to geometry
• definition, what, where, how.
• Parallel projection
• definition, uniqueness.

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Lecture Tutorial

• Cerný Jaroslav Geometry
• (manuscript of FCE CTU, 1996)
• http//mat.fsv.cvut.cz/veckova

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Content of the course

• Projection parallel (Monge projection, oblique
  projection, axonometry), central (perspective)
• Lighting (parallel)
• Helix Helicoidal Surfaces
• Curves calculation, tangent, normal line
• Quadrics parameterisation, sketching,
  presentation in software

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1. Projections
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2. Lighting
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3. Helix Helicoidal Surface
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4. Curves
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5. Quadrics
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Mascot of Faculty of Nuclear Science Temelak
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Pig creating may be funny
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Two Dogs
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Bart Simpson Bee Maja
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Introduction
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Introduction

• Constructive Geometry
• is the science of graphic and numeric
  representation and solution of space problems.
• What?
• Points-Lines-Planes-Curves-Solids-Surfaces ...
• Where?
• Coordinate System
• How?
• Calculation (analytic and differential geometry)
• Construction (projections)

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What

• Pyramid and Cone
• Pyramid director polygon vertex gt pyramidal
  surface
• Cone director curve vertex gt conical surface
• regular pyramid, cone of revolution (right
  circular cone)

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What

• Prism and Cylinder
• Prism director polygon direction gt prismatic
  surface
• Cylinder director curve direction gt
  cylindrical surface
• regular prism, cylinder of revolution (right
  circular cylinder)

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What solids have you already known?
oblique hexagonal prism
cube
octahedron
dodecahedron
regular quadrangular pyramid
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What solids have you already known?
right triangular prism
regular pentangular pyramid
icosahedron
sphere
right circular cone
right circular cylinder
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Where

• Right-handed Cartesian coordinate system
• One fixed point O (origin) and three fixed
  mutually perpendicular lines x, y, z, (axes).
• p(x,y) horizontal (the first) plane
• ?(x,z) vertical / frontal (the second) plane
• µ(y,z) side (the third) plane
• Position of a point M in the coordinate system
• Coordinates are distances of M from three
  coordinate planes.
• xMMM3, M3- side view of M,
• yMMM2, M2- front view of M,
• zMMM1, M1- top view of M,
• MxM, yM, zM.

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Where

• Right-handed Cartesian coordinate system

Left-handed Cartesian coordinate system
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Where Coordinate System

• Note If some coordinates are missing some
  equivalent information must be known instead of
  them.
• Ex1 M2,0,?, M lies on the given cone K, zMgt0
  .
• Ex2 M?,3,?, M lies in plane a x/4 z/21 and
  in plane ß x/2 z/41.
• Ex3 Make up your own example.

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How Projections!

• Projection P is a mapping of space E3 onto plane
  ? called image plane.
• P E3 ? E2 A ? A
• Specific attributes of a projection depend on its
  type. We distinguish two basic types parallel
  and central projection.

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Parallel Projection
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Parallel projection

• Image (projection) plane ?
• Direction line of the parallel projection s (s
  ? ?)
• s ? ? orthogonal projection
• s ? ? oblique projection
• A is the projection of point A.

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Parallel projection - features

• Projection of a point is
• Projection of a line is
• Projection of a plane is

a point.
a line or
a point.
the whole image plane or
a line.
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Parallel projection - features

• Projection of a pair of parallel equal segments
  is a pair of parallel equal segments (if they are
  not parallel to direction s).

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Parallel projection - features

• It keeps ratios on the lines not parallel to the
  direction s.

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Parallel projection - features

• It keeps conjugate diameters.

Conjugate diameters Two diameters are called
conjugate if the tangents at the endpoints of one
are parallel to the other one.
If an ellipse is the image of a circle in a
parallel projection then the images of any
conjugate diameters of the circle are the
conjugate diameters of the ellipse.
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Parallel projection - uniqueness
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Parallel projection - uniqueness
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Parallel projection uniquenessSolutions

• Orthographic Projection onto One Plane (with Spot
  Heights)
• E3 ? (?, ?)
• A ? (A1, kA)

spot height
Disadvantages line 2 points with spot
heights plane 3 points with spot heights solids
(polyhedron) surfaces
maps (contour lines)
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Parallel projection uniquenessSolutions

• Orthographic Projection onto Two Planes (Monge)
• E3 ? ?
• E3 ? ?
• ? ? ?
• E3 ? ? x ?

Describes 3D objects in 2D by means of two
orthogonal views of the objects.
Front view orthogonal projection onto the ?(x,z)
plane (your sheet of paper)
Top view orthogonal projection into the ?(x,y)
and its revolution about x-axis by 90into the
?(x,z) plane
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Parallel projection uniquenessSolutions

• Axonometry
• E3 ? ? or ? or ? (at least two of them)
• ? ? ?
• ? ? ?
• ? ? ?
• E3 ? ? x ?

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Thank you for your attention.

• Next lecture
• Orthographic projection
• Special lines in plane
• General axonometry
• Oblique projection of basic planar object and
  solids